786 Changing the Family Structure in Developed and Developing...
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Transcript of 786 Changing the Family Structure in Developed and Developing...
Changing the Family Structure in Developed and
Developing Countries - Should We Care? A Theoretical
Analysis and Empirical Evidence.
Md. Masum Emran
1st
Year PhD Student
Department of Economics
University of Birmingham
Abstract: A family is considered as an initial institution of social relation, which is a fundamental
wheel of all economic activities as well. Unfortunately this primary economic and social institution is
facing severe challenge in both developed and developing countries although the nature and
reasons of its fragility are different. In this paper I have tried to develop a theoretical reason of
changing family structure in developed and developing countries via simple utility maximisation
principle and then add some empirical evidences to support my propositions. My conclusion says
that advanced social security support and high wage rate in developed countries motivates elderly
generation to live in a nuclear family setting while least social security and low wage rate in
developing countries we come across this tendency in working generation. This conclusion has been
empirically supported using cross-section pooled data from developed, newly developed, and least
developed seventy five countries through over the world.
Introduction:
Definition of a family structure: on December 10, 1948, the United Nations General
Assembly, speaking to the World through the ‘Universal Declaration of Human Rights’,
declared on its Article 16 item 3 that ‘The Family is the natural and fundamental group unit
of society and is entitled to protection by society and state’ (GA Resolution 217a(III)). Family
is an institution that is constituted by people of both sexes (male and female) from a
generation or several generations. Joint family or traditional family, as I mean, usually is
constructed by at least an active (working) generation and one passive (retired/elderly)
generation living together while a nuclear family or modern family is constructed by only an
active generation. Joint family primarily relies on (i) altruistic belief that means ‘one
generation’s utility function depends positively on the well-being of other generation’
(Becker G.S. 1981) (ii) cooperative support which implies that without any formal contract of
cooperation, each generation believes that other generations always look after one’s
problem without seeking any return (Hamoudi A. et. al. 2006), and (iii) exchange of
resources between generations which suggests that when a generation faces any forms of
(financial, moral, or physical) hurdles then other generations within the family try to pull up
the vulnerable generation by giving all forms (financial, moral, physical) of supports hoping
that it will be exchanged back when other generations face the similar crisis (Herskovits,
1965; Posner, 1980). Nuclear family basically behaves on Samuelson’s (1958) ‘Consumption-
Loan Theory’ like competitive market exchange between generations, which infers that in
nuclear family structure, each generation is self-lover, individualistic and cares about
promoting personal interest (Boswell, 1959).
Benefits of Joint family: familial relation is not always motivated by Samuelson’s type of
profit maximization attitude. Many studies showed that generations co-residing in a joint
family exchange financial asset like Samuelson’s ‘Consumption Loan’ theory as well as they
also exchange in-kind transfers such as abstract human instincts, wisdom of experience,
affection and love, family chores etc (Pas Sarmistha, 2007) which are not exchanged in the
market mechanism. Joint family helps to cope uncertainty regarding financial, physical and
natural shocks (Posner R, 1980). It is, therefore, viewed that joint family structure is a partial
substitute of imperfect capital market and social security system by transferring resources
between generations - either altruistic motive or altruistic exchange motive. It is usually
thought that joint family is needed to increase the well-being of elderly or passive
generation. In countries with poor or ill-established pension system or social security system,
passive generation is usually more benefited than working or active generation although
passive generation in developing countries make substantial contributions to family well-
being, in ways ranging from socialization to housekeeping and childcare which helps active
generation to get more free time for paid employment (Hashimoto, 1991; Apt, 1992). Based
on Census 2000 data in the United States, nearly six million grandparents lived with their
grandchildren who were 18 years old or younger and over 40 percent of these co-resident
grandparents had primary responsibility for the care of their grandchildren (Simmons and
Dye 2003). Several researchers, therefore, found that both generations are mutually
benefited from joint family structure (Casterline and others, 1991; Chan, 1997). Joint family
is the usual structure of developing countries while nuclear family is the usual feature of
developed countries1.
1 Studies show many developed countries like Japan still follow joint family structure for the betterment of
both generations while in some developing countries nuclear family structure is predominant.
Breaking Joint Family structure: Becker G.S (1981) focused that in developed countries the
growth of the welfare activities such as expenditure on social security, unemployment
benefits, free health and caring benefit for the elderly, aids to mother with dependent
children, food stamps etc. have been a powerful force of encouraging younger generation to
live independently not only in nuclear family but also without family responsibility (living
together) or in homosexual family. While increasing percentage of elderly people, declining
fertility rate, increasing life expectancy, rising migration, changing social norms, rising
unemployment rate and social frustration etc. have eroded the joint family structure in
developing countries, looking at the Table-1, we see that 73% of elderly people in North
America and 69% of elderly people in Europe lived alone or in couple typed family structure
while in Asia and Africa it is only 23% and 17% respectively. The population Census -2001 of
Bangladesh reports that ‘the proportion of households with seven members and above
shows a decreasing trend since 1981 and the percentage of household of five members or
less has shown increasing trend and such proportions are 33.7%, 37.8% and 50.6%
respectively in 1981, 1991 and 2001’. This statistics indicates that ‘large and joint families
have been breaking down gradually and has a tendency to be nuclear family’ (The
population Census -2001, Bangladesh). So undoubtedly, we can say that all over the world
joint family is changing to nuclear family parallel with economic development and time.
Consequences of breaking joint family structure: It is believed that multigenerational bonds
are increasingly becoming more important than nuclear family ties for the well-being and
support of individuals over the course of their entire life (Bengston 2001). Breaking of joint
family structure in such a way will lead to severe calamity both for the developed and
developing countries in near future although the nature of disaster might be different.
Literature on grandparent-grandchild co-residence is rare, and virtually all such literature in
the U.S. is motivated by the concern of children in non-traditional family structures. In the
1990s, researchers and policymakers began to focus their attention on children raised solely
by grandparents. Among all household types, "grandparent maintained families"
experienced the fastest growth among all household types since 1990, as a result of the
rapid rise in single-parent households, drug abuse among parents, mental and physical
illnesses, child abuse and neglect, and incarceration- to name a few (Casper and Bryson
1998). Children in grandparent maintained, no parent present families are substantially
more likely to live in poverty and the most likely to be uninsured than the children in any
other family type (Casper and Bryson 1998). Despite the odds, Solomon and Marx (1995)
found that children raised solely by grandparents performed better across several
dimensions, including some academic criteria, than did those who were raised by a single
parent. ‘The 2001 Youth Risk Behaviour Survey’ shows among US students enrolled in grades
9-12 in 2001, 47.1% drank alcohol, 23.9% used marijuana and 45.6% had sexual intercourse
although their residence information is not available - this is the feature of teenager
generation around most developed countries. Avert – an international AIDS charity - reports
that according to the National Survey of Sexual Attitudes and Lifestyles (NATSAL) surveys in
the UK clearly indicates that women having a same sex sexual experience has increased from
2.8% in 1990 to 9.7% in 2000 while that for men has increased from 5.3% to 8.4%. In the USA
an estimated homosexual population is 5% (more than 10 million) of the total U.S.
population over 18 years of age. The Human Rights Campaign (2001) suggests that 3,136,921
gay or lesbian people are living in the United States in committed relationships in the same
residence (Smith D.M. et. al. 2001). They also want equal right in all aspects of social
activities even right to be parents (adoptive or natural) of the next generation. Since they
have limited or no reproduction capacity and responsibility towards next generation, thus,
after three to four decades, there would be severe shortage of active generation in these
developed countries. Therefore, social security system, pension system, and any public
programme, moreover, overall economic growth would face severe catastrophe in financial
sector (Barro, 1978) along with epidemic disease like AIDS. The other side of the world -
developing countries where severe uncertainty prevails on all these economic programmes
viz. weak social security programme provided by the government, imperfect and insufficient
credit facilities, inefficient insurance facility, increase in the percentage of elderly people,
decaling fertility rate, increasing life expectancy, changing social norms, rising
unemployment rate and social frustration etc. have already been facing the crucial impacts
on social norms, values and activities such as eradication of familial support for the elderly,
rising separation rate of newly married couples, increasing number of children cared by
single parent (Mason K. O) - a few mentioned social activities due to changing family
structure. Traditional joint family structure of these developing countries plays protection
against such uncertainty by providing altruistic assistance among generations (Herskovits,
1965; Posner, 1980). Changing family structure encourages active generation of these
countries to bear less altruistic responsibility towards passive generation. Therefore, passive
generations (elderly generation and children) in developing countries becomes more
vulnerable compared to their counterpart in developed countries because developing
countries have already started facing these consequences of changing family structure but
governments of these countries have less economic sovereignty and ability, no time to get
ready and moreover have yet to start any policy measure to tackle this problem.
Motivations of study: The Proclamation of the year of 1994 as the ‘International Year of the
Family’ by the UN/GA Resolution 44/82 of December 8, 1989, reflected growing
international recognition of and concern for the family perspectives. Although little is
known about reasons of breaking joint family is due to the motivation for the transfers, the
effects of the transfers and the volume and direction of the transfers or something else.
Ongoing research in Asia is beginning to reveal the complexity of familial exchange, not just
between parents and children, but among wider family and social networks as well (Agree,
Biddlecom and Valente, 1999). Knowing more about the familial exchange network may be
useful in helping anticipate the nature and cause of changing family structure in developing
society. There is a need for more research on the preferences and attitudes of both active
and passive generations in terms of their living arrangements (Kinsella, 1990; Myers, 1996)
as well. It is, therefore, reasonable for researchers to find out - what are the theoretical
reasons of breaking joint family structure in developed and developing countries? What are
the economic structures (individualistic market exchange mechanism, altruistic feeling,
altruistic exchange motive or a combination) that improve social wellbeing? What are the
relevant theories that can explain such intergenerational or intra-generational family
relation of an economy? All these questions have been attempted to be answered in this
study.
Contributions of this study: The paper is novel in several ways. Firstly, I have constructed
a theoretical model of overlapping generations using four consecutive stages beginning from
individualistic behaviour to individualistic family behaviour to mono-dimensional altruistic
family behaviour to duo-dimensional altruistic family behaviour. Moreover, in each stage I
have tried to compare individual choice decision and social planner’s choice decision. It is
unique in a sense that the theory combines Dimond P. (1965)’s competitive ‘Overlapping
Generation Model’ to Becker G.S (1974)’s mono-dimensional ‘Altruistic Intergeneration
Mobility Model’ to duo-dimensional ‘Altruistic Bargaining Model’ (Manser and Brown 1980)
by using simple maximization principle. Secondly, I have developed several propositions in
every stage and tried to accommodate those propositions in the context of developed and
developing countries. Thirdly, the study is basically motivated to develop theoretical
framework for a utility maximizing family using simple optimisation principle but it could be
adaptable in many complex scenarios also. Along with have some empirical supports also
been provided. The empirical model introduced in this paper is a new idea and is suitably
fitted with a cross-sectional pooled data from 75 developed and developing countries all
over the world, which is also an important contribution of this paper. In this paper I have
considered only household sector-one wheel of a two-wheel economy. So this analysis is
fully based on partial static framework. Another thing I need to mention is that data
availability of familial emotions, transfer, and direction are very much scarce all over the
world especially in developing countries.
