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.

Key concepts

Altruism in Economics

Altruism plays a fascinating role in economics, especially when considering family dynamics. Traditionally, altruism refers to the concern for the well-being of others, even at a cost to oneself. In an economic context, this is expressed through models that try to capture how individuals might prioritize, support, or benefit others to optimize overall welfare.
Becker's Rotten Kid Theorem explores this notion by considering how a parent's altruism influences a child's behavior. The theorem postulates that even if a child acts selfishly, a parent's altruism can lead to outcomes that maximize the welfare of both parties. This is because the parent's actions indirectly motivate the child to make decisions that enhance the family's total income.

• The parent considers the child's utility in their own utility maximization, leading to intergenerational transfers, like bequests.
• Altruistic behavior ensures that the child's self-interested actions align with maximizing total family welfare.

Through this lens, altruism can transform potentially selfish actions into outcomes that serve the greater good, showcasing its integral role in economic theory.

Game Theory

Game theory provides a robust framework for analyzing strategic interactions between rational decision-makers. It involves the study of mathematical models to understand how individuals or entities make decisions, influencing each other's outcomes.
In the context of the Rotten Kid Theorem, game theory is employed to understand the strategic decisions made by a child and a parent when resources and utilities are involved. The game model includes two main players - a child and a parent - both seeking to maximize their respective utilities.

• The child makes the first move, determining actions that maximize personal income, directly affecting both own and parental income.
• The parent responds by deciding on a bequest, factoring in their altruistic sentiment towards the child's welfare.

This strategic setup illustrates a classic game theory model, where both players anticipate each other's actions and possible reactions. By considering these strategic choices, game theory helps predict the equilibrium where both parties' utilities are maximized despite conflicting interests.

Utility Maximization

Utility maximization is a fundamental concept in economics, focusing on how individuals make choices to achieve the greatest satisfaction possible. Each decision-maker aims to allocate resources to gain the highest possible personal utility.
In the Rotten Kid Theorem's scenario, both the child and the parent aim to maximize utility, albeit in different ways. The child focuses solely on maximizing personal income and utility, while the parent considers the overall well-being, including the child's utility.

• The child's utility depends on their own income and any bequest received, so they select an action that boosts their earnings.
• Meanwhile, the parent balances their utility derived from personal income retention against the altruistic value of the child's utility through bequests.

Even though their approaches differ, both participants pursue their respective utility objectives. The parent's altruistic consideration ensures that the child's utility maximization inadvertently aligns with maximizing the sum of family income, exemplifying how utility-driven decisions can harmonize within economic systems.