Abstract
We develop a model of the joint determination of education and health investments under four hypotheses of self-interest, filial altruism, parental altruism, and reciprocal altruism. Three-period-lived agents in our overlapping generations model optimally choose fractions of time devoted to investments in children’s education and old parents’ health. The agents with parental altruism spend the longest time in education, hence the economy’s growth rate is the highest among the four hypotheses. However, their life expectancy is predicted to be the second lowest. The agents with filial altruism invest in health the most and enjoy the longest lifespan. Under this hypothesis, the economy grows at the slowest rate because they substitute away from education investments. The self-interest and reciprocal altruism hypotheses yield the results with more balanced investments between the two kinds of human capital. The models are calibrated to fit Japanese economy to examine effects of an expansion of Pay-As-You-Go social security on the macroeconomy. We find that raises in the contribution rate make the economy grow faster but negatively affect life expectancy of old agents under all hypotheses. Social welfare increases by the expansion of social security under the hypotheses of self-interest, parental altruism, and reciprocal altruism but decreases under the hypothesis of filial altruism.
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Notes
See Hayashi (1986).
Blackburn and Cipriani (2005) is the only theoretical paper discussing two-sided (reciprocal) altruism.
One-sided (parental) altruism is frequently employed in the theoretical literature, but the idea that parents unconditionally care well-being of their children, who do not care that of their parents at all, is rather odd.
As shown in our results, households devote similar fractions of time to education investments under two hypotheses of self-interest and reciprocal altruism because two diverging effects from individuals’ filial and parental altruistic traits tend to cancel each other [see Eq. (24)]. This similarity may disguise households connected via reciprocal altruism as selfish ones.
For example, Lau et al. (2014) suggests that per capita health care expenditure of the EU countries is under the process of convergence towards the very high levels of its northern members.
According to Ehrlich and Lui (1991), various empirical evidences support this view of the family relationships in the case of the United States (see their note 1). Analogous to our previous argument that family members should be connected via both filial and parental altruism, they are likely to be connected via not only emotional but also material interdependencies.
An amount of financial supports is proportional to beneficiaries’ health level because in this study the health level of old agents is an equivalent of their consumption capacity or period lifespan (see our note 25).
Filial altruism is discussed in Cigno and Rosati (1996), Boldrin and Jones (2002), and Blackburn and Cipriani (2005). Endogenous longevity is discussed in van Zon and Muysken (2001), Blackburn and Cipriani (2002), Chakraborty (2004), and Barro (2013). However, the two subjects have not been brought together in one study.
Mustafa and Lines (2012) investigates how paternalism in Asian societies influences younger family members’ behaviors towards their parents.
The concept of social norms is expressed by setting time devoted to health investments in different periods equal, \( v_{t} = v_{t + 1} = v_{t + 2} \) [see Eq. (3) and note 20]. It has to be noted that norms of filial care do not specify a fraction of time that agents spend on old-age care services.
In contrast to the concept of social norms mentioned in note 10, we measure agents’ degrees of filial and parental altruism by parameters attached to utilities derived from their parents’ and children’s old-age consumption, respectively [see Eq. (1)].
Family relationships might have moved from those with filial altruism, where people spend a significant amount of time to take care of their cohabiting old parents, to those with parental altruism, where young parents carefully raise their sole child in nuclear family setups. However, considering the current high old-age dependency rates in developed economies, family relationships have to move back to filial altruism. If we take our social security contributions into account, total financial supports for the elderly parents have not declined during the post war period. See Ehrlich and Lui (1991).
Although the relationship between filial altruism and endogenous longevity (health investments) has not studied yet, there is a large literature on the similarly important relationship between parental altruism and endogenous growth (education investments).
The fixed amount of gifts shelled out by selfish children might be prescribed by their family rules. See Cigno and Rosati (1996).
An old parent’s labor force participation can be exogenously or endogenously determined. If she is forced to retire, for example, due to a legal retirement age (exogenous retirement), an addition of utility from leisure to Eq. (1) does not affect our main results shown in Eqs. (24) and (25). If she can freely choose whether she retires or not (endogenous retirement), the new problem requires a complex two-stage optimization procedure [see Matsuyama (2008) and Aisa et al. (2012)]. We are currently working on an issue of endogenous retirement and longevity in a separate study under the simplest hypothesis of self-interest but assume away from it in this study.
The first term in the right hand side of Eq. (1) is the utility from consumption of a young parent (superscript y) at time t (subscript t), and, analogically, the second term is the utility from consumption of an old parent (superscript o) at time t + 1 (subscript t + 1).
The measure of well-being is not necessarily agents’ consumption level. In Ehrlich and Lui (1991), their companionship function has only two arguments, which are the number of surviving children and their human capital stock (the quantity and quality of children). In our study, the quality of children and the quantity of the elderly included in old-age consumption are their counterparts. Because an agent’s own old-age consumption at time t + 1 is already in place in Eq. (1), the choice of old-age consumption as the measure of well-being significantly facilitates the derivation of first-order conditions as well.