Content: This paper is organised as follows. The next section briefly discusses related
literature review. Section three has developed model and corresponding propositions and
discussions. The empirical supports of these theories are covered in section four. Section five
is for conclusions and policy implications.
Literature Reviews: ‘Lifecycle consumption’ hypothesis (Modiogliani F,1949; Friedman M
1956) discusses how an individual maximises life time utility within budget constraint. The
‘Consumption-loan’ theory (Samuelsson PA, 1958) broadens lifelong utility maximisation to
more than one generation’s context. Diamond PA (1965) gives different flavour of lifelong
utility maximisation problem of an individual and calls it ‘Overlapping Generation (OLG)’
utility maximisation. There are huge literatures discussing different aspects of economic
problems regarding production, consumption, tax, transfer, trade, growth- almost all aspects
of economic problems and almost all these literatures try to introduce some new ways of
analysis or technique with this basic principle such as simple calculus techniques of
optimisation in different perspectives of economic problem are used by Arrow KJ, 1971;
Feldstien M, 1974; Samuelson PA, 1975; Deardorff AV, 1976 in their articles; dynamic
optimisation techniques with demographic elements such as birth rate, life expectancy etc.
are developed by Brian AW and Geoffrey Mc. Nicoll, 1977, 1978; Lee RD, 1980; Willis RJ,
1988; Lee RD and Lapkoff Shelley, 1988; Lee RD and Millier T, 1994; Lee RD et al., 2000;
Bommier A and Lee Rd, 2003; Mason A and Lee RD, 2004; Mason A et al.,2006; while Becker
GS, (1960, 1974, 1976) Becker GS and Lewis HG (1973), Becker GS and Tomes N (1979)
Bernheim BD et al., (1985), Cox D (1987), Becker GS and Barro RJ, (1988), Altonji JG et al.,
(1992), Altonji JG et al. (1997), Frankenberg E et al. (2002), Sloan FA et al., (2002) introduced
altruistic emotions, transfers, generational mobility and direction and overall social well-
being; the idea of uncertainty in life and capital market in the OLG framework was
introduced by Blanchard OJ in 1985 while game theory concept was incorporated by Manser
and Brown (1980), Browing et al., (1994), Ermish and Chiara, (2008). Geanakoplos J D and
Heraklis M P (1991) discussed in detail and showed conditions of the efficiency and existence
of competitive equilibriums of the overlapping generation model. Dutta J. and P Michel
(1998) developed a theory about the wealth distribution of an imperfect altruistic dynastic
family. According to their theory, when a generation lives for infinite time and one
generation may or may not be altruistic to other generations, then a stationary value of
wealth distribution exists there and imposing some restrictions, it is Pareto optimum. In this
paper I have tried to make mathematical analysis as simple as possible by using the OLG
model of Diamond PA to altruistic OLG model of Becker GS at first, then mono-dimensional
altruistic model of Bernheim BD and others and finally duo-dimensional altruistic model to
reach a basic conclusion of changing family structure. There are some literatures discussing
co-residence topic in the context of game theoretic frame (Weiss Y and Willis R.J, 1985;
Chiappori P.A., 1992; Browning M. et al., 2005; Rangel M.A., 2006; Sakudo M, 2007; Ermisch
J and Chiappori P, 2008). The remarkable thing in this paper is that although I have
introduced and developed propositions in easy techniques, one can make it complex
according to her/his requirement such as one can introduce different size of generations,
imperfect altruism among generations, uncertainty in lifetime, as well as bargaining
techniques into the family utility maximisation behaviour.
There are a few empirical works regarding co-residence behaviour (Manacorda M and
Moretti E., 2002), altruistic emotions and generational transfers (Altonji JG, et al., 1997;
Lillard L.A. and Willis R.J., 1997; Park C., 2001; Hamoudi A and Thomas D., 2006; Pal S, 2007)
but almost all of these have used developed countries’ data. The context and data of
developing countries are analysed in hardly a few.
Model , Propositions and Discussions:
1.Life Cycle Behaviour: Assume that an individual �, lives two periods �, ��� �� + 1�. In
which, period- �, s/he works and earns competitive wage �� and period-�� + 1� he retires
and has no income. Her/his endowments for periods � ��� �� + 1� are
�� ��� 0 respectively. With these endowments s/he has to manage her/his of both periods’
consumptions ��� ��� ����� respectively. So s/he splits her/his � -period earning into
consumption ���and saving ��� and uses this saving for the consumption of period-�� + 1�,
when her/his endowment is zero. He bears a constant degree of patience (or generation in
period–t), ���[0,1], for the consumption of period-�� + 1�. Assume the time invariant
market rate of interest for saving in period - � is �� . The individual’s problem is to maximise
life time utility which is written mathematically as following.
������,��� � ,!��"# = "�%���& + ��"�%����� &
Budget constrain for time-t: ��� + ��� = ��
And budget constraint for time –�� + 1�: ����� = �1 + ������
The Life time budget constraint is: ����� = �1 + ���� �� − ����
Or, ����� = ��1 + ��� �� − �1 + ������
The marginal rate of substitution 2(�(���� ,� ���� between current consumption in time-t and
future consumption in time-�� + 1� is:
)*��+�)*�� � = ,����-��
,� .�, )*��)*�� � = ���1 + ��� − − − −�/
The equation- �/ is the optimum condition of an individual maximising life time utility and
also be considered as optimum condition of utility maximisation for two overlapping active
and passive generations, where each generation is constituted by single member (Dimond P.
1965). The condition implies that the ratio of the marginal utility of consumption of an
individual between two periods (or two overlapping generations) must be equal to the
product of individual’s (or generation’s) degree of patient and the absolute value of the
slope of individual’s life time (or overall) budget constraint. If we assume that the individual
(or generations) is rational and wants to smooth her/his consumption over the life time
(both generations) then the optimum consumption and saving would be a function of simply
wage income and degree of patient (of each generation) as follows:
���∗ = 1�����+�� ; �����∗ = ���-��+�1��
���+�� ��� ���∗ = +�1�����+��
Taking an additive, separable logarithmic utility function for both time periods of
consumption (or each generation) the optimum condition becomes as following.
2 Calculation shows in mathematical appendix: 1.1
The marginal rate of substitution: �(���� ,� ��� =
*��3�
*�� �= ,
,4 = ��� ���� = ���1 + ���
Optimum consumption for period – � (or generation- �) is: ��� = 1�����+��
Optimum consumption for period – �� + 1� (or generation- � + 1 ) is: ����� =+����-�� �1��
���+��
Optimum saving in period – � (or generation- � ) is: ��� = +�1�����+��
If we assume that utility function is linear Cobb-Douglas form as follows:
"� = ����5+������%+�& 6ℎ8�8 ���[0,1]
Assume that individual wants to smooth consumption between two periods (or generation)
so that assigning �� = �9 share of consumption in each period (or generation).
The optimum values of consumptions for two periods (or generations) and saving in period-t
(or active generation) are following.
���∗ = �1 − ��� ��; �����∗ = �1 + ���%��& ����� ���∗ = ���� ��
The consumption and saving are for both forms of utility function.
Proposition -1: When an individual following logarithmic, additive and separable utility
function splits his/her life time into infinite numbers of small period, : ∈ [<, =]>=? = →∞, and the degree of patience for future consumption is greater than zero but less than
one [A < CD < 1] ,that means future consumption is less important than current
consumption, then the optimum consumption of current period becomes same as that of
the linear Cobb-Douglas utility function that logarithmic utility function becomes Cobb-
Douglas utility function (like the CES function transforms Cobb-Douglas function).3
Proof: If individual splits his/her life time into two-period then optimum consumption of
current period is- ���∗ = 1�����+�� .
If individual splits his/her life time into three-period then optimum consumption of current
period is- ���∗ = 1��[��+���+��4] .
If individual splits his/her life time into four-period then optimum consumption of current
period is- ���∗ = 1��[��+��%+�&4��+��E] .
If individual splits his/her life time into n-period then optimum consumption of current
period is- ���∗ = 1��[��+��%+�&4�%+�&E………��+��GH ] .
If � → ∞ ��� 0 < � < 1then the current period consumption is –
���∗ = ��[1 + �� + ����9 + ����I … … … + ����J5�] = �1 − ��� ��
This is nothing but the optimum consumption of current period if an individual follows linear
Cobb-Douglas from of utility function.
3Using the L’Hopital rule Constant Elasticity of Substitution( CES) utility function becomes Cobb-Douglas utility
function with limited value of zero. See derivation in appendix 1.3
Proposition-1.a: When an individual following logarithmic, additive and separable utility
function splits his/her life time into infinite numbers of small period, : ∈ [<, =]>=? = →∞, and the degree of patience for future consumption is one [CD = <] ,that means future
consumption gets equal important with current consumption, the optimum consumption
of current period is simply the income of current period,K:D divided by the numbers split,
=.
Proof: If individual splits his/her life time into two-period then optimum consumption of
current period is- ���∗ = 1��[��+�] .
If individual splits his/her life time into three-period then optimum consumption of current
period is- ���∗ = 1��[��+���+��4] .
If individual splits his/her life time into four-period then optimum consumption of current
period is- ���∗ = 1��[��+���+��4��+��E] .
If individual splits his/her life time into � − L8��.� ��� � → ∞, then optimum consumption
of current period is- ���∗ = 1��[��+���+��4��+��E……..�+���GH �] .
When the degree of patience for future consumption is one [�� = 1] then,
The optimum consumption of current period is- M:D∗ = K:D[<�=5<] = K:D
= .
