Sibling rivalry and strategic parental transfers.

• Author: Chang, Yang-Ming

1. Introduction

   Beginning with the seminal works of Becker (1974, 1981), considerable attention has been paid to analyzing the relationship between parental transfers and the recipient child's earnings. Several empirical studies of inter vivos transfers (i.e., transfers between living persons) have documented that financial resources are more likely to be transferred to children with low rather than high income (McGarry and Schoeni, 1995; Hochguertel and Ohlsson, 2000). This inverse relationship between transfer amounts and recipient earnings lends support to Becker's model of purely altruistic transfers.

   Another prominent study on parents and children is by Becker and Tomes (1976). They examine investment in the human capital of children and its implications for earnings capabilities and for the intrafamily distribution of income among children. Specifically, they show that parents invest more in children with larger endowments to achieve "efficiency" in human capital investment and then use transfers (e.g., inter vivos gifts) to achieve "equity" in income distribution.

   The seminal works of Becker (1974, 1976, 1981) and Becker and Tomes (1976), however, do not allow for "merit goods" that children render to their parents (e.g., services, visits, or parental care). The pioneering studies of Bernheim, Shleifer, and Summers (1985) and Cox (1987) further argue that intergenerational transfers are related to the exchange between parent and a child for family-specific merit goods such as child companionship or services. This consideration parallels Pollak's (1988) finding that parental transfers are tied to a child's consumption of particular goods or services that parents value. Kotlikoff and Morris (1989) contend that parents attempt to manipulate the behavior of their children by offering a "bribe" to induce them to provide services. Moreover, Cox and Rank (1992) contend that parental transfers can be interpreted as a way of "buying" child services. Empirical studies of inter vivos transfers and strategic exchange provide evidence that transfers and recipient earnings may be positively related (e.g., Cox 1987; Cox and Rank 1992; Cox and Jakubson 1995; Lillard and Willis 1997).

   Most theoretical models of intergenerational transfers emphasize the relationship between parents and children without explicitly considering interactions between the children. In this paper, we develop an alternative approach that emphasizes not only intergenerational interactions between parents and children, but also strategic intragenerational interactions between siblings. We incorporate into the analysis a "contest success function," which is common in the rent-seeking literature, to examine "transfer-seeking" activities by children within the family. We attempt to examine the role of parental transfers in redistributing income among siblings and in affecting offspring behavior. We also characterize the parents' choice of a financial transfer and link it to parents' income, altruism, and children's supply of merit goods to parents.

   Based on the underlying premise that siblings are equally altruistic toward their parents, the sibling-rivalry model developed in this study implies that children whose earning capabilities or earnings are higher supply less services (or exert less effort) in acquiring financial resources from their parents. In other words, parents transfer more resources to children with lower earnings. In this case, parental transfers are compensatory in the Beckerian sense, despite the fact that the models of Becker (1974, 1981) and Becker and Tomes (1976) do not allow for merit goods. We compare differences in income between siblings before and after parental transfers and find that in equilibrium the income differential is reduced. This finding supports the proposition that the family as an institution serves as an "income equalizer." If earnings capability is a reasonably good proxy for a child's ability, other factors (e.g., good fortune in labor markets) being equal, then parents provide a type of "insurance" for the lower ability child by transferring proportionately more resources to that child. We also discuss conditions under which the transfer amount and a recipient child's earnings may be positively related.

   In addition to examining sibling rivalry for parental transfers, we also analyze utility maximizing altruistic financial transfers by the parents. We further compare the noncooperative Nash equilibrium with the cooperative solution in transfers and children's services. Special attention is paid to moral hazard problems with merit goods.

   The remainder of the paper is organized as follows. Section 2 develops a simple model of sibling competition for parental resources in a noncooperative Nash game. In this section, we discuss implications of parental transfers for income differentials between siblings. In section 3, we characterize the endogeneity of parents' financial transfers and compare the outcomes of the two alternative games that parents and children may play. Section 4 summarizes and concludes.

2. A Nash Model of Sibling Rivalry for Parental Transfers

   Consider a family in which two siblings compete for financial transfers from their parents. The parents have a total amount of M dollars to distribute to the siblings. The parents, however, do not make their transfers unconditionally. Rather, the parents divide the "prize" M according to the proportion of time that each sibling expends in rendering services to their parents. Specifically, let [e.sub.i] denote the amount of time that sibling i, i = 1, 2, devotes to his parents. (1) For sibling 1, the share of the prize is [p.sub.1] ([e.sub.1],[e.sub.2]), and for sibling 2, it is [p.sub.2]([e.sub.1],[e.sub.2]) = 1 - [p.sub.1] ([e.sub.1],[e.sub.2]), where

   (1) [p.sub.1]([e.sub.1], [e.sub.2]) = [e.sub.1]/[e.sub.1] + [e.sub.2].

