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

In conclusion, the Rotten Kid Theorem shows that, under certain conditions, a self-interested child will make decisions that also maximize the sum of the two players' incomes. This is due to the fact that the child's optimal choice of \(r\) leads to a situation where \(Y_1'(r)=-Y_2'(r)\), thus maximizing the sum \(Y_1+Y_2\). The proof of this theorem is achieved through backward induction in a sequential game with the parent and the child, first analyzing the parent's problem, then applying the implicit function rule to find the derivative of the optimal bequest \(L^*\) with respect to \(r\), and finally proving that the child chooses \(r\) to maximize \(Y_1+Y_2\).

Step-by-step solution

Analyze the parent's problem

The parent's utility function is given by: \(U_2(Y_2-L)+\alpha U_1(Y_1+L)\). To find the best response of the parent, we need to find the optimal value of \(L\). To do this, we will differentiate the parent's utility function with respect to \(L\) and set it equal to zero. \(U_2'(Y_2-L)\cdot(-1)+\alpha U_1'(Y_1+L)=0\)

Apply the implicit function rule

Although we cannot find an explicit expression for \(L^*\), we can find the derivative of \(L^*\) with respect to \(r\) using the implicit function rule. To do this, we need to differentiate both sides of the equation found in step 1 with respect to \(r\): \((-U_2''(Y_2-L)\cdot(-Y_2'(r))+\alpha U_1''(Y_1+L)\cdot Y_1'(r))\frac{dL^*}{dr}=0\) From the equation above, we can solve for \(\frac{dL^*}{dr}\): \(\frac{dL^*}{dr}=\frac{U_2''(Y_2-L)\cdot(-Y_2'(r))}{\alpha U_1''(Y_1+L)\cdot Y_1'(r)}\)

Prove the child chooses \(r\) to maximize \(Y