2.a Family Behaviour with selfish (individualistic) attitude and different marginal utility of
money: Now assume that at any time-t there is a family in the economy consisting two
generations such as parent, P and adult-child, A. There is only one member of each
generation within the family. Currently adult-child is active generation so s/he is working in
the labour market and earning wage income, �N while parent is passive generation or retire
generation so currently s/he does not supply his/her labour in the market and not earn any
wage income but s/he consumes at time-t from his/her saving4, ��O = +P1�H P���+P� that s/he has
made when s/he was active. Although both members of two generations reside in a joint
family but they independently maximize their own utility and passive generation will die
after period-t, and active generation will become passive generation in period-t+1. This
implies that family survives infinite time periods and each and every time there are always
two generations in a joint family. It is assumed that family utility function is the summation
of both generations’ utility function. The utility maximization problem of the family is
written as follow:
"Q��R,��PSTU = "N���N� + "O���O�
Subject to budget constraint for adult-child: ��N + ��N = �N
.� ��N = �N − ��N = �N − �N�1 + �N� = �N �N�1 + �N�
Similarly, budget constraint for parent is:
��O = �1 + �����O = �1 + ����O �5�O�1 + �O�
When each generation has different marginal utility of money, the marginal rate of
substitution (�(���,P��R� between the consumption of two generations is )*RV)*PV = ,R
,P the
4 Individual’s life time utility maximisation is same as overlapping generation model. The optimum saving used
here is derived from logarithmic, additive separable utility function in previous section.
optimum condition which implies that the ratio of marginal utility of consumption between
two generations is equal to the ratio of marginal utility of money between two generations.
When marginal utility of money between two generations is equal then the optimum
condition says that marginal utility of consumption must be equal for both generations to
maximise family utility.
Now assume that each generation’s utility function is logarithmic form and each generation
always maximise life cycle utility and also assume that the degree of patient and wage rate
vary between the generations. As well as, marginal utility of money is different between
generations. Solving family problem we get optimum consumptions for both generations
and Marginal rate of substitutions (�(���,P��R� between two generations as follows.
��N∗ = 1�R���+R� ��� ��O∗ = +P���-��1�H P
���+P�
The marginal rate of substitution (�(���,P��R) between adult child and parent is: �(��P,�R =
��P∗��R∗ = ,R∗
,P∗ =+P���-��1�H P
���+P�W
1�R %��+R&W
= XO∗�1 + �N��O�1 + ��� �5�O = XN∗�1 + �O� �N
Axiom-2a-i: In a joint family individualistic parent generation is more patient than
individualistic adult-child generation if wage rate and marginal utility of money are equal
for both generations and rate of interest is positive.
Axiom-2a-ii: When degree of patience A < CY = CZ < 1 and marginal utility of money are
equal for both generations and rate of interest is positive then wage rate of adult
generation must be greater than that of the parent generation to maximise family utility.
2b. Family Behaviour with selfish (individualistic) attitude of each generation and same
marginal utility of money (Social Planner’s point of view): Now assume that each
generation belongs the same value of marginal utility of money that implies that there is a
social planner who likes to allocate resources for the consumption of adult child generation
and parent generation such a way that maximises a joint family’s utility. Therefore, the
planner combines the budget constraints of both generations into a family budget constraint
and maximizes family utility subject to family budget constraint.
The family budget constraint is: ��N + ��O = 1�R���+R� + �1 + ��� +P1�H P
���+P�
The family problem becomes as follows-
"Q��R,��P,,V,STU = "N���N� + "O���O� + XQ [ �N�1 + �N� + �1 + ����O �5�O�1 + �O� − ��N − ��O\
The marginal rate of substitution5 (�(���,P��R� between two generations is:
"�NQ"�OQ = 1 .� "�NQ = "�OQ
This implies that the marginal utility of consumption for each generation must be equal to
maximize family utility.
We also find that, 1�R%��+R& + ���-��+P1�H P
���+P� = ��N + ��O
5 Derivation is given in mathematical appendix-1.5.
Since a social planner gives equal importance on the consumption of both generations so
optimum consumption for each generation is nothing but simply equal distribution of
family’s total income likes as follows.
��N∗ = ��O∗ = 12 [ �N�1 + �N� + �1 + ����O �5�O
�1 + �O� ]
For a logarithmic utility function, which is additive to get family utility function the optimum
values of consumption and marginal utility of money for both generations are-
��N∗ = ��O∗ = �1 + �O� �N + �1 + ����O�1 + �N� �5�O2�1 + �N��1 + �O�
XQ∗ = 2�1 + �N��1 + �O��1 + �O� �N + �1 + ����O�1 + �N� �5�O
The optimum values of consumptions imply that consumption of each generation depend
not only its own parameters values (wage rate, degree of patience) but also the parameter
values of other generation. Alternatively we can say each generation plays a game in the
family setting to maximize its own utility.
Corollary-2b-i: Selfish/individualistic generations in a joint family play a ‘Cournot’ type
game to maximise own consumption but equilibrium consumption will be same for both
generations.
3a.Family Behaviour with Mono-Dimensional Altruism and Different Marginal Utility of
Money: Now assume that parent generation is altruistic to adult child generation. Due to
altruistic feeling parent gifts a certain amount of its income,^ = _ �5�O ; _ ∈ [0,1], to
increase his/her adult-child’s endowment that helps to increase the consumption of adult
generation although the adult generation is selfish. In this situation family’s behaviour
becomes as follow-
The utility function of selfish adult child is: "N = "N���N�
The utility function of altruistic parent is: "O = "O���O� + ^"N���N�
"Q��R,��PSTU = �1 + ^�"N���N� + "O���O�
Subject to budget constraint for adult-child: ��N + ��N = �N + ^
.� ��N = �N + ^ − ��N = �N + ^ − �N�1 + �N� = �N �N�1 + �N� + ^
��N = �N�1 + �N�
Similarly, budget constraint for parent is:
��O = �1 + �����O − ^ = �1 + ����O �5�O�1 + �O� − ^
The marginal rate of substitution6 (�(���,P��R� between the consumption of two generations
is as follow. ���`�)*RV)*PV = ,R
,P .� )*RV)*PV = ,R
���`�,P
This is the optimum condition which implies that marginal utility of money to parent is lower
than that of his/her child, if their marginal utility of consumption held constant and vice
versa (the marginal utility of child consumption lower than that of parent so the total utility
of child will be higher than that of the parent.
6 Derivation is given in mathematical appendix-1.6
Using logarithmic utility function for both generations, we find the optimum consumptions
and the values of marginal utility of money for both generations are7-
The optimum value of consumption and marginal utility of money for selfish adult-child
generation are
��N∗ = �N + �1 + �N�^�1 + �N� ��� XN∗ = �1 + �N��1 + ^�
�N + �1 + �N�^
The optimum value of consumption and marginal utility of money for altruistic parent-
generation are:
��O∗ = �1 + ����O �5�O − �1 + �O�^�1 + �O� ���
XO∗ = �1 + �O��1 + ����O �5�O − �1 + �O�^
Proposition-3a-i: When marginal utility of consumption, degree of patient and wage
income for both generations remain constant then marginal utility of money to parent is
lower than that of his/her child.
Proof: ���`�)*RV
)*PV = ,R,P .� )*RV
)*PV = ,R���`�,P a., XN∗ = 1 + ^ ��� XO∗ = 1
bℎca, XN∗ > XO∗ a��e8 ^ > 0
XN∗ = �1 + �N��1 + ^� �N + �1 + �N�^ > �1 + �O�
�1 + ����O �5�O − �1 + �O�^ = XO∗
7 Derivation is given in mathematical appendix-1.6
Proposition-3a-ii: When marginal utility money, degree of patient and wage income for
both generations remains constant then total utility of child generation is higher than that
of parent generation.
Proof: )*RV)*PV = ,R
���`�,P a��e8XN∗ > XO∗ a��e8 ^ > 0 �ℎ8� "�NQ < "�OQ
Therefore, total utility of adult-child generation is higher than that of parent generation.
��N∗ = 1�R�%��+R&`���+R� > ���-��+P1�H P 5%��+P&`
���+P� = ��O∗
3b.Family Behaviour with Mono-Dimensional Altruism and same Marginal Utility of Money
(Social Planner’s Point of View): Assumed that marginal utility of money is same for both
generations that is family utility is maximized according to the social planner’s point of view.
Then the model becomes-
"Q��R,��PSTU = �1 + ^�"N���N� + "O���O�
Subject to, Family budget constraint: the budget constraint:
�N�1 + �N� + �1 + ����O �5�O�1 + �O� = ��N + ��O
The marginal rate of substitution (�(���,P��R� between the consumption of two generations
is - ���`�)*RV)*PV = 1 .� )*RV
)*PV = ����`�
This implies that the marginal utility of consumption for child is less than that of the parent.
It can be inferred that in this case total consumption of parent is lower than that of the
selfish parent but the opposite is the case for child’s consumption because marginal utility of
child consumption decreases means total utility increases.
Using the logarithmic utility function, the optimum values of consumption for both
generations are-
Optimum value of consumption for selfish adult-child is:
��N∗ = �1 + ^�[�1 + �O� �N + �1 + ����O�1 + �N� �5�O ]�2 + ^��1 + �N��1 + �O�
Optimum value of consumption for altruistic parent is:
��O∗ = [�1 + �O� �N + �1 + ����O�1 + �N� �5�O ]�2 + ^��1 + �N��1 + �O�
Proposition-3b-i: According to social planer’s point of view total consumption of an
altruistic parent generation is always lower than that of the selfish adult child generation.
Proof: �(���,P��R = )*RV)*PV = �
���`� < 1 a��e8 ^ > 0.
"�NQ < "�OQ �ℎ8�8f.�8, ��N∗ > ��O∗
From logarithmic utility function we get
��N∗ = �1 + ^�[�1 + �O� �N + �1 + ����O�1 + �N� �5�O ]�2 + ^��1 + �N��1 + �O�
> [�1 + �O� �N + �1 + ����O�1 + �N� �5�O ]�2 + ^��1 + �N��1 + �O� = ��O∗
Proposition-3b-ii: According to social planer’s point of view and mono-dimensional
altruistic family enjoys more total utility compared to selfish family.
Proof: The ratio of marginal utility of substitution between two generations of selfish family
to mono-dimensional altruistic family is-
�(���,P��R .f a8gf�aℎ f�h�gi�(���,P��R .f h.�. − ��h8�a�.��g �g��c�a��e f�h�gi =
11�1 + ^�
= �1 + ^� > 1.
This implies that, �(���,P��R of mono-dimensional altruistic family is lower than that of the
selfish family. Therefore, the total utility of mono-dimensional altruistic family is higher than
that of the selfish family. This proposition supports almost all mono-dimensional altruistic
literatures that altruistic behaviour improves family wellbeing. (Backer G.S, et.al 1979,
Altoniji et.al. 1997).
4a.Family Behaviour with Duo-Dimensional Altruisms and Different Marginal Utility of
Money: Now assume that both generations in a joint family are altruistic to each other but
their degree of altruism and marginal utility of money might be different. Further assume
that parent generation is altruistically spending ^ = _ �5�O ; _ ∈ [0,1] amount of his/her
income for its adult-child generation, on the other hand, adult-child generation is also
altruistically giving up j = k �N; k ∈ [0,1] amount of his/her income for looking after
his/her parent generation. The utility maximisation problem of this joint family becomes as
follow.
