Rotten spouses, family transfers, and public goods

Abstract

We show that once interfamily exchanges are considered, Becker’s rotten kids mechanism has some remarkable, hitherto unnoticed, implications. Specifically, Cornes and Silva’s (J Polit Econ 107(5):1034–1040, 1999) result of efficiency in the contribution game amongst siblings extends to a setting where the contributors (spouses) belong to different families. More strikingly still, the mechanism may also have dramatic redistributive implications. In particular, we show that the rotten kids mechanism combined with a contribution game to a household public good may lead to an astonishing equalization of consumptions between and within families, even when their parents’ wealth levels differ. The most striking results obtain when wages are equal and when parents’ initial wealth levels are not too different. For very large wealth differences, the mechanism must be supplemented by a (mandatory) transfer that brings them back into the relevant range. When wages differ but are similar, the outcome will be near efficient (and near egalitarian).

Appendix

1.1 Proof of Proposition 1

Assume w ₁ = w ₂ = w and consider a given level of x ₂>0 (and continue to assume without loss of generality that x ₁ ≤ x ₂). From Eq. 18, we can see that a corner solution, $(g_{1}^{\ast }=0,g_{2}^{\ast }>0)$, prevails if

$$ 2\varphi^{\prime}(G^{\ast})=2\varphi^{\prime}(\widetilde{g}_{2}(0))<u^{\prime }(d_{1}^{\ast})w, $$

(26)

where $G^{\ast }=g_{2}^{\ast }=\widetilde {g}_{2}(0).$ For x ₁ = x ₂, we have

$$u^{\prime}(d_{1}^{\ast})w=u^{\prime}\left( \frac{(\tau-a_{1}^{\ast})w+x_{1} }{2}\right) w<u^{\prime}\left( \frac{(\tau-a_{2}^{\ast}-g_{2}^{\ast} )w+x_{2}}{2}\right) w=2\varphi^{\prime}(g_{2}^{\ast}), $$

so that condition (26) does nothold. Since $u^{\prime }(d_{1}^{\ast })$ increases as x ₁ decreases, there exists at most one $\widehat {x}_{1}$ defined by $\widehat {x}_{1}=x_{2}-g_{2}^{\ast }w_{2}$ (yielding $d_{1}^{\ast }=d_{2}^{\ast }$) with $g_{2}^{\ast }=\widetilde {g}_{2}(0)$ and $g_{1}^{\ast }=0$ for which Eq. 26 holds as equality. When $x_{1}<\widehat {x}_{1}$, there exist then a corner solution (with only type 2 contributing). And since $\widetilde {g}_{2}(g_{1})$ is decreasing, it is plain that there cannot also be an interior equilibrium (which would require $d_{1}^{\ast }=d_{2}^{\ast }$). When $x_{1}>\widehat {x}_{1}$, condition (26) is violated and the equilibrium can only be interior. Observe that for $x_{1}=\widehat {x}_{1}$, we have $g_{1}^{\ast }=0$ and $g_{2}^{\ast }>0$ but these levels also satisfy the conditions for an interior solution (the constraint that g ₁≥0 hold with equality but is not binding). This is where the “transition” between corner and interior solution occurs.

To complete the proof, it remains to show that an interior equilibrium is unique. Observe that the slopes of the reaction functions are (in absolute values) smaller than one. Substituting Eq. 17 into Eq. 18 and differentiating yields

$$\frac{\mathrm{d}g_{i}}{\mathrm{d}g_{\text{-}i}}=-\frac{2\varphi^{\prime \prime} (G)}{u^{\prime \prime}(d_{i})\frac{w_{i}}{2}+2\varphi^{\prime \prime}(G)} \in(-1,0). $$

This means that the best reply map is a contraction which immediately implies uniqueness (see Vives 2001, pages 47–48).

1.2 Proof of Proposition 3

To determine the optimal transfers, (T ₁,T ₂) (the ones that implement the utilitarian Pareto efficient solution), we have to revisit the different stages of the game. In stage 2, parents leave a bequest to their children. This bequest is chosen so as to equalize consumption between the parent and the child,

$$m_{i}=d_{i}=\frac{(\tau-a_{i}-g_{i})w+h(a_{i})+x_{i}+T_{i}}{2}\quad \forall\,i. $$

Note that as long as bequests are interior, it is irrelevant whether the lump-sum transfer is paid by the children or by the parent.²³ With T _i set so that T ₁ = −T ₂≡T, if follows from Eqs. 19 and 20 that the best-response functions of spouses 1 and 2 are implicitly defined by

$$\begin{array}{@{}rcl@{}} u^{\prime}\left( \frac{(\tau-a_{1}^{\ast}-g_{1}^{\ast})w_{1}+h(a_{1}^{\ast })+x_{1}+T}{2}\right) w_{1} & =&2\varphi^{\prime}(G^{\ast}), \end{array} $$

(27)

$$\begin{array}{@{}rcl@{}} u^{\prime}\left( \frac{(\tau-a_{2}^{\ast}-g_{2}^{\ast})w_{2}+h(a_{2}^{\ast })+x_{2}-T}{2}\right) w_{2} & =&2\varphi^{\prime}(G^{\ast}). \end{array} $$

(28)

The transfer must be chosen such that an interior solution for both $g_{1}^{\ast }$ and $g_{2}^{\ast }$ is guaranteed. At an interior solution, we have $d_{1}^{\ast }=d_{2}^{\ast }$, implying

$$\frac{(\tau-a_{1}^{\ast}-g_{1}^{\ast})w_{1}+h(a_{1}^{\ast})+x_{1}+T}{2} =\frac{(\tau-a_{2}^{\ast}-g_{2}^{\ast})w_{2}+h(a_{2}^{\ast})+x_{2}-T}{2}. $$

Since w ₁ = w ₂≡w, we have $a_{1}^{\ast }=a_{2}^{\ast }$ and the above equation reduces to

$$x_{1}+T-g_{1}^{\ast}w=x_{2}-T-g_{2}^{\ast}w. $$

At an interior solution, $(g_{1}^{\ast },g_{2}^{\ast })\in (0,1)\times (0,1)$, the overall public good production, $g_{1}^{\ast }+g_{2}^{\ast }$, is uniquely determined by G ^e. That is, we can write

$$T=\frac{x_{2}-x_{1}+(2g_{1}^{\ast}-G^{e})w}{2}. $$

Since $g_{1}^{\ast }\in (0,G^{e}),$ the optimal transfer is in the interval as stated in Proposition 2.

1.3 Proof of Proposition 4

The transfer across families must be chosen such that $d_{1}^{\ast } =d_{2}^{\ast }$, then from Eqs. 19 and 20, it can be seen that only the spouse with the lower-wage rate (spouse i) contributes to the family public good implying $g_{i}^{\ast }\equiv G^{e}$ and $g_{j}^{\ast }=0$ (i,j = 1,2; i≠j). The transfer T must thus be chosen such that

$$\frac{(\tau-a_{1}^{\ast}-g_{1}^{\ast})w_{1}+h(a_{1}^{\ast})+x_{1}+T}{2} =\frac{(\tau-a_{2}^{\ast}-g_{2}^{\ast})w_{2}+h(a_{2}^{\ast})+x_{2}-T}{2}. $$

Solving for T yields expression (25) in Proposition 4.