Questions by ggorczany - Page 34
Rotten Kid Theorem In A Treatise on the Family (Cambridge, MA: Harvard Uni- versity Press, 1981), Nobel laureate Gary Becker proposes his famous Rotten Kid Theorem as a sequential game between the potentially rotten child (player 1) and the child's parent (player 2). The child moves first, choosing an action r that affects both his own income Y (r) and the income of his par- ent Y (r), where Y' (r) > 0 and Y(r) < 0. Later, the parent moves, leaving a monetary bequest L to the child. The child cares only for his own utility, U (Y + L), but the parent max- imizes U(Y L) + U, where a > 0 reflects the parent's altruism toward the child. Prove that, in a subgame-perfect equilibrium, the child will opt for the value of r that maxi- mizes Y + Y even though he has no altruistic intentions. Hint: Apply backward induction to the parent's problem first, which will give a first-order condition that implicitly deter- mines L*; although an explicit solution for L* cannot be found, the derivative of L* with respect to r-required in the child's first-stage optimization problem-can be found using the implicit function rule.