The utility function of child is: "�N = "N���N� + j"�O
The utility function of parent is: "�O = "O���O� + ^"�N
For simplicity it is omitted the cross effects on total utility of both generations because the
higher order products of A and G trends to zero. Becker & Tomes (1979) shows estimated
time lag that has positive effect on consumption. So the utility functions I assign for child and
parent are -
The utility function of child is: "�N = "N���N� + j"O���O�
The utility function of child is: "�O = "O���O� + ^"N���N�
Assuming additive utility, family function becomes –
"Q = �1 + ^�"N���N� + �1 + j�"O���O�
Budget constraint for child generation: ��N + ��N + j = �N + ^
Or, ��N = 1�R���+R� + ^ − j
Budget constraint for parent generation: ��O + ^ = �1 + ��� +P1�H P���+P� + j
"Q��R,��P,,R,,PSTU = �1 + ^�"N���N� + �1 + j�"O���O� + XN [ �N�1 + �N� + ^ − j − ��N\ + XO[�1
+ ��� �O �5�O�1 + �O� + j − ^ − ��O]
The marginal rate of substitution8 (�(���,P��R� between the consumption of two generations
is - ���`�)*RV���N�)*PV = ,R
,P = )*RV)*PV = ���N�,R
���`�,P
The optimum consumptions are-
8 Derivation is shown in mathematical appendix-1.8
An altruistic adult-child is-
��N∗ = �N�1 + �N� + ^ − j
An altruistic parent is-
��O∗ = �1 + ��� �O �5�O�1 + �O� + j − ^
Using logarithmic utility function the family utility maximisation problem is-
"Q��R,��P,,R,,PSTU = �1 + ^�g���N + �1 + j�g���O + XN [ �N�1 + �N� + ^ − j − ��N\ + XO[�1
+ ��� �O �5�O�1 + �O� + j − ^ − ��O]
The optimum consumption of an altruistic adult-child generation is-
��N∗ = XO�1 + ^��1 + ����O �5�OXN�1 + j��1 + �O� + XO�1 + ^��j − ^�
XN�1 + j�
Optimum marginal utility of money for adult-child is
XN∗ = �1 + ^��1 + �N� �N + �1 + �N��^ − j�
The optimum consumption of an altruistic parent generation is-
��O∗ = XN�1 + j� �NXO�1 + ^��1 + �N� + XN�1 + j��^ − j�XO�1 + ^�
Optimum marginal utility of money for parent is
XO∗ = �1 + j��1 + �O��1 + ����O �5�O + �1 + �O��j − ^�
Proposition-4a-i: If the degrees of patient, amount of transfer and wage income of both
generations are equal then values of marginal utility of money for both generations must
be equal.
Proof: �(���,P��R = ���`�)*RV���N�)*PV = ,R
,P = )*RV)*PV = ���N�,R
���`�,P �f j = ^ ��� "�NQ = "�OQ
ThenXN = XO. Using logarithmic function,
XN∗ = �1 + ^��1 + �N� �N + �1 + �N��^ − j� = �1 + j��1 + �O�
�1 + ����O �5�O + �1 + �O��j − ^� = XO∗
Proposition-4a-ii: If values of marginal utility of money and the altruistic transfers are
equal then total utility of consumption for both generations is same.
Proof: �(���,P��R = ���`�)*RV���N�)*PV = ,R
,P = )*RV)*PV = ���N�,R
���`�,P = 1�f XN = XO��� j = ^
Then, "�NQ = "�OQ the marginal utility of consumption equal means total utility of both
generations must be equal. Using logarithmic utility function,
��N∗ = XO�1 + ^��1 + ����O �5�OXN�1 + j��1 + �O� + XO�1 + ^��j − ^�
XN�1 + j�= XN�1 + j� �NXO�1 + ^��1 + �N� + XN�1 + j��^ − j�
XO�1 + ^� = ��O∗
Proposition -4a-iii: When marginal utility of money and wage income for both generations
is same then altruistic family behaviour would be same as individualistic family behaviour
if mutual transfers from both generations are identical that means A=G.
Proof: �f XN = XO��� j = ^
The ratio of ��(���,P��R� between the consumption of two generations of
selfish/individualistic model to duo-dimensional altruistic model is-���`����N�=1. So there is no
difference between duo-altruistic model and individualistic family utility maximisation.
Altruistic adult child consumption is -��N∗ = ,P���`����-��+P1�H P,R���N����+P� + ,P���`��N5`�
,R���N�
��N∗ = ���-��+P1�H P���+P� = 1�R
���+R� = ���∗= Individualistic adult child consumption.
Altruistic parent consumption is –��O∗ = ,R���N�1�R,P���`�%��+R& + ,R���N��`5N�
,P���`�
��O∗ = ,R���N�1�R,P���`�%��+R& = ���-��+P1�H P
���+P� = ��5��∗ =Individualistic parent consumption.
4b.Family Behaviour with Duo-Dimensional Altruisms having same Marginal Utility of
Money (Social Planner Point of View):
Now assume that both generations are altruistic to each other and their degree of altruism
might be different but marginal utility of money is same. The family problem becomes now
as following.
"Q��R,��P,,V,STU = �1 + ^�"N���N� + �1 + j�"O���O�
+ XQ [ �N�1 + �N� + �1 + ��� �O �5�O�1 + �O� − ��N − ��O\
The marginal rate of substitution9 (�(���,P��R� between the consumption of two generations
is - ���`�)*RV���N�)*PV = 1 = )*RV
)*PV = ���N����`�. This implies that the marginal rate of substitution depends
on the value of altruistic transfers of both generations A and G. When A>G, marginal utility
of child’s consumption is higher than that of the parent’s. So parent’s total utility is higher
than that of child’s and the opposite would be true if G>A.
If social planner gives equal importance on the consumption of both generation then
optimum consumption is half of total family income, which has no difference from
individualistic case.
��N∗ = ��O∗ = 12 [ �N�1 + �N� + �O �5�O
�1 + �O�]
Using logarithmic utility function we get optimum consumption of both generations as
follows.
The optimal consumption of altruistic adult generation is as follows:
��N∗ = �1 + ^��1 + �O� �N + �1 + ����O�1 + �N� �5�O�2 + ^ + j��1 + �N��1 + �O�
The optimal consumption of altruistic old (parent) generation is as follows:
��O∗ = �1 + j��1 + �O� �N + �1 + ����O�1 + �N� �5�O�2 + ^ + j��1 + �N��1 + �O�
We also get the marginal utility of money for both generations is as follows:
XQ∗ = �2 + ^ + j��1 + �N��1 + �O��1 + �O� �N + �1 + ����O�1 + �N� �5�O
9 Derivation is given mathematical appendix-1.9
Proposition-4b-i: When altruistic transfer of adult-child towards parent is bigger than that
of parent towards adult-child, Y > ^, marginal utility of child’s consumption is higher than
that of the parent’s. So parent’s total utility is higher than that of child’s and the opposite
would be true if the condition revised, that is G>A.
Proof: �(���,P��R = ���`�)*RV���N�)*PV = 1 = )*RV
)*PV = ���N����`�
�f j > ^ �ℎ8�, "�NQ > "�OQ a. �.��g c��g��i .f ��cg� − eℎ�g� < b.��g c��g��i .f L��8��
The ratio of optimal consumption is:��R∗��P∗ =
� �l�m �3Pno�R�� �p��3P� �3R�o�H P�4�l�R�% �3R&� �3P�
� �R�% �3P&o�R�� �p��3P� �3R�o�H P�4�l�R�% �3R&� �3P�
= ��`��N
�f j > ^ �ℎ8�, ��N∗ < ��O∗ and, �f j < ^ �ℎ8�, ��N∗ > ��O∗
Why is joint family structure breaking in developed and developing countries?
In developing countries:
It is usual that child altruistic transfer to parent (A) must not be equal to the parent’s
altruistic transfer to child (G). Becker & Tomes (1979) show that parent’s transfer to child
increases child’s productivity which leads to increase child wage income along with
economic growth enhances child’s wage income compare to his/her parent. So child’s
transfer to his/her parent (A) might be bigger than that of the parent’s (G). Then the
�(���,P��R is greater than one and total consumption of child is less than that of his/her
parent. Parent is better off than child. It is also assumed that parent’s transfer to child (G)
and Child’s transfer to parent (A) are respectively _ ��� k [�0,1]share of corresponding
generation’s wage income. If a child wants to increase his/her total consumption in the
family s/he needs to reduce the value of degree of altruismk, which leads the child to
behave selfishly and motivate to stay separately not in a family. That implies that economic
growth of developing countries and altruistic transfers from parent leads child-generation to
break co-residing family structure and to construct nuclear family live.
In developed Countries:
If parent transfer to child (G) is bigger than child’s transfer (A) due to high value of his/her
degree of altruism _, and marginal utility of money remains same for both generations then
total consumption of parent would be lower than that of his/her child-generation. So parent-
generation wants to reduce the value of _ , to increase his/her total consumption in the
family. This leads to parent-generation to break family structure and reside selfishly in a
nuclear family. In developed countries where social security system is very much advanced
and reliable for both child-generation and parent-generation then both generation want to
reduce the values of their corresponding degree of altruistic feeling _ and k, to increase the
share of their total consumption in a joint family setting. The continuous reduction of the
value of altruistic emotion leads each generation to behave selfishly and construct nuclear
family structure.
Empirical Evidence:
Model selection: It is usual to convert theoretical model into a suitable empirical model
which is accommodated with relevant and available data set. I theoretically develop the
conclusion that joint family structure of a society is changing mainly because of wage rate
differential and difference of social and cultural emotions i.e. altruistic emotions between
generations. According to my definition of joint family, an elderly generation must reside
with a younger generation. I, therefore, choose the percentage of elderly men or women
living with at least a child or grandchild as a proxy of measuring the number of joint family in
a society because it captures at least two generations co-residing in a family. According to
my theory, increasing a generation’s income leads to construct nuclear family by breaking
the joint family to maximise that generation’s total utility. I, therefore, assume that per
capita GDP of a country might be a good proxy variable10 to empirically evaluate a
generation’s income and the relationship between these two variables might be negative i.e.
when per capita GDP of a generation increases then that generation wants to live
independently and less likely to stay with elderly people. It is natural to assume that each
and every generation by born grasps some degree of altruistic emotion toward other
generation and needs co-operation among themselves in several stages in a life cycle, so
each generation likes to stay in a joint family structure. This implies annual population
growth rate leads to increase the probability of new generation stays in a joint family. But
when the population growth is due to increase in the stock of migrated people then
depending on other socio-economic circumstances these people may or may not stay in a
joint family structure. So it does not always give us a clear picture of living arrangement of a
society11. Thus the stock of migrated people either positively or negatively affects the living
arrangement of a society. It is usual that development of urban amenities and stresses will
lead a generation to construct a nuclear family because of high residential cost,
unemployment rate, severe competition in living style etc. (Manacorda and Moretti, 2002).
