Question

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 \(\Upsilon_{1}(r)\left[Y_{1}^{\prime}(r)>0\right]\) and the income of the parent \(\Upsilon_{2}(r)\left[\Upsilon_{2}^{\prime}(r)<0\right] .\) Later, the parent moves, leaving a monetary bequest \(L\) to the child. The child cares only for his own utility, \(U_{1}\left(\Upsilon_{1}+L\right),\) but the parent maximizes \(U_{2}\left(\Upsilon_{2}-L\right)+\alpha U_{1},\) where \(\alpha>0\) 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 \(\Upsilon_{1}+\Upsilon_{2}\) 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.

Short answer

Answer: Yes, the child will choose the value of r that maximizes the sum of their own and the parent's income in a subgame-perfect equilibrium, even if they have no altruistic intentions.

Step-by-step solution

Analyze the parent's optimization problem

By applying backward induction, we first analyze the parent's optimization problem, which is given by: \(maximize\) \(U_{2}(\Upsilon_2 - L) + \alpha U_1\) Subject to the constraint that \(L \geq 0\) , as the bequest is non-negative. By applying the first-order condition for maximization, we get the following equation that implicitly determines the optimal bequest \(L^*\) : \(\frac{\partial U_{2}}{\partial L}(\Upsilon_2 - L^*) -\alpha \frac{\partial U_{1}}{\partial L} = 0\) Explicitly solving for \(L^*\) is not possible, but we can obtain the derivative of \(L^*\) with respect to \(r\) using the implicit function rule.

Applying the implicit function rule

Using the implicit function rule, we differentiate both sides of the above equation with respect to \(r\): \(\frac{\partial^2 U_{2}}{\partial L \partial r}(\Upsilon_2 - L^*) -\alpha \frac{\partial^2 U_{1}}{\partial L \partial r} + \frac{\partial U_{2}}{\partial L}(\frac{\partial L^*}{\partial r}) - (\frac{\partial \Upsilon_2}{\partial r}) - \frac{\partial U_{1}}{\partial L}(\frac{\partial L^*}{\partial r}) = 0\) The child's optimization problem is to maximize \(U_{1}(\Upsilon_{1}+L^*)\), and by applying the first-order condition, we get: \(\frac{\partial U_{1}}{\partial(\Upsilon_1 + L^*)}(\frac{\partial r}{\partial(\Upsilon_1 + L^*)}) = 0\) Now, we want to solve for the value of \(r\) that maximizes \(U_{1}(\Upsilon_{1}+L^*)\). To do this, insert the first-order derivative of \(L^*\) obtained in Step 2 into the child's optimization problem. The new first-order condition for the child's optimization problem will be: \(\frac{\partial U_{1}}{\partial(\Upsilon_1 + L^*)}[\frac{\partial \Upsilon_1}{\partial r} + \frac{\partial L^*}{\partial r}] = 0\)

Proving the theorem

According to the Rotten Kid Theorem, the child will choose a value of \(r\) that maximizes both his and his parent's income. Therefore, we want to show that: \(\frac{\partial U_{1}}{\partial(\Upsilon_1 + L^*)}[\frac{\partial \Upsilon_1}{\partial r} + \frac{\partial \Upsilon_2}{\partial r}] = 0\) Comparing this equation to the first-order condition obtained in Step 2 for the child's optimization problem, we see that: \(\frac{\partial L^*}{\partial r} = \frac{\partial \Upsilon_2}{\partial r}\) Thus, if the child maximizes both his and his parent's income, his own income will also be maximized. Since the child is acting rationally, we can conclude that in a subgame-perfect equilibrium, the child will choose the value of r that maximizes the sum of their own and the parent's income, despite having no altruistic intentions.

Key concepts

Sequential Game Theory

Sequential game theory explores strategic decision-making situations where players move one after the other. In these games, the order of moves is crucial since the decisions of one player can affect the choices of the other player and vice versa. In the context of the Rotten Kid Theorem, the sequence starts with the child selecting an action that affects incomes. After observing this action, the parent decides on the monetary bequest.

This setup contrasts with simultaneous games, where players make decisions without knowing the opponent's choice. Sequential games are best represented using decision trees, where nodes represent players' decision points and branches represent possible actions. These trees help visualize the consequences of each player's choice and illustrate how one player's decision influences subsequent actions in the game.

Here are some key aspects of sequential games:

• Player Moves: Decisions are made in a specific order, with earlier moves being visible to later-moving players.
• Information Set: Later players choose actions with knowledge of previous moves, which significantly impacts their strategy.
• Payoffs: Each player aims to maximize their payoff, considering both their own actions and the anticipated response of others.

Backward Induction

Backward induction is a method used to solve sequential games by working from the end of the game towards the beginning. In other words, players consider the last move first, then deduce backwards to determine the optimal actions at each preceding decision point.

In the Rotten Kid Theorem, backward induction is applied to the parent's bequest decision problem. The parent, knowing their altruism towards the child would lead them to adjust the bequest, optimizes their decision considering the child's earlier choice. This inductive reasoning helps in identifying the subgame-perfect equilibrium, ensuring that each player's strategy is optimal given the others' strategies.

The backward induction steps in this context can be outlined as follows:

• Start at the final decision point – the parent's choice of bequest.
• Determine the best action, which maximizes their objective, given the strategies of all preceding players.
• Consider the child's decision next and how it impacts the parent's subsequent response.
• Repeat this process until arriving at the initial decision point, ensuring optimal strategies at every step.

By using backward induction, players can assure each action aligns with maximizing their utility through all points in the game.

Subgame-Perfect Equilibrium

Subgame-perfect equilibrium is a refinement of Nash equilibrium applied to sequential games. This concept ensures that players' strategies form a Nash equilibrium in every subgame of the original game, effectively addressing the problem of credible threats.

In the context of the Rotten Kid Theorem, subgame-perfect equilibrium involves analyzing the game tree and deciding the optimal choice at each node, ensuring no player can benefit by changing their strategy at any stage. This requires that the strategy chosen is the best response, not only in the initial game but in any game after any possible player's move.

Here's how subgame-perfect equilibrium works:

• Break Down the Game: Analyze every possible point in the game where a player must make a decision, called a subgame.
• Optimal Strategy in Subgames: Determine the best strategy for each player in every subgame, ensuring it leads to a Nash equilibrium.
• Credible Strategies: Only strategies that are credible at every point in the game tree are considered part of the equilibrium.

This concept guarantees the child's choice that maximizes the sum of incomes is part of a strategy that remains optimal throughout, even without altruistic considerations.

Altruism in Economics

Altruism in economics refers to the behavior that reflects concern for the welfare of others, impacting the utility of economic agents and influencing their decisions. In the Rotten Kid Theorem, a parent's altruism is quantified by the parameter \( \alpha \). The parent values not only their utility but also the child's, leading to a decision that considers the child's welfare.

This illustrates how altruistic behaviors can shape economic outcomes, influencing decision-making in situations such as inheritance, public goods provision, and family economics. In this theorem, even though the child is self-interested, the parent's altruism alters the strategic setup and affects the resulting equilibrium.

Key points on altruism in economics include:

• Motive for Decisions: Altruism affects decisions by adding another layer of utility based on others' welfare.
• Economic Dynamics: Can lead to cooperative outcomes in situations traditionally seen as competitive.
• Behavioral Influence: Alters traditional models based on self-interested utility maximization by introducing shared welfare considerations.

Thus, even a self-interested child may act in a manner beneficial to the parent when the parent incorporates the child's utility into their decision-making process.