In A Treatise on the Family (Cambridge, MA: Harvard University 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 his own income !1ðrÞ½!01ðrÞ > 0+ and the income of the parent !2ðrÞ½!02ðrÞ < 0+. Later, the parent moves, leaving a monetary bequest L to the child. The child cares only for his own utility, U1ð!1 þ LÞ, but the parent maximizes U2ð!2 ! LÞ þ aU1, 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 maximizes !1 þ !2 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 determines 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.