Even more than the economic and demographic points of view, living arrangement mainly
depend on social norms, values, religions, traditions, rules and obligations etc. Each region in
10 Some literatures say changing growth rate might be a good proxy of per capita GDP in mostly for developing
countries but I do believe that later one is more stable than the former and less affected by short run shock. 11 Some literatures include the number of migrated people in co-residence behaviour analysis such as Cheolsung
Park (2001).
the world such as Asia, Europe or Africa has some distinctive sorts of social characteristics
and these characteristics must directly influence living style of that region. Above all, the
state of development of a country must influence co-residence behaviour of the people. It is
assumed that in a developed society joint family structure is less important than that of a
developing society because ‘market insurance is used instead of kin insurance, market
schools instead of family schools, and examinations and contracts instead of family
certification’ (Becker, 1981) and public social security transfer benefits instead of familial
social security transfers. One, therefore, may conclude that the more developed a country
the less joint family structure is existed in the society.
To make the empirical model easily estimable I, therefore, choose three economic variables,
two demographic variables and a regional dummy variable discussed in earlier paragraph.
Three economic variables are: (i) per capita GDP, (ii) nature of development of a country
which is a dummy variable such as developed country, newly developed countries and least
developed countries, and (iii) proportion of total population lives in urban region of a
country; two demographic variables are: (i) population growth rate and (ii) percentage of
stock of migrated people of a country; and the regional dummy variable includes to cover
common effect of social characteristics. All these variables are included as independent
variables in a linear additive functional form of multivariate regression model and the
dependent variable is percentage of elderly people living with a child or grandchild. I follow
the step regression system by starting with one numerical variable and two dummy variables
as explanatory variables and step by step include other three numerical variables with the
first one. This method of estimation helps me to specify the effect of each explanatory
numerical variable in the estimation process and to diagnose statistical significance of the
variable in the linear regression process – which is a common way of specifying an empirical
model. I estimate the linear functional form of following equations both by
heteroscadasticity adjusted ‘OLS’ estimation technique and simple OLS technique to specify
proper empirical model that supports my theoretical model. Each functional form is
estimated twice – one for elderly men and another for elderly women.
qrq��� = q^sq� + jt(u� + j�uj� + r"(v� + wbj�� + wsrx� + ysrx� + "� − − − 1
qrq� � = q^sq� + jt(u� + j�uj� + r"(v� + wbj�� + wsrx� + ysrx� + "� − − − 2
qrq��� = q^sq� + q��q� + jt(u� + j�uj� + r"(v� + wbj�� + wsrx� + ysrx� + "�− − − 3
qrq� � = q^sq� + q��q� + jt(u� + j�uj� + r"(v� + wbj�� + wsrx� + ysrx� + "�− − − 4
qrq��� = q^sq� + jq^(� + jt(u� + j�uj� + r"(v� + wbj�� + wsrx� + ysrx� + "�− − − 5
qrq� � = q^sq� + jq^(� + jt(u� + j�uj� + r"(v� + wbj�� + wsrx� + ysrx� + "�− − − 6
qrq��� = q^sq� + q"qy� + jt(u� + j�uj� + r"(v� + wbj�� + wsrx� + ysrx� + "�− − − 7
qrq� � = q^sq� + q"qy� + jt(u� + j�uj� + r"(v� + wbj�� + wsrx� + ysrx� + "�− − − 8
qrq��� = q^sq� + q��q� + jq^(� + jt(u� + j�uj� + r"(v� + wbj�� + wsrx�+ ysrx� + "� − − − 9
qrq� � = q^sq� + q��q� + jq^(� + jt(u� + j�uj� + r"(v� + wbj�� + wsrx�+ ysrx� + "� − − − 10
qrq��� = q^sq� + q��q� + q"qy� + jt(u� + j�uj� + r"(v� + wbj�� + wsrx�+ ysrx� + "� − − − 11
qrq� � = q^sq� + q��q� + q"qy� + jt(u� + j�uj� + r"(v� + wbj�� + wsrx�+ ysrx� + "� − − − 12
qrq��� = q^sq� + q"qy� + jq^(� + jt(u� + j�uj� + r"(v� + wbj�� + wsrx�+ ysrx� + "� − − − 13
qrq� � = q^sq� + q"qy� + jq^(� + jt(u� + j�uj� + r"(v� + wbj�� + wsrx�+ ysrx� + "� − − − 14
qrq��� = q^sq� + q��q� + jq^(� + q"qy� + jt(u� + j�uj� + r"(v� + wbj��+ wsrx� + ysrx� + "� − − − 15
qrq� � = q^sq� + q��q� + jq^(� + q"qy� + jt(u� + j�uj� + r"(v� + wbj��+ wsrx� + ysrx� + "� − − − 16
Where,
qrq���= The percentage of elderly men (60 years and above) living with at least a child or a
grandchild of the country-i.
qrq� �= The percentage of elderly women (60 years and above) living with at least a child
or a grandchild of the country-i.
q^sq�= Per capita GDP of the country-i, which might be inversely related with the
dependent variable.
q��q�= Percentage of stock of migrated people in the country-i, which may or may not be
positively related with dependent variable.
jq^(�= Annual population growth rate of the country-i, which might be directly related
with dependent variable.
q"qy�= Percentage of urban population of the country-i, which is expected to be negatively
related with dependent variable.
jt(u�= It is a regional dummy variable included in the regression model to cover social
norms, values, rules, religions, etc of that region. Here, it takes value=1 if the country is
situated in African region otherwise it takes value = 0.
j�uj�= It is a regional dummy, it takes value =1 if the country is situated in Asian region
otherwise it takes value = 0.
r"(v�= It is a regional dummy, it takes value=1 if the country is located in European region
otherwise it takes value = 0.
wbj��= It is a regional dummy, it takes value=1 if the country is situated in Latin America or
the Caribbean otherwise it takes value = 0.
wsrx�= It is a dummy variable covering least developed status of a country compared to the
others. It takes value =1 value if the per capita GDP of a country-i is lower than $5000 in the
PPP measurement of 1995, otherwise it takes value =0.
ysrx�= It is also dummy variable indicating newly developed status of a country, It takes =1
value if the per capita GDP of the country-i is lower than $12000 but higher than $5000 in
the PPP measurement of 1995 otherwise =0.
"�= Disturbance term of a country-i’s observation which is assumed to be normally,
independently distributed with zero means and constant variance.
Sources of Data: The percentage of the elderly people living with a child or grandchild is
collected from ‘the Executive Summary - the Annex I:A, iv.1:’ of the United Nations
Population Divisions, Department of Economic and Social Affairs in a title ‘Living
Arrangements of Older Persons Around the World’ which covers 75 countries of which 20
countries are from African region, 20 countries are from Asian region, 18 and 16 countries
are from European and Latin America and the Caribbean respectively and the United States
of America. All most all observations of dependent variable are from period 1998, 1999 or
2000 but there are some countries’ information which has been reported between 1991 and
1998 due to unavailability of 1999 or 2000 information12. Data on four independent
numerical variables are collected from the ‘Country Profile’ of the United Nations Population
Division for the year 2000.The percentage of stock of migrated people is calculated by using
this formula.
12 I think most of countries use 10 years time difference between two population censuses. So It is justifiable to
use one census information until the next census information would be available.
q8�e8����8 .f a�.e� .f h�����8� L8.Lg8= b.��g ��.e� .f ���8�����.��g h�����8� L8.Lg8
b.��g L.Lcg���.� �100
Then I pool data of dependent variable with that of independent variables. I think it is
reasonable to call this data set as cross section polled data of cross country not as cross
section panel data of cross country because all observations of all independent variables of
each country are basically in the year 2000 point except the dependent variable. There is no
observation on another point of time.
Results and Discussions:
The above 16 estimated regression equations indicate that dependent variable and
independent variables are nearly perfectly causally related because each of these estimated
equations has both robust and adjusted (9 −value more than 0.80. As well as all variables
have expected sign and majority of them are significant at below 5% error level. In the
following table I just have reported two estimated equations – one is for elderly men living
with a child or grandchild and another is for elderly women by including all independent
numerical variables and dummy variables using ‘Robust OLS estimation’ technique in Table-2
and Table-3 of appendix all estimated results have provided. From these reported estimated
equations one can easily say that elderly people is less likely to co-reside with a child or
grandchild in a country having high per capita GDP (PGDP) and the argument is statistically
significant below 5% error level in case of elderly women but the elderly men’s case has
expected sign although not statistically significant at 5% error level. The estimated co-
efficient of PGDP variable in case elderly women says that one percent increase in per capita
GDP of a country will decrease 0.09% probability of elderly women’s co-residing status in a
joint family. The percentage of stock of migrated population (PSMP) is statistically
insignificant in both reported estimated equations but the positive sign in both cases implies
that migrated people are like to co-reside in a joint family.13 The annual population growth
rate (APGR) variable is significant below 10% error level in both reported equations and has
expected positive sign. The estimated value of co-efficient for this variable tell us that 1%
increase in population growth rate will rise 3.40% and 1.89% of co-residing possibility of
elderly men and women respectively. The percentage of population living in urban area
(PUPN) is statistically significant at 10% error level in case elderly men but in significant for
women. In both equations the PUPN variable has expected negative sign which implies that
urban people are more likely to live in a nuclear family setting rather than rural people in
every country. The estimated co-efficient of the variable PUPN for men indicates that the
0.14% probability decreases of elderly men co-residence attitude due to 1% percent increase
in urban population of a country. All the regional dummy variables (AFRI, ASIA, EURO and
LTAM) are highly significant and positive sign expect the European region which has negative
sign and insignificant co-efficient value. The positive sign of these regional dummy indicates
that sociological characteristics are very much influential to a generation to determine its
residence decision. The highest positive value of estimated coefficient among these four
regional dummies is 45.28 and 37.90 which is obtained from Asian region from the
estimated equation of elderly men and women respectively. The highest score indicates
both that Asian people are more likely to stay in join family and that they are more
influenced by social characteristics compared to other regions. The negative sign of regional
dummy for Europe although not statistically significant indicates that European sociological
characteristics support nuclear family attitude. Two dummy variables (LDEV and NDEV)
13 We know omission of a relevant explanatory variable is more harmful than inclusion of an irrelevant variable.
Since the sign of estimated coefficient explain some behaviour I include this variable in reported equation.
indicating development status are statistically insignificant although both belong expected
negative sign. The negative sign of these dummy variables imply that economic development
of a country motivates a generation to construct nuclear family leaving from joint family. In
any regression equation having dummy explanatory variable constant co-efficient is very
important causal relation with dependent variable if estimation process is not forcefully
excluded constant term. This term covers estimated value of base dummy which, in my
estimated equations, is the USA. The co-efficient of constant term is highly significant in all
cases and positive sign. The overall consequence of regional dummy, development dummy
and constant term gives positive estimated co-efficient for all estimated equations both
elderly men and women cases. These might have several interpretations such as (i)
sociological characteristics are more influential than economic development indicator of
country; and (ii) the probability of overall co-residence decision is improve due to a unit
change of these dummy variables. These empirical findings support the conclusion that co-
residence decision of a generation depends on both economic and social variables.
