## Abstract

Based on the notions of parental altruism, sibling competition, and family constitution, we present a self-enforcing model where heterogeneous children have economic incentives to supply family-specific merit goods (e.g., companionship) to their parents for securing inheritable wealth and the altruistic parents decide on division rules according to an optimizing behavior. In our analysis of intergenerational cooperation and intragenerational competition, the altruistic parents care about the efficiency of the children-provided merit goods and the equity of the children’s incomes. For an optimal allocation of wealth, the parents strategically partition it into two pools: one to be distributed equally whereas the other unequally according to their children’s supply of merit goods. We look at motivation of the parents in allocating their wealth to the two different pools. The analysis of endogenous division rules has implications for the compatibility between equal postmortem transfers and unequal inter vivos gifts, both of which are consistent with parental altruism.

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## Notes

For a systematic review of empirical studies on this issue, see Stark and Zhang (2002).

A family constitution in connection with a wealth division rule is said to be self-enforcing if it is in each family member’s self-interest to comply with the rule.

On the first page of his seminal book, A Treatise on the Family, Becker (1981) remarks that “Conflict between the generations has become more open, and parents are now less confident that they can guide the behavior of their children.” In the present paper, “conflict” refers to situations in which parents and children make their decisions independently and noncooperatively. Buchanan (1983) is the first to introduce the notion of rent seeking into the analysis of family transfers. Cox (2003) stresses elements of conflict in economic analysis of family transfers and interactions.

The results we present below can easily be generalized into scenarios with more than two children.

We borrow this division rule from Noh (1999) who analyzes the endogeneity of sharing rules in intragroup competition when players allocate their resources between productive and appropriative activities. For studies on endogenous sharing rules in the theory of contest or rent-seeking, see, e.g., Nitzan (1994), Lee (1995), and Baik and Lee (2000). Our model departs from these studies, however. We consider endogenous sharing rules within the family in which “selfish” children as intragroup competitors allocate their time between two

*productive*activities: providing services inside the family and working outside the family, and their parents are altruistic in making strategic transfers to their children in exchange for services.It is trivial to talk about the case of homogeneous children in the framework with an endogenous division because the children will allocate the same amounts of time to serving their parents and to labor market participation. This naturally leads to an equal division of inheritable wealth among the children.

For the case with

*N*children competing for wealth transfers, we show in the Appendix that a model with only two children always yields an interior solution in terms of service time rendered to their parents.Lundholm and Ohlsson (2000) use a quadratic cost function to capture a dislike of inequality in bequests. We follow their approach by using a quadratic cost function to capture a dislike of post-transfer income inequality. As pointed out by an anonymous referee, a more general approach to modeling such an income inequality should consider the concavity of children’s consumption utility or convexity of their effort costs functions. This is an interesting question for future research.

Our definition of the acquisition ratio is the inverse of the value ratio defined by Margolis (1984, p. 37).

Cigno (2006a) shows that parents-to-children transfers are related to certain types of “political equilibrium” such as a self-enforcing family constitution or representative democracies. In analyzing mutually beneficial cooperation across generations, Cigno (2006b) further stresses norms or institutions in enhancing intra-family transfers and intergenerational bonds.

See Joulfaian (2005) for an analysis of how parents choose between gifts and bequests in response to gift taxes and bequest taxes. His analysis suggests that gifts and bequests be treated as two different modes of transfers in a utility-maximizing distribution of inheritable wealth.

Faith et al. (2008) note that in ancient times, wealth that children received came from the possession of hereditary rights. The authors explain why primogeniture was the preferred method of inheritance during the Middle Ages in Europe, particularly in areas dominated by the Roman Catholic Church. The primary reason, according to their paper, was that primogeniture made wealth distribution across families less dispersed and hence lowered the Church’s information costs of collecting taxes.

