Transfers within a three generations family: when the rotten kids turn into altruistic parents

We study exchanges between three overlapping generations with
non-dynastic altruism. The middleaged choose informal care provided
to their parents and education expenditures for their children. The
young enjoy their education, while the old may leave a bequest to
their children. Within each period the three generations play a
“game” inspired by Becker’s (1974, 1991) rotten kids framework,
with the added features that the rotten kids turn into the
altruistic parent in the next period and that parents invest in the
education of their children. We show that Becker’s rotten kids
theorem holds for the single period game in that informal aid is
set according to an efficient rule. However, education is distorted
upwards. In the stationary equilibrium the levels of both transfers
are inefficient: education is too large and informal aid is too
low.