   Equation 1 is a "contest success function" similar to those commonly used in the rent-seeking literature (e.g., Tullock 1980; Skaperdas 1996). In other words, parents orchestrate a "transfer-seeking contest" between siblings to induce their supply of services and to determine the distribution of the financial transfer.

   According to Equation 1, sibling l's share of the prize, M, depends positively on his time of services, [e.sub.1], and negatively on sibling 2's time of services, [e.sub.2]. Similarly, sibling 2's share depends positively on [e.sub.2] and negatively on [e.sub.1]. It can easily be verified that the marginal effect of [e.sub.i] on [p.sub.i], [p'.sub.i] [equivalent to] [differential][p.sub.i]([e.sub.1],[e.sub.2])/[differential][e.sub.i] = [e.sub.j]/ [([e.sub.1] + [e.sub.2]).sup.2], is positive but is subject to diminishing returns, where i, j = 1, 2, i [not equal to] j.

   For analytical simplicity, each sibling is assumed to be risk neutral and has T units of time available for working outside of the family and for providing services to the parents. (2) Earning capabilities of the siblings are reflected by the wage rates they command in the labor markets. Let the market wage rate for sibling i be [w.sub.i] > 0.

   The siblings choose their service allocations to maximize their individual expected incomes, which are given by

   (2) [Y.sub.1] = (T - [e.sub.1])[w.sub.1] + [p.sub.1]([e.sub.1], [e.sub.2])M + [[alpha].sub.1][e.sub.1],

   and

   (3) [Y.sub.2] = (T - [e.sub.2])[w.sub.2] + [1 - [p.sub.1]([e.sub.1], [e.sub.2])M + [[alpha].sub.2][e.sub.2]

   where the altruism coefficient, [[alpha].sub.i], represents the monetary valuation that sibling i places on each unit of time spent with the parents. Note that if [[alpha].sub.i] > 0, sibling i "enjoys" spending time with the parents. We assume that 0 [less than or equal to] [[alpha].sub.i] 0). The first-order conditions (FOCs) for sibling 1's and sibling 2's optimization problems are given respectively by

   (4) [differential][Y.sub.1]/[differential][e.sub.1] = [e.sub.2]/[([e.sub.1] + [e.sub.2]).sup.2] M - [w.sub.1] + [[alpha].sub.1] = 0,

   and

   (5) [differential][Y.sub.2]/[differential][e.sub.2] = [e.sub.1]/ [([e.sub.1] + [e.sub.2]).sup.2] M - [w.sub.2] + [[alpha].sub.2] = 0

   The FOCs indicate that each sibling's service time is optimally chosen so that the expected marginal benefit of exerting one more unit of service time equals its marginal cost (in terms of wage income forgone) net of the altruistic coefficient (i.e., [p'.sub.i]M = [w.sub.i] - [[alpha].sub.i]). The sufficient, second-order conditions for a maximum are satisfied as a result of the strict concavity of the contest success functions.

   It follows from Equations 4 and 5 that

   (6) [e.sub.2]/[e.sub.1] = [w.sub.1] - [[alpha].sub.1]/[w.sub.2] - [[alpha].sub.2].

   Equation 6 implies that, given the altruism coefficients, the relative amount of services supplied by the siblings is negatively related to their relative market wage. Furthermore, it follows from Equations 4 and 5 that if [w.sub.1] - [[alpha].sub.1] > [w.sub.2] - [[alpha].sub.2] then

   (7) [e.sub.2]/[([e.sub.1] + [e.sub.2]).sup.2]M > [e.sub.1]/ [([e.sub.1] + [e.sub.2]).sup.2]M,

   which implies that

   (8) [e.sub.1]

   Hence, the amount of child services is inversely related to the market wage (net of the altruism coefficient).

   Next, we examine the equilibrium behavior of the siblings and its economic implications. The FOCs in Equations 4 and 5 implicitly define the reaction functions for sibling 1 and sibling 2, respectively, [e.sub.1] = [e.sub.1]([e.sub.2];M, [w.sub.1],[[alpha].sub.1]) and [e.sub.2] = [e.sub.2] ([e.sub.1];M,[w.sub.2],[[alpha].sub.2]). The two reaction functions jointly determine the noncooperative Nash equilibrium solution, denoted by the two-tuple ([e.sup.*.sub.1],[e.sup.*.sub.2]). Using Equations 4 and 5, we solve for the Nash equilibrium levels of child services:

   (9) [e.sup.*.sub.1] = [w.sub.2] - [[alpha].sub.2]/[[([w.sub.1] - [[alpha].sub.1]) + ([w.sub.2] + [[alpha].sub.2])].sup.2]M,

   and

   (10) [e.sup.*.sub.2] = [w.sub.1] - [[alpha].sub.1]/[[([w.sub.1] -...