The Rotten Kid Theorem and Almost Transferable Utility

We assume like Bergstrom (1989) and Dijkstra (2007) that each child’s utility is treated as a normal good in the altruistic head’s utility function, and show that if utility functions lead to Almost Transferable Utility children can manipulate the tradeoff between their own utility and the parent’s utility through their own actions, but they have an incentive to maximize the altruistic head’s utility if the altruistic head also considers children’s utilities as Hicksian substitutes and hence the rotten kid theorem holds. A special class of such altruistic utility functions that treat utilities of children as normal and Hicksian substitutes are the Generalized Utilitarian Welfare functions.