Question

In A Treatise on the Family (Cambridge; Harvard University Press, 1981 ), G. Becker proposes his famous Rotten Kid theorem as a game between a (potentially rotten) child, \(A\), and his or her parent, \(B . A\) moves first and chooses an action, \(r,\) that affects his or her own income \(Y_{A}(r)\) \(\left(Y_{A}^{\prime}>0\right)\) and the income of the parent \(Y_{s}(r)\left(Y_{B}^{\prime}<0\right) .\) In the second stage of the game, the parent leaves a monetary bequest of \(L\) to the child. The child cares only for his or her own utility, \(U_{A}\left(Y_{A}+L\right),\) but the parent maximizes \(U_{B}\left(Y_{B}-L\right)+X U_{A},\) where \(A>0\) reflects the parent's altruism toward the child. Prove that the child will opt for that value of \(r\) that maximizes \(Y_{A}+Y_{B}\) even though he or she has no altruistic intentions. Plint: You must first find the parent's optimal bequest, then solve for the child's optimal strategy, given this subsequent parental behavior.)

Short answer

In conclusion, the Rotten Kid theorem states that a selfish child will choose a strategy that maximizes the sum of the parent's and their own income in an optimal bequest scenario. By finding the parent's optimal bequest and the child's optimal strategy that maximizes their utility, we proved the theorem. The child's optimal strategy involves choosing a value for r that satisfies \(Y_A'(r) + Y_B'(r) = 0\), meaning that the child will maximize the sum of incomes, \(Y_A + Y_B\), even without altruistic motives.

Step-by-step solution

Find the parent's optimal bequest

To find the parent's optimal bequest, we need to take the first-order derivative of their utility function with respect to L, and set it equal to zero. This will help us determine the optimal value of L that maximizes the parent's utility. The parent's utility function is given by: \(U_B(Y_B - L) + XU_A\) Taking the derivative with respect to L gives: \(\frac{dU_B}{dL} = -U_B'(Y_B - L) + XU_A'(Y_A + L)\) Now, we need to find the value of L that sets this derivative to zero: \(-U_B'(Y_B - L) + XU_A'(Y_A + L) = 0\) Rearranging to solve for L: \(L = \frac{U_B'(Y_B - L)}{XU_A'(Y_A + L)}\)

Find the child's optimal strategy

Now, we need to find the child's optimal strategy, which means determining the value of r that maximizes their utility. The child's utility function is given by: \(U_A(Y_A + L)\) We already found the optimal bequest, L, in step 1. Plugging the value of L from step 1 into the child's utility function gives: \(U_A \left(Y_A + \frac{U_B'(Y_B - L)}{XU_A'(Y_A + L)}\right)\) Now we need to find the optimal r by taking the derivative of this function with respect to r and setting it equal to zero: \(\frac{dU_A}{dr} = Y_A'(r) + \frac{U_B''(Y_B - L)Y_B'(r) - XU_A''(Y_A + L)Y_A'(r)}{[XU_A'(Y_A + L)]^2} = 0\) This is a complicated expression, but we can notice that the denominator will not affect the sign of the derivative, so we can ignore it for the maximization problem. We can rearrange the remaining terms to solve for the optimal r: \(Y_A'(r) = XU_A''(Y_A + L)Y_A'(r) - U_B''(Y_B - L)Y_B'(r)\) According to the problem statement, we have \(Y_A'(r)>0\) and \(Y_B'(r)<0\). The equation above implies that the optimal r must have \(Y_A'(r) + Y_B'(r) = 0\), which means that the child's optimal strategy is choosing r to maximize the sum of incomes, \(Y_A + Y_B\), even though they have no altruistic intentions.