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

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.

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

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

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:

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

(22)

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

(23)

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

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

(24)

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

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

(25)

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, 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.