While Ogawa and Retherford (1993) suggest that norms of filial care was stable during the post war period in Japan, the assumption that agents of three consecutive generations devote the exactly same fraction of time to health care services is rather of theoretical necessity and for simplicity. However, if we relax this assumption and change it to a more realistic one such that agents spend \( \left( {1 - \sigma } \right) \times 100\,\% \) less time for health investments than ones belong to a directly preceding generation, \( \sigma^{2} v_{t} = \sigma v_{t + 1} = v_{t + 2} , \, \, 0 < \sigma < 1 \), this does not affect first-order conditions (12) and (13).
The earning capacity of the young parent at time t is \( wE_{t}^{y} H_{t}^{y} \), where w is the wage rate per effective labor, to be precise. As we normalized both w and \( H_{t}^{y} \) equal to one, the earning capacity is stated as \( E_{t}^{y} \).
If the compensation rate is endogenously determined as in Ehrlich and Lui (1991), the new problem requires a two-stage maximization procedure under the hypotheses without filial altruism. In the first stage, taking the compensation rate as given, a young agent chooses the optimal values of education and health investments, which are functions of the compensation rate. In the second stage, by maximizing her children’s utility, the young agent chooses an optimal value of the compensation rate, which applies in all time period. However, even for the self-interest model we have no closed-form solution in the second stage. Though to endogenize the compensation rate is more straightforward under the hypotheses with filial altruism, we assume that the rate is exogenously given in order to derive consistent results from all the four hypotheses and compare them.
A young parent may provide her child with a fixed fraction of time or a fixed amount of financial supports in order to raise the child [see Ehrlich and Lui (1991) for time supports and Cigno and Rosati (1996) for financial supports]. However, because the young parent always has one child in this study, the fixed cost of raising a child does not affect first-order conditions (12) and (13). As discussed in note 22, the amount of financial supports has to be determined exogenously in order to compare results from the four hypotheses.
These are the extreme cases for the clearer demonstration, but as discussed briefly in our conclusion the actual values of the two parameters should be somewhere between zero and one. For the more general cases of \( 0 < \alpha^{F} < 1 \) and \( 0 < \alpha^{P} < 1 \), the solutions for u t and v t are given by: \( \left( {1 + 2\beta + \alpha^{F} + 2\beta^{2} \alpha^{P} } \right)u_{t}^{2} + \left[ {\left( {1 + \beta + \alpha^{F} + 2\beta^{2} \alpha^{P} } \right)\left( {1 - \delta } \right)\frac{1}{{\gamma \bar{z}}} - \left( {\beta + \beta^{2} \alpha^{P} } \right)} \right]u_{t} - \beta^{2} \alpha^{P} \left( {1 - \delta } \right)\frac{1}{{\gamma \bar{z}}} = 0 \) and \( v_{t} = \frac{{\beta + \alpha^{F} + \beta^{2} \alpha^{P} }}{{\left( {1 + \beta + \alpha^{F} + \beta^{2} \alpha^{P} } \right)\left( {1 + \bar{z}\psi } \right)}}\left( {1 - u_{t} } \right) \).
As the one period in this model spans 30 years, the annual growth rate is given by: \( \left( {1 + g^{X} } \right)^{\frac{1}{30}} - 1 \), and life expectancy at birth of the agents is given by: \( 30 + 30 + l^{X} \).
See Barro (2013).
To achieve the solution below, we assume that the social security benefit is a function of the contribution of the old agent’s own child, and this motivates the young agent to invest in her child’s education capital. In case of Japan, PAYG benefits are related to agents’ previous earnings, that is, \( S_{t + 1} = \phi E_{t}^{y} \), where the parameter ϕ is called as the replacement rate (see Social Security Administration 2013). If the benefit is recognized as a fraction of the old agent’s own earning capacity when she was young, she has no additional incentive to invest in her child’s education capital. As discussed in Ehrlich and Lui (1998), a higher PAYG compensation rate lowers the return on education capital and slows economic growth.
In this case, the annual depreciation rate is 0.04. The depreciation rate of education capital is also discussed in Sala-I-Martin (1996).
In Ehrlich and Lui (1991), the compensation rate is 0.18 and the equilibrium value of savings rate is 0.16. We add up these two rates as our compensation rate.
In our study, the relevant parameter values are given by: \( \alpha^{F} = \beta^{2} \;\alpha^{P} = 0.6 \).
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Koda, Y., Uruyos, M. Altruism and four shades of family relationships. Eurasian Econ Rev 5, 345–365 (2015). https://doi.org/10.1007/s40822-015-0027-4
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DOI: https://doi.org/10.1007/s40822-015-0027-4