Reported Regression Results
Estima
ted
Dept.
Variab
le
Const.
Co-
efft.
Co-efft.
of PGDP
Co-
efft.
of
PSM
P
Co-
efft.
of
APG
R
Co-
efft.
of
PUPN
Co-
efft.
of
AFRI
Co-
efft.
of
ASIA
Co-
efft.
of
EUR
O
Co-
efft.
of
LTAM
Co-
efft.
of
LDEV
Co-
efft.
of
NDE
V
(9
Valu
e
qr�q�� 38.30�12.70�∗ −0.00046�0.00044� 0.048 �0.179
3.40∗∗�1.89� −0.14�0.08�
∗ 37.24�8.90�∗∗ 45.28�8.09�
∗∗ 5.47�5.93� 35.61�8.69�∗∗ −3.23�8.46�
∗ −2.85�6.94� 0.88
(std.
err)
qr�q � 53.21�11.90�∗ −0.0009�0.00042� 0.057 �0.159
1.89∗∗�1.12� −0.12
(0.08
)
29.55�8.78�∗∗ 37.90�7.79�
∗∗ −2.59�5.72� 30.02�8.24�∗∗ −11.76�8.18� −5.70�6.77� 0.88
(std.
err)
Conclusions: Changing family structure all over the world is not occurring due to a uniform
reason and the nature and type of change are not also unique in a sense that currently,
family structure in developed countries moves from nuclear family to living together system
(between heterogeneous sex or homogeneous sex) and in developing countries it turns from
traditional joint family to nuclear family. There must be several distinct reasons of changing
family structure in these two worlds. In this paper, I have theoretically as well as empirically
found out that changing degree of altruistic emotion along with wage rate differential
among generations, which leads to change in family structure. The most important thing
that I have proved in this paper is that in an infinite time horizon over lapping generation
model, the optimum current period consumption getting from a logarithmic utility function
is equal to that of the Cobb-Douglas utility function when degree of patience is less than
unity. Another important feature of this paper is that the maximum utility of a duo-
dimensional altruistic family consisting two generations would be equal to
individualistic/selfish exchange family when the amount of altruistic transfer between
generations is equal although mono-dimensional altruistic family produces more total utility
than selfish exchange family. This conclusion leads me to say that any social planner should
follow mono-dimensional altruistic familial transfer no matter what the direction of transfer
may be, (working generation to retire generation for developed countries and retire
generation to working generation for developing countries) to enhance social well-being.
The empirical results suggest that per capita GDP, annual population growth rate,
percentage of urban population and the most important sociological characteristics are
statistically significant determinants of co-residence decision of a country which supports my
theoretical propositions.
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Mathematical Appendix:
1.1 (Individual problem or overlapping generations problem):
The individual would like to maximize both periods’ consumptions as follows:
"#���,��� �STU = "�%���& + ��"�%����� & + X�[�1 + ��� �� − �1 + ������ − ����� ]
First Order conditions are:
�"#����
= "��� = X��1 + ��� − − − − − �
�"#������ = ��"��� � = X� − − − − − ��
�)��,� = �1 + ��� �� − �1 + ������ − ����� = 0 − �ii
1.2 Example of logarithmic utility function:
"� = g���� + ��g��������,��� �TU
Subject to, budget constraint for period- �: ��� + ��� = �� Budget constraint for period - �� + 1�: ����� = �1 + �������� Using Lagrange Method,
w��,��� ,!�,, ,,4�TU = g���� + ��g������ + X�% �� − ��� − ���& + X9{�1 + �������� − ����� }
First Order Conditions are:
�w����
= 1���
= X� − − − − − 1.1
�w������ = ��
����� = X9 − − − − − 1.2
�w����
= −X� + �1 + �����X9 = 0 − − − − − 1.3
�w�X� = �� − ��� − ��� = 0 − − − − − 1.4
�w�X9 = �1 + �������� − ����� = 0 − − − − − 1.5
�(���� ,� ��� =1�����
�����= X�X9 = �����
���= ���1 + ���
Optimum consumption for period – � is : ��� = 1�����+��
Optimum consumption for period – �� + 1� is : ����� = +����-�� �1�����+��
Optimum saving in period – � is : ��� = +�1�����+��
Cobb-Douglas Utility function: If we assume that utility function is linear Cobb-Douglas form as follows:
"� = ����5+������%+�& 6ℎ8�8 ���[0,1]
jaach8 �ℎ�� �ee.����� �. e.�achL��.� ah..�ℎ��� ��
= 12 �ℎ�� h8��a e.�ach8� ��/8a 8�c�g 68��ℎ� �. �.�ℎ L8��.� e.�achL��.�a
The budget constraint is as before: �1 + ��� �� − �1 + ������ − �����
The Lagrange function is:
"#���,��� �STU = m����5+��n m�����+��n + X�[�1 + ��� �� − �1 + ������ − ����� ]
First Order Conditions are:
�"#����
= �1 − ��� m��5+�n m�����+��n = �1 + ���X�
�"#������ = �� m����5+��n m�����+�5��n = X�
�"#�X� = �1 + ��� �� − �1 + ������ − ����� = 0
The marginal rate of substitution (�(���,� ��� � � between the consumption of two generations is as follows.
�1 − ��� m���5+��n �����+� ����� m����5+��n ������+�5��� = �1 + ��� = �����
���= ���1 + ���
�1 − ���
The optimum values of consumptions for two periods (or generations) and saving in period-t (or active
generation) are following.
���∗ = �1 − ��� ��; �����∗ = �1 + ����� ����� ���∗ = �� ��
1.3. Derivation for more than two periods:
"� = g���� + ��g������ + ����9g����9���,��� ,*��4�TU
Subject to, budget constraint for period- �: ��� + ��� + ��� = �� where ���, ��� are respectively saving
and bond purchased in period-� to use for the consumption of period-�� + 1� and �� + 2) respectively.
Budget constraint for period - �� + 1�: ����� = �1 + �������� Budget constraint for period - �� + 2�: ���9� = �1 + ���9���
Using Lagrange Method, we can solve the individual problem as follows:
w��,��� ,���4,!�,�� , ,,4,E�TU= g���� + ��g������ + ����9g����9� + X�% �� − ��� − ���& + X9��1 + �������� − ����� �+ XI[�1 + ���9��� − ���9� ]
First Order Conditions are: �w����
= 1���
= X� − − − − − 1.1
�w������ = ��
����� = X9 − − − − − 1.2
�w����9� = ����9
���9� = X9 − − − − − 1.3
�w����
= −X� + �1 + �����X9 = 0 − − − − − 1.4
�w����
= −X� + �1 + ���9XI = 0 − − − − − 1.5
�w�X� = �� − ��� − ��� = 0 − − − − − 1.6
�w�X9 = �1 + �������� − ����� = 0 − − − − − 1.7
�w�XI = �1 + ���9��� − ���9� = 0 − − − − − 1.8
Solving these equations we will get optimum value of consumptions for different time periods are –
���∗ = ��[1 + �� + ����9] ; �����∗ = ���1 + ����� ��[1 + �� + ����9] ��� ���9�∗ = ����9�1 + ���9 ��[1 + �� + ����9]
1.4 Family problem (both generations are selfish/individualistic):
The utility maximization problem of the family is written as follow:
"Q��R,��PSTU = "N���N� + "O���O�
Subject to budget constraint for adult-child: ��N + ��N = �N
.� ��N = �N − ��N = �N − �N�1 + �N� = �N �N�1 + �N�
Similarly, budget constraint for parent is:
��O = �1 + �����O = �1 + ����O �5�O�1 + �O�
"Q��R,��P,,R,,P,STU = "���N� + "���O� + XN [ �N�1 + �N� − ��N\ + XO [�1 + ����O �5�O�1 + �O� − ��O\
First Order conditions for optimization are:
�"Q���N
= "�NQ = XN − − − 2. �
�"Q���O
= "�OQ = XO − − − 2. ��
�"Q�XN = �N�1 + �N� − ��N = 0 = �N�1 + �N� = ��N − − − 2. ���
�"Q�XO = �1 + ����O �5�O
�1 + �O� − ��O = 0 = �1 + ����O �5�O�1 + �O� = ��O − − − 2. �/
The marginal rate of substitution (�(���,P��R� between the consumption of two generations is as follows.
"�NQ"�OQ = XN
XO
Logarithmic Utility function:
"Q��R,��P�TU = g���N + g���O
Budget constraint for adult child: ��N + ��N = �N
Budget constraint for parent: ��O = �1 + �����O
Saving function of adult child is
��N = �N �N�1 + �N�
And, saving function of parent is
��O = �O �5�O�1 + �O�
(individual point of view):
wQ��R,��P,!�R,R,,P�TU = ln ��N + g���O + XN � �N − ��N − �N �N�1 + �N�� + XO{�1 + ��� �O �5�O�1 + �O� − ��O}
wQ��R,��P,,R,,P�TU = ln ��N + g���O + XN � �N�1 + �N� − ��N� + XO{�1 + ��� �O �5�O�1 + �O� − ��O}
First Order Conditions are:
�wQ���N
= 1��N
= XN − − − − − 2.1
�wQ���O
= 1��O
= XO − − − −2.2
�wQ�XN = �N�1 + �N� − ��N = 0 − − − 2.3
�wQ�XO = �1 + ��� �O �5�O
�1 + �O� − ��O = 0 − − − −2.4
For adult generation optimum values are:
��N∗ = 1�R���+R� ��� XN∗ = ���+R�
1�R
For parent generation optimum values are:
��O∗ = +P���-��1�H P���+P� ��� XO∗ = ���+P�
+P���-��1�H P
The marginal rate of substitution (MRS) between adult child and parent is : �(��P,�R = ��P∗��R∗ = ,R∗
,P∗ =+P���-��1�H P
%��+P&W1�R %��+R&W
1.5 Utility maximisation of selfish/individualistic joint family- A Social planner’s view:
The family budget constraint is: ��N + ��O = 1�R���+R� + �1 + ��� +P1�H P
%��+P&
"Q��R,��P,,V,STU = "N���N� + "O���O� + XQ [ �N�1 + �N� + �1 + ����O �5�O�1 + �O� − ��N − ��O\
First Order Conditions:
�"Q���N
= "�NQ = XQ − − − 2j. �
�"Q���O
= "�OQ = XQ − − − 2j. ��
�"Q�XQ = �N�1 + �N� + �1 + ����O �5�O
�1 + �O� − ��N − ��O = 0 − − − 2j. ���
For Logarithmic Utility Function:
For a logarithmic utility function, which is additive to get family utility function. The Lagrange Function
becomes as follow:
wQ��R,��P,,V�TU = ln ��N + g���O + XQ � �N�1 + �N� + �1 + ��� �O �5�O�1 + �O� − ��N − ��O�
First Order Conditions for utility maximization are:
�wQ���N
= 1��N
= XQ − − − − − 2j. 1
�wQ���O
= 1��O
= XQ − − − − − 2j. 2
�wQ�XQ = �N�1 + �N� �1 + ��� �O �5�O
�1 + �O� − ��N − ��O = 0 − − − 2j. 3
The optimum value of consumption for each generation as follows.
