## 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).

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## Notes

An excellent overview of this literature is given by Laferrère and Wolff (2006).

For example, Hirshleifer (1977) was the first to point out that the efficient solution is implemented only when the parents have the last word, that is, when they move last. In a similar vein, Bruce and Waldman (1990) show that in a model of intertemporal consumption decisions, the rotten kid mechanism leads to the so-called Samaritan’s dilemma in that the child does not save enough.

For instance, in a recent paper, Cremer and Roeder (2013) show that when there are several goods, including family aid (and long-term care services in general), the outcome is likely to be inefficient. Still, the rotten kid mechanism is at work and ensures that a positive level of aid is provided as long as the bequest motive is operative.

Efficiency is, however, only guaranteed if the solution to the kids problem is interior, that is, if all children make contributions to the family public good. Chiappori and Werning (2002) provide examples when this is or is not the case.

A notable exception is Cornes et al. (2012) who consider two families and different scenarios of contributors to a (general) public good. They focus on the neutrality result by Warr (1983) and show that it continues to hold in their setting. This result says that lump-sum redistributions between participants in a Nash game of private provision of a public good are allocatively neutral when all participants make positive contributions and have the same productivity in producing the public good.

And recall that this comes on top of the efficiency property which is at odds with conventional results and in particular with Bergstrom et al. (1986).

Our result is also related to Caplan et al. (2000). These authors consider a federation where regional governments contribute to the public good. This setting is somewhat similar to a rotten kids model. The federation plays a role similar to the altruistic parents, while the regions replace the rotten kids. But it also introduces new specific features like labor mobility. The major difference to our approach lies once again in the interfamily aspect which has no direct counterpart in the (single) federation setting.

Recall that Bernheim and Bagwell’s point was to take dynastic models

*ad absurdum*by showing that the interdynasty links would have policy implications that are obviously unrealistic.Provided that parent’s wealth differences are not too large.

See also Bernheim et al. (1985), who present econometric and other evidence that strongly suggests that bequests are often used as compensation for services rendered by beneficiaries.

In other words, we assume descending altruism and truncate the analysis after the parent’s generation.

As long as it occurs sufficiently late to be consistent with the timing of our model. Put differently, its timing is such that it can represent a “payment” for informal care.

Throughout the paper, we assume that the time constraint

*a*_{ i }+*g*_{ i }≤*τ*will be never binding.The level of

*G*will (in general) vary across Pareto-efficient allocations.For equal wages (

*w*_{1}=*w*_{2}),*G*^{e}is determined by$$-w_{i}u^{\prime}\left( \frac{x_{1}+x_{2}+2w_{i}(\tau-{a_{i}^{e}})+2h(a_{i}^{e})-w_{i}G^{e}}{4}\right) +2\varphi^{\prime}(G^{e})=0. $$Differentiating yields

$$\frac{\mathrm{d}G^{e}}{\mathrm{d}x_{i}} =\frac{\frac{w_{i}}{4}u^{\prime \prime}({d_{i}^{e}})}{\frac{{w_{i}^{2}}}{4}u^{\prime \prime}({d_{i}^{e}})+2\varphi^{\prime \prime}(G^{e})}=\frac{1}{w_{i}+\frac{8\varphi^{\prime \prime}(G^{e} )}{w_{i}u^{\prime \prime}({d_{i}^{e}})}}>0. $$Recall that bequests are restricted to be nonnegative, and one obtains from Eq. 12

$$b_{i} >0\quad \Longleftrightarrow \quad x_{i}+h(a_{i})>w_{i}(\tau -a_{i}-g_{i})\quad \forall \,i. $$In words, the net resources of the parents (including the monetary value of informal aid, if any) must be larger than that of the children otherwise the bequest motive is not operative.

Strategy spaces are compact sets and each player’s utility is continuous and quasi-concave in his own strategic variable.

So that the social benefit is exactly twice the individual benefit.

To see this, observe that the benchmark level of

*x*_{1}, \(\widehat {x}_{1}\equiv x_{2}-\widetilde {g}_{2}(0)w_{2}\) is decreasing in \(\widetilde {g}_{2}(0)\), which will be, the larger is*φ*(*G*).This follows because the term \(\partial b_{i}^{\ast }/\partial g_{i}\) appears in Eq. 16. In words, the adjustment in bequests is formally equivalent to a subsidy on contributions which is well known to enhance provision (recall that individual contributions are strategic substitutes).

Wealth is of course not the only conceivable source of heterogeneity across parents. They could also differ in their degree of altruism. This would be a departure from the rotten kid hypothesis, which relies on perfect altruism, and the full efficiency and equalization results would no longer obtain. Still, as long as the parent is altruistic (even though not perfectly), the rotten kid mechanism is at work and inter- and intra-generational transfers are higher (closer to the first-best) than without an operative bequest motive.

This requires some additional technical conditions, but since our best-reponse functions are “well-behaved,” it is plain that the continuity applies in our setting.

With operative bequests, Ricardian equivalence holds for the transfers.

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## Acknowledgments

Financial support from the Chaire “Marché des risques et creation de valeur” of the FdR/SCOR is gratefully acknowledged. We thank two referees and the editor, Sandro Cigno, for their insightful and constructive comments.

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## Appendix

### 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

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

so that condition (26) does *not*hold. 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

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,

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.^{Footnote 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

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

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

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

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

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

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Cremer, H., Roeder, K. Rotten spouses, family transfers, and public goods.
*J Popul Econ* **30**, 141–161 (2017). https://doi.org/10.1007/s00148-015-0564-x

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DOI: https://doi.org/10.1007/s00148-015-0564-x