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Rotten spouses, family transfers, and public goods

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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

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

  2. 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.

  3. 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.

  4. 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.

  5. 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.

  6. 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).

  7. 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.

  8. 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.

  9. Provided that parent’s wealth differences are not too large.

  10. 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.

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

  12. 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.

  13. Throughout the paper, we assume that the time constraint a i +g i τ will be never binding.

  14. The level of G will (in general) vary across Pareto-efficient allocations.

  15. 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

    $$\frac{\mathrm{d}G^{e}}{\mathrm{d}x_{i}} =\frac{\frac{w_{i}}{4}u^{\prime \prime}({d_{i}^{e}})}{\frac{{w_{i}^{2}}}{4}u^{\prime \prime}({d_{i}^{e}})+2\varphi^{\prime \prime}(G^{e})}=\frac{1}{w_{i}+\frac{8\varphi^{\prime \prime}(G^{e} )}{w_{i}u^{\prime \prime}({d_{i}^{e}})}}>0. $$
  16. 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.

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

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

  19. 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).

  20. 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).

  21. 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.

  22. 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.

  23. With operative bequests, Ricardian equivalence holds for the transfers.

References

  • Becker GS (1974) A theory of social interactions. J Polit Econ 82(6):1063–1093

    Article  Google Scholar 

  • Becker GS (1991) A treatise on the family. Harvard University Press, Cambridge

    Google Scholar 

  • Bergstrom TC, Blume L, Varian H (1986) On the private provision of public goods. J Public Econ 29:25–49

    Article  Google Scholar 

  • Bergstrom TC (1989) A fresh look at the rotten kids theorem–and other household mysteries. J Polit Econ 97(5):1138–1159

    Article  Google Scholar 

  • Bernheim BD, Bagwell K (1988) Is everything neutral? J Polit Econ:308–338

  • Bernheim BD, Shleifer A, Summers LH (1985) The strategic bequest motive. J Polit Econ 93(6):1045–1076

    Article  Google Scholar 

  • Bruce N, Waldman M (1990) The rotten-kid theorem meets the Samaritan’s dilemma. Q J Econ 105:155–165

    Article  Google Scholar 

  • Chiappori P-A, Werning I (2002) Comment on ‘Rotten Kids, Purity, and Perfection’. J Polit Econ 110(2):475–480

    Article  Google Scholar 

  • Caplan AJ, Cornes RC, Silva ECD (2000) Pure public goods and income redistribution in a federation with decentralized leadership and imperfect labor mobility. J Public Econ 77:265–284

    Article  Google Scholar 

  • Cigno A (2014) Conflict and cooperation within the family, and between the state and the family, in the provision of old-age security, CHILD WP No 22

  • Cornes RC, Itaya J, Tanaka A (2012) Private provision of public goods between families. J Popul Econ 24:1451–1480

    Article  Google Scholar 

  • Cornes RC, Silva ECD (1999) Rotten kids, purity, and perfection. J Polit Econ 107(5):1034–1040

    Article  Google Scholar 

  • Cremer H, Pestieau P, Ponthière G (2012) The economics of long-term care: a survey. Nordic Economic Policy Review 2:107–148

    Google Scholar 

  • Cremer H, Roeder K (2013) Long-term care and lazy rotten kids, IZA Discussion Paper No 7565

  • Greenwood J, Guner N, Kocharkov G, Santos C (2014) Marry your like: assortative mating and income inequality, IZA Discussion Paper, No 7895

  • Hirshleifer J (1977) Shakespeare vs. Becker on altruism: the importance of having the last word. J Econ Lit 15(2):500–502

    Google Scholar 

  • Laferrère A, Wolff F-C (2006) Microeconomic models of family transfers. In: Kolm S-C, Ythier JM (eds) Handbook of the economics of giving, altruism and reciprocity, vol 1. Elsevier, Amsterdam, pp 879–969

    Google Scholar 

  • Norton EC (2000) Long-term care. In: Culyer AJ, Newhouse JP (eds) Handbook of health economics, vol 1B. Elsevier, Amsterdam, pp 956–94

    Google Scholar 

  • Norton E, Nicholas L, Huang SS-H (2014) Informal care and inter-vivos transfers: results from the National Longitudinal Survey of Mature Women. The B.E. Journal of Economics Analysis & Policy 14(2):377–400

    Google Scholar 

  • Schwartz CR (2010) Earnings inequality and the changing association between spouses’ earnings. Am J Sociol 115(5):1524–1557

    Article  Google Scholar 

  • Schwartz CR, Mare RD (2005) Trends in educational assortative marriage from 1940 to 2003. Demography 42(4):621–646

    Article  Google Scholar 

  • Torche F (2010) Educational assortative mating and economic inequality: a comparative analysis of three Latin American countries. Demography 47(2):481–502

    Article  Google Scholar 

  • Vives X (2001) Oligopoly pricing: old ideas and new tools. MIT Press

  • Warr PG (1983) The private provision of a public good is independent of the distribution of income. Econ Lett 13(2):207–211

    Article  Google Scholar 

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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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Correspondence to Helmuth Cremer.

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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 1x 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.Footnote 23 With T i set so that T 1 = −T 2T, 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 2w, 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; ij). 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.

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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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