## References

Baik KH, Lee S (2000) Two-stage rent-seeking contests with carryovers. Public Choice 103:285–96

Barro R (1974) Are government bond net worth? J Polit Econ 82:1092–1117

Becker GS (1974) A theory of social interactions. J Polit Econ 82:1063–93

Becker GS (1981) A treatise on the family. Harvard University Press, Cambridge

Becker GS, Tomes N (1979) An equilibrium theory of the distribution of income and intergenerational mobility. J Polit Econ 87:1153–1189

Behrman JR (1997) Intrahousehold distribution and the family. In: Rosenzweig M R, Stark O (eds) Handbook of population and family economics. North-Holland

Bernheim BD (1991) How strong are bequest motives? Evidence based on estimates of the demand for life insurance and annuities. J Polit Econ 99:899–927

Bernheim BD, Shleifer A, Summers L (1985) The strategic bequest motive. J Polit Econ 93:1045–76

Bernheim BD, Severinov S (2003) Bequests as signals: an explanation for the equal division puzzle. J Polit Econ 111:733–764

Buchanan JM (1983) Rent seeking, noncompensated transfers, and laws of succession. J Law Econ 26:71–85

Chang Y-M (2007) Transfers and bequests: a portfolio analysis in a Nash game. Ann Financ 3:277–295

Chang Y-M (2009) Strategic altruistic transfers and rent seeking within the family. J Popul Econ 22:1081–1098

Chang Y-M (2012) Strategic altruistic transfers, redistributive fiscal policies, and family bonds: a micro-economic analysis. J Popul Econ 25:1481–1502

Chang Y-M, Weisman DL (2005) Sibling rivalry and strategic parental transfers. South Econ J 71:821–836

Cigno A (1993) Intergenerational transfers without altruism: family, market and state. Eur J Polit Econ 9:505–518

Cigno A (2006a) A constitutional theory of the family. J Popul Econ 19:259–283

Cigno A (2006b) The political economy of intergenerational cooperation. In: Kolm SC, Ythier JM (eds) Handbook of the economics of giving, altruism and reciprocity. North-Holland

Cox D (1987) Motives for private income transfers. J Polit Econ 95:508–46

Cox D (2003) Private transfers within the family: mothers, fathers, sons and daughters. In: Munnell AH, Sundén A (eds) Death and dollars: the role of gifts and bequests in America. The Brookings Institution, Washington, DC, pp 167–197

Dunn TA, Phillips JW (1997) The timing and division of parental transfers to children. Econ Lett 54:135–137

Faith RL, Goff BL, Tollison RD (2008) Bequests, sibling rivalry, and rent seeking. Public choice 136:397–409

Farmer A, Horowitz AW (2010) Mobility, information, and bequest: the “other side” of the equal division puzzle. J Popul Econ 23:121–138

Joulfaian D (2005) Choosing between gifts and bequests: how taxes affect the timing of wealth transfers. NBER Working Papers No. 11025, National Bureau of Economic Research

Kohli M, Künemund H (2003) Intergenerational transfers in the family: what motivates giving? In: Bengtson VL, Lowenstein A (eds) Global aging and challenges to families. Aldine de Gruyter, New York, pp 123–142

Konrad KA, Harald A, Nemund K, Lommerud KE, Robledo JR (2002) Geography of the family. Am Econ Rev 92:981–998

Kotlikoff LJ, Morris JN (1989) How much care do the aged receive from their children? In: Wise DA (ed) The economics of aging. University of Chicago, Chicago, pp 149–172

Lee S (1995) Endogenous sharing rules in collective-group rent-seeking. Public Choice 85:31–44

Light A, McGarry K (2004) Why parents play favorites: explanations for unequal bequests. Am Econ Rev 94:1669–1681

Lundholm M, Ohlsson H (2000) Post mortem reputation, compensatory gifts and equal bequests. Econ Lett 68:165–71

Margolis H (1984) Selfishness, altruism and rationality. University of Chicago, Chicago

