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4)). However, only a stronger father's altruism can secure a distribution which is at once con¯ictfree and more favorable. 8) are replaced by VF …CF † ˆ CF …1X7H † VS …CS † ˆ CS …1X8H † and for any 0 `  ` 1. 12) follows through as before, except that the optimal consumption ratios now appear as 2 3 ~F C 1 À F ˆ ~ F CS …1X10H † F and 2 3 ~F C ~S C S ˆ S Y 1 À S 21 …1X11H † Altruism and beyond ~F ˆ C1À and C ~S ˆ C1À . 1, except that for CF CS . ~F C ~S C substitutes Two remarks are in order.

Our model expands the human capital framework to incorporate the case where the prolongation of life expectancy of cohort t induces more human capital formation by cohort t ‡ 1 because of the resulting postponement of the transfer of familial productive assets. We prove 41 Altruism and beyond that, as a consequence, per capita income in the economy with the longer-lived generation t is higher. The model expands the human capital framework in yet another way: the acquisition of human capital serves not only as a means of enhancing productivity but also as a form of insurance.

Altruism and beyond From the ®rst order condition we thus obtain   CF 1 À F ˆ Y CS F F …1X10† where the subscript F indicates that this is the optimal consumption ratio arising from the father's optimization. 11) it follows that     CF CF 1 À F S b D b D F ‡ S ` 1Y …1X12† CS F CS S F 1 À S since the right-hand side inequality indeed holds, we conclude that the father's optimal allocation is such that he wishes to consume a larger proportion of corn than his son wishes him to consume.

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