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Endogenous division rules as a family constitution: strategic altruistic transfers and sibling competition


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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  1. See, e.g., Becker (1974), Barro (1974), Becker and Tomes (1979), Tomes (1981), Bernheim et al. (1985), Cox (1987), Kotlikoff and Morris (1989), Behrman (1997), Dunn and Phillips (1997), McGarry (1999), and Light and McGarry (2004).

  2. See, e.g., Menchik (1980, 1988), Wilhelm (1996), Dunn and Phillips (1997), and McGarry (1999).

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

  4. 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.

  5. 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.

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

  7. Although transfers or bequests may be “accidental,” we focus the analysis on planned transfers that arise from altruism and exchange motives (Masson and Pestieau 1997). Kohli and Künemund (2003) indicate that accidental transfers are “not really motives per se in terms of purposeful action.”

  8. 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.

  9. 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.

  10. 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.

  11. 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.

  12. It is obvious that when this condition does not hold, the problem becomes trivial. We prove below that {M , β } in Eqs. (7) and (9) constitute the unique interior solution to the parents’ utility maximization problem.

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

  14. 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.

  15. 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.

  16. 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.


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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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Correspondence to Yang-Ming Chang.

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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

$$S( {\beta ,1-\beta ,A_{i} ,...,A_{n}} )=\frac{\beta} {N}+(1-\beta )\frac{A_{i}} {\sum {A_{i}} } .$$

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

$$Y_{i} =( {1-A_{i}} )\mathit{w}_{i} +\left[ {\frac{\beta} {N}+(1-\beta )\frac{A_{i}} {\sum {A_{i}} } } \right]M.$$

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

$$\frac{\partial Y_{i}} {\partial A_{i}} =-\mathit{w}_{i} +(1-\beta )\frac{\sum {A_{-i}} } {\left( {\sum {A_{i}} } \right)^{2}}M=0,\mathrm{} A_{i} > 0$$
$$\frac{\partial Y_{i}} {\partial A_{i}} =-\mathit{w}_{i} +(1-\beta )\frac{\sum {A_{-i}} } {\left( {\sum {A_{i}} } \right)^{2}}M < 0,\mathrm{} A_{i} =0$$

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

$$\begin{array}{@{}rcl@{}} A_{i} &=&\sum {A_{i}} -\frac{\left( {\sum {A_{i}} } \right)^{2}\mathit{w}_{i}} {(1-\beta )M};\notag\\ \sum {\frac{\partial Y_{i}} {\partial A_{i}} } &=&-\sum {\mathit{w}_{i}} +(1-\beta )M\frac{N-P-1}{\sum {A_{i}} } =0; \end{array}$$

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

$$A_{i} =\frac{(1-\beta )(N-P-1)M}{\sum {\mathit{w}_{i}} } \left[ {1-\frac{(N-P-1)\mathit{w}_{i}} {\sum {\mathit{w}_{i}} } } \right].$$

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

$$\frac{(1-\beta )}{\sum {A_{i}} } M < \mathit{w}_{i} .$$

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

$$w_{l} \ge \frac{\sum {\mathit{w}_{i}} } {N-P-1},$$

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

$$\sum {\mathit{w}_{i}} \ge \frac{\sum {\mathit{w}_{i}} } {N-P-1}.$$

Thus, PN − 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).

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

JEL Classifications

  • D1
  • D6
  • C7