Masson A, Pestieau P (1997) Bequests motives and models of inheritance: a survey of the literature. In: Erreygers G, Vandevelde T (eds) Is inheritance legitimate? Springer-Verlag, Berlin, pp 54–88

McGarry K (1999) Inter vivos transfers and intended bequests. J Public Econ 73:321–351

Menchik PL (1980) Primogeniture, equal sharing, and the U.S. distribution of wealth. Q J Econ 94:299–316

Menchik PL (1988) Unequal estate division: is it altruism, reverse bequests, or simply noise? In: Kessler D, Masson A (eds) Modelling the accumulation and distribution of wealth. Oxford University, New York

Menchik PL, David MH (1983) Income distribution, lifetime savings, and bequests. Ame Econ Rev 73:672–690

Nitzan S (1994) Modelling rent-seeking contests. Eur J Polit Econ 10:41–60

Noh SJ (1999) A general equilibrium model of two group conflict with endogenous intra-group sharing rules. Public Choice 98:251–267

Shorrocks AF (1979) On the structure of inter-generational transfers between families. Economica 46:415–426

Stark O (1998) Equal bequests and parental altruism: compatibility or orthogonality? Econ Lett 60:167–171

Stark O, Zhang J (2002) Counter-compensatory inter-vivos transfers and parental altruism: compatibility or orthogonality? J Econ Behav Organ 47:19–25

Tomes N (1981) The family, inheritance and the intergenerational transmission of inequality. J Poli Econ 89:928–958

Wilhelm MO (1996) Bequest behavior and the effects of heirs’ earnings: testing the altruistic model of bequests. Am Econ Rev 86:874–892

## Acknowledgments

We thank the editor, Alessandro Cigno, and anonymous referees for valuable comments and suggestions which have significantly improved the quality of the paper. An earlier version of this paper was presented at the 80th Annual Southern Economic Association Conferences, Atlanta, Georgia, November 20–22, 2010. We thank Susan Carter, Carolina Castilla, Shane Sanders, Daru Zhang, and conference participants for valuable comments. The usual caveats apply.

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*Responsible editor*: Alessandro Cigno

## Appendix

### Appendix

An interior solution exists as long as there are two children competing for parental wealth

In the case of *N* children, the endogenous division rule is given by

Given wage rates *w*
_{
i
} and parents’ overall transfer *M*, we have children *i*’s disposable income as given by

Allowing for the possibility of a corner solution, we have the FOCs for the children as follows:

We first solve for *A*
_{
i
} for the subset of children who provide services to their parents. From Eq. (22), we have

where *P* is the number of children not providing services to the parents. Solving for *A*
_{
i
} yields

We now solve for the corner solution for *P* children who do not render service to parents. We use subscript *l* to represent these *P* children, and *i* for the rest. From Eq. (23), we solve for necessary condition for those children who do not render services. When (23) holds, we have *A*
_{
i
} = 0 which implies that \(\sum {A_{i}} =\sum {A_{-i}} \) and thus

Together with (24) and (25), we find that if the following condition is satisfied:

then *A*
_{
i
} = 0. This is the marginal condition to have one more child that does not render service to the parents. To obtain the maximum number of *P*, we set \(w_{l} =\frac {\sum {\mathit {w}_{i}} } {N-P-1}\) for all *P* children. Assuming that if we have at the least one more child who does not render services to the parents, we must have

Thus, *P* ≤ *N* − 2 is the sufficient condition to have at least one more child who does not render services to the parents. In other words, if only two children compete for parental transfers, an interior solution always exists.

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Chang, YM., Luo, Z. Endogenous division rules as a family constitution: strategic altruistic transfers and sibling competition.
*J Popul Econ* **28**, 173–194 (2015). https://doi.org/10.1007/s00148-013-0501-9

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DOI: https://doi.org/10.1007/s00148-013-0501-9

### Keywords

- Parental altruism
- Endogenous division rules
- Sibling competition
- Family constitution

### JEL Classifications

- D1
- D6
- C7