��N∗ = ��O∗ = �1 + �O� �N + �1 + ����O�1 + �N� �5�O2�1 + �N��1 + �O�
XQ∗ = 2�1 + �N��1 + �O��1 + �O� �N + �1 + ����O�1 + �N� �5�O
1 .6Mono-dimensional Altruism and different marginal utility of money:
The utility function of selfish adult child is: "N = "N���N�
The utility function of altruistic parent is: "O = "O���O� + ^"N���N�
"Q��R,��PSTU = �1 + ^�"N���N� + "O���O�
Subject to budget constraint for adult-child: ��N + ��N = �N + ^
.� ��N = �N + ^ − ��N = �N + ^ − �N�1 + �N� = �N �N�1 + �N� + ^
Similarly, budget constraint for parent is:
��O = �1 + �����O − ^ = �1 + ����O �5�O�1 + �O�
"Q��R,��P,,R,,P,STU = �1 + ^�"N���N� + "O���O� + XN [ �N�1 + �N� + ^ − ��N\ + XO [�1 + ����O �5�O�1 + �O� − ^ − ��O\
First Order conditions for optimization are:
�"Q���N
= �1 + ^�"�NQ = XN − − − 3. �
�"Q���O
= "�OQ = XO − − − 3. ��
�"Q�XN = �N�1 + �N� + ^ − ��N = 0 = �N�1 + �N� + ^ = ��N − − − 3. ���
�"Q�XO = �1 + ����O �5�O
�1 + �O� − ^ − ��O = 0 = �1 + ����O �5�O�1 + �O� − ^ = ��O − −3. �/
The marginal rate of substitution (�(���,P��R� between the consumption of two generations is as
follows. ���`�)*RV)*PV = ,R
,P .� )*RV)*PV = ,R
���`�,P
Logarithmic utility function:
Utility function of adult child: "�N���N� = g���N
Utility function of altruistic Parent: "�O���O� = g���O + ^g���N
Thus the family utility function at time-t is:"�Q = �1 + ^�g���N + g���O
Budget constraint for adult child is: ��N + ��N = �N + ^
When ��N = 1�R���+R� �ℎ8�, ��N = 1�R
���+R� + ^
Budget constraint for altruistic parent is: ��O = �1 + ��� +P1�H P���+P� − ^
wQ��R,��P,,R,,P�TU = �1 + ^�g���N + g���O + XN � �N�1 + �N� + ^−��N� + XO{�1 + ��� �O �5�O�1 + �O� − ^ − ��O}
First Order Conditions are:
�wQ���N
= �1 + ^���N
= XN − − − − − 3.1
�wQ���O
= 1��N
= XO − − − − − 3.2
�wQ�XN = �N�1 + �N� + ^−��N = 0 − − − − − 3.3
�wQ�XO = �1 + ��� �O �5�O
�1 + �O� − ^ − ��O = 0 − − − − − 3.4
The optimum value of consumption and marginal utility of money for adult child generation are:
��N∗ = �N + �1 + �N�^�1 + �N� ��� XN∗ = �1 + �N��1 + ^�
�N + �1 + �N�^
The optimum value of consumption and marginal utility of money for altruistic parent-generation are:
��O∗ = �1 + ����O �5�O − �1 + �O�^�1 + �O� ���
XO∗ = �1 + �O��1 + ����O �5�O − �1 + �O�^
Mono-dimensional Altruism according to social planner’s view:
"Q��R,��PSTU = �1 + ^�"N���N� + "O���O�
Subject to, the budget constraint for adult-child is: ��N + ��N = �N + ^
.�, ��N = �N + ^ − ��N = �N + ^ − �N�1 + �N� = �N �N�1 + �N� + ^
Similarly, budget constraint for parent is:
��O = �1 + �����O − ^ = �1 + ����O �5�O�1 + �O�
"Q��R,��P,,V,STU = �1 + ^�"N���N� + "O���O� + XQ [ �N�1 + �N� + �1 + ����O �5�O�1 + �O� − ��N − ��O\
First Order Conditions for optimization are:
�"Q���N
= �1 + ^�"�NQ = XQ − − − 3j. �
�"Q���O
= "�OQ = XQ − − − 3j. ��
�"Q�XQ = �N�1 + �N� + �1 + ����O �5�O
�1 + �O� − ��N − ��O = 0 − −3j. ���
.�, �N�1 + �N� + �1 + ����O �5�O�1 + �O� = ��N + ��O − − − 3j. �/
The marginal rate of substitution (�(���,P��R� between the consumption of two generations is - ���`�)*RV)*PV =
1 .� )*RV)*PV = �
���`�
��N∗ = ��O∗ = 12 [ �N�1 + �N� + �1 + ����O �5�O
�1 + �O� ]
Using Logarithmic Function:
wQ��R,��P,,V�T� = �1 + ^� ln ��N + g���O + XQ[ 1�R���+R� + �1 + ��� +P1�H P
%��+P& − ��N − ��O]
First Order Conditions are:
�wQ���N
= �1 + ^���N
= XQ − − − − − −3.1.1
�wQ���O
= 1��O
= XQ − − − − − −3.1.2
�wQ�XQ = �N�1 + �N� + �1 + ��� �O �5�O
�1 + �O� − ��N − ��O = 0 − − − − − −3.1.3
⇒ �N�1 + �N� + �1 + ��� �O �5�O�1 + �O� = 2 + ^
XQ
XQ∗ = �2 + ^��1 + �N��1 + �O��1 + �O� �N + �1 + ����O�1 + �N� �5�O
Optimum value of consumption for adult child is:
��N∗ = �1 + ^�[�1 + �O� �N + �1 + ����O�1 + �N� �5�O ]�2 + ^��1 + �N��1 + �O�
Optimum value of consumption for altruistic parent is:
��O∗ = [�1 + �O� �N + �1 + ����O�1 + �N� �5�O ]�2 + ^��1 + �N��1 + �O�
1.8 A Duo-Dimensional Altruistic joint family with different marginal utility of money:
The utility function of child is: "�N = "N���N� + j"�O
The utility function of parent is: "�O = "O���O� + ^"�N
The utility function of child is: "�N = "N���N� + j"O���O�
The utility function of child is: "�O = "O���O� + ^"N���N�
Assuming additive utility, family function becomes –
"Q = �1 + ^�"N���N� + �1 + j�"O���O�
Budget constraint for child generation: ��N + ��N + j = �N + ^
Or, ��N = 1�R���+R� + ^ − j
Budget constraint for parent generation: ��O + ^ = �1 + ��� +P1�H P���+P� + j
"Q��R,��P,,R,,PSTU = �1 + ^�"N���N� + �1 + j�"O���O� + XN [ �N�1 + �N� + ^ − j − ��N\ + XO[�1 + ��� �O �5�O�1 + �O�
+ j − ^ − ��O]
First Order Conditions are:
�"Q���N
= �1 + ^�"�NQ = XN − − − 4. �
�"Q���O
= �1 + j�"�OQ = XO − − − 4. ��
�"Q�XN = [ �N�1 + �N� + ^ − j − ��N\ = 0 − −4. ���
��N∗ = �N�1 + �N� + ^ − j
�"Q�XO = �1 + ��� �O �5�O
�1 + �O� + j − ^ − ��O = 0 − −4. �/
��O∗ = �1 + ��� �O �5�O�1 + �O� + j − ^
The marginal rate of substitution (�(���,P��R� between the consumption of two generations is - ���`�)*RV���N�)*PV =
,R,P = )*RV
)*PV = ���N�,R���`�,P
Logarithmic utility function:
"��g��i fc�e��.� f.� ��cg�: "N = g���N + jg���O
And,
"��g��i fc�e��.� f.� L��8��: "O = g���O + ^g���N
Budget constraint for adult generation: ��N + ��N + j = �N + ^
Or, ��N = 1�R���+R� + ^ − j
Budget constraint for parent generation: ��O + ^ = �1 + ��� +P1�H P���+P� + j
The family utility function becomes,
"Q = �1 + ^�g���N + �1 + j�g���O
wQ = �1 + ^�g���N + �1 + j�g���O + XN [ �N�1 + �N� + ^ − j − ��N\ + XO[�1 + ��� �O �5�O�1 + �O� + j − ^ − ��O]
First Order Conditions are:
�wQ���N
= �1 + ^���N
= XN − − − − − − − 4.1
�wQ���O
= �1 + j���O
= XO − − − − − − − 4.2
�wQ�XN = �N�1 + �N� + ^ − j − ��N = 0 − − − − − − − 4.3
�N�1 + �N� + ^ − j = ��N − − − −4.3�
�wQ�XO = �1 + ��� �O �5�O
�1 + �O� + j − ^ − ��O = 0 − − − − − − − 4.4
�1 + ��� �O �5�O�1 + �O� + j − ^ = ��O − − − −4.4�
The marginal rate of substitution between two generations is
�(��P,�R =�1 + ^� ��NW�1 + j� ��OW = XN
XO − − − −4.5
After mathematical manipulation we get the optimum values of consumptions and marginal utility of money
are as follows.
��N∗ = XO�1 + ^��1 + ����O �5�OXN�1 + j��1 + �O� + XO�1 + ^��j − ^�
XN�1 + j�
��O∗ = XN�1 + j� �NXO�1 + ^��1 + �N� + XN�1 + j��^ − j�XO�1 + ^�
Optimum marginal utility of money for adult-child is
XN∗ = �1 + ^��1 + �N� �N + �1 + �N��^ − j�
Optimum marginal utility of money for parent is
XO∗ = �1 + j��1 + �O��1 + ����O �5�O + �1 + �O��j − ^�
4A.Family Behaviour with Two Dimensional Altruisms having same Marginal Utility of Money (Social Planner
Point of View):
"Q��R,��P,,V,STU = �1 + ^�"N���N� + �1 + j�"O���O� + XQ [ �N�1 + �N� + �1 + ��� �O �5�O�1 + �O� − ��N − ��O\
First Order Conditions are:
�"Q���N
= �1 + ^�"�NQ = XQ − − − 4j. �
�"Q���O
= �1 + j�"�OQ = XQ − − − 4j. ��
�"Q�XQ = [ �N�1 + �N� + �1 + ��� �O �5�O
�1 + �O� − ��N − ��O\ = 0 − −4j. �/
��N + ��O = �N�1 + �N� + �O �5�O�1 + �O�
The marginal rate of substitution (�(���,P��R� between the consumption of two generations is - ���`�)*RV���N�)*PV =
1 = )*RV)*PV = ���N�
���`�.
��N∗ = ��O∗ = 12 [ �N�1 + �N� + �O �5�O
�1 + �O�]
Logarithmic utility function:
"��g��i fc�e��.� f.� ��cg�: "N = g���N + jg���O
And,
"��g��i fc�e��.� f.� L��8��: "O = g���O + ^g���N
Budget constraint for adult generation: ��N + ��N + j = �N + ^
Or, ��N = 1�R���+R� + ^ − j
Budget constraint for parent generation: ��O + ^ = �1 + ��� +P1�H P���+P� + j
The family utility function becomes,
"Q = �1 + ^�g���N + �1 + j�g���O
Now the marginal utility of money is same for both generations, so the Lagrange function can be written as
follow:
wQ = �1 + ^�g���N + �1 + j�g���O + XQ[ �N�1 + �N� + �1 + ��� �O �5�O�1 + �O� − ��N − ��O]
First Order Conditions:
�wQ���N
= �1 + ^���N
= XQ − − − − − − − −7.1
�wQ���O
= �1 + j���O
= XQ − − − − − − − −7.2
�wQ�XQ = �N�1 + �N� + �1 + ��� �O �5�O
�1 + �O� − ��N − ��O = 0 − − − − − − − 7.3
Solving equations 7.1, 7.2 and 7.3 we get the marginal utility of money for both generations is as follows:
XQ = �2 + ^ + j��1 + �N��1 + �O��1 + �O� �N + �1 + ����O�1 + �N� �5�O
The optimal consumption of altruistic adult generation is as follows:
��N = �1 + ^��1 + �O� �N + �1 + ����O�1 + �N� �5�O�2 + ^ + j��1 + �N��1 + �O�
The optimal consumption of altruistic old (parent) generation is as follows:
��O = �1 + j��1 + �O� �N + �1 + ����O�1 + �N� �5�O�2 + ^ + j��1 + �N��1 + �O�
Table 1
Living Arrangement of Elderly Population (aged 60 and above) by Selected Regions and Sub-regions
of the World
Living Arrangement
Region/Subregion
Alone
Couple
only
With
children or
grandchildr
en
With other
relatives or
nonrelatives
Total
Asia 7 16 74 4 100
East 9 20 70 1 100
▪Southeast 6 13 73 9 100
South-central 4 9 83 - 100
Africa 8 9 74 8
Europe 26 43 26 4 100
South 19 39 38 4 100
Latin America &
Caribbean
9 16 62 14 100
North America 26 47 19 8 100
Source: UN 2005, Table II.5, p 36
Table-2: The dependent variable is percentage of elderly men living with a child or grandchild
(PECPM)
Const.
Co-efft. Co-efft.
of PGDP Co-
efft.
of
PSMP
Co-
efft.
of
APG
R
Co-
efft. of
PUPN
Co-efft.
of AFRI Co-efft.
of ASIA Co-
efft.
of
EUR
O
Co-efft.
of
LTAM
Co-
efft.
of
LDEV
Co-
efft.
of
NDE
V
(9
Value
32.37�14.030�∗ −0.0005�0.0004�
∗
45.79�12.60�∗∗ 48.48�12.29�
∗∗ 2.78
(10.7
8)
38.00�12.20�∗∗ −0.85�6.94�
∗ −3.75�5.57�∗
0.85
(std.e
rr)
33.38�15.05�∗∗∗ −0.0005�0.0004�
∗ −0.07 �0.20�
45.43�8.24�∗∗∗ 48.37�8.12�
∗∗∗ 2.48
(6.50
)
37.54�7.71�∗∗∗ −1.31�9.65�
∗ −4.10�7.66�∗
0.85
(std.e
rr)
31.55�13.06�∗∗∗ −0.0006�0.0004�
∗
4.06∗∗ �1.71� 35.12�8.28�∗∗∗ 43.04�7.70�
∗∗∗ 4.98
(6.03
)
30.91�7.23�∗∗∗ −0.40�8.5�
∗ −1.94�6.91�∗
0.87
(std.e
rr)
38.30�12.70�∗∗∗ −0.00046�0.00044� −0.21�0.08�
∗∗ 45.98�7.38�∗∗∗ 50.33�7.47�
∗∗∗ 3.66
(5.75
)
42.63�6.65�∗∗∗ −5.31�9.44�
∗ −5.02�7.51�∗
0.86
(std.e
rr)
32.35�13.79�∗∗∗ −0.0006�0.0004�
∗ −0.05 �0.15�
4.05∗∗∗�1.70� 34.86�8.70�
∗∗∗ 42.97�7.91�∗∗∗ 4.74
(6.17
)
30.57�7.79�∗∗∗ −0.77�8.75�
∗ −2.23�7.01�∗
0.87
(std.e
rr)
42.46�14.05�∗∗∗ −0.0003�0.0004�
∗ 0.094 �0.21�
−0.22�0.07�∗∗ 46.53�7.73�
∗∗∗ 50.65�7.67�∗∗∗ 4.19
(6.04
)
43.71�7.66�∗∗∗ −5.04�9.42�
∗ −4.61�7.48�∗
0.86
(std.e
rr)
38.52�12.51�∗∗∗ −0.0005�0.0004�
∗
3.44∗∗∗�1.72� −0.13�0.07�
∗∗ 36.86�8.15�∗∗∗ 45.06�7.64�
∗∗∗ 5.21
(5.77
)
34.97�7.32�∗∗∗ −3.35�8.41�
∗ −3.03�6.93�∗
0.88
(std.e
rr)
38.30�12.70�∗∗∗ −0.00046�0.00044� 0.048 �0.179�
3.40∗∗ �1.89� −0.14�0.08�∗∗ 37.24�8.90�
∗∗∗ 45.28�8.09�∗∗∗ 2.48
(6.50
)
35.61�8.69�∗∗∗ −3.23�8.46�
∗ −2.85�6.94�∗
0.88
(std.e
rr)
• *Indicates expected sign but not significant.
** Indicates expected sign and significant at 10% error level.
*** Indicates expected sign and significant at 5% error level.
Table-3: The dependent variable is percentage of elderly women living with a child or grandchild
(PECPW)
Const.
Co-efft. Co-efft.
of PGDP Co-
efft.
of
PSMP
Co-
efft.
of
APG
R
Co-
efft. of
PUPN
Co-
efft. of
AFRI
Co-
efft. of
ASIA
Co-
efft.
of
EUR
O
Co-
efft. of
LTAM
Co-
efft.
of
LDEV
Co-
efft.
of
NDEV
(9
Value
48.18�12.71�∗∗ −0.00095�0.0004�
∗∗
34.08�7.43�∗∗ 39.24�7.25�
∗∗ −4.40(5.8
9)
30.17�6.70�∗∗ −9.72�8.22�
∗
−6.10�6.66�∗
0.87
(std.e
rr)
40.75�13.47�∗∗ −0.00095�0.0004�
∗∗ −0.36 �0.14�
33.88�7.76�∗∗ 39.19�7.44�
∗∗ −4.58(6.0
3)
29.92�7.12�∗∗ −9.98�8.50�
∗
−6.30�6.84�∗
0.87
(std.e
rr)
47.69�12.50�∗∗ −0.001�0.0004�
∗∗∗
2.44∗∗∗�1.05�
27.67�8.38�∗∗ 35.97�7.55�
∗∗ −3.09(5.8
1)
25.91�7.14�∗∗ −9.45�8.17�
∗
−5.01�6.61�∗
0.87
(std.e
rr)
55.96�12.07�∗∗ −0.0008�0.0004�
∗∗∗ −0.15�0.07�
∗∗∗ 34.22�7.37�∗∗ 40.80�7.22�
∗∗ −3.76(5.5
4)
33.57�6.82�∗∗ −12.99�8.26�
∗ −7.03�6.73�∗
0.87
(std.e
rr)
48.13�13.12�∗∗ −0.001�0.0004�
∗∗∗ −0.03 �0.13�
2.43∗∗∗�1.03�
27.52�8.58�∗∗ 35.93�7.67�
∗∗ −3.
22
(5.9
1)
25.72�7.41�∗∗ −9.66�8.46�
∗
−5.17�6.84�∗
0.88
(std.e
rr)
55.51�12.24�∗∗ −0.0008�0.0004�
∗∗∗ 0.08 �0.16�
−0.17�0.07�∗∗∗ 34.69�7.52�
∗∗ 40.88�7.27�∗∗ −3.31
(5.7
3)
34.51�7.40�∗∗ −12.76�8.31�
∗ −6.68�6.78�∗
0.87
(std.e
rr)
53.46�11.84�∗∗ −0.0009�0.0004�
∗∗∗ 1.93∗∗ �1.05�
−0.11�0.065�∗ 29.11�8.38�
∗∗ 37.64�7.62�∗∗ −2.89
(5.5
4)
29.27�7.43�∗∗ −11.9�8.09�
∗
−5.92�6.67�∗
0.88
(std.e
rr)
53.21�11.90�∗∗ −0.0009�0.00042�
∗ 0.057 �0.159�
1.89∗∗ �1.12�
−0.12
(0.08)
29.55�8.78�∗∗ 37.90�7.79�
∗∗ −2.59(5.7
2)
30.02�8.24�∗∗ −11.76�8.18�
∗ −5.70�6.77�∗
0.88
(std.e
rr)
• *Indicates expected sign but not significant.
** Indicates expected sign and significant at 10% error level.
*** Indicates expected sign and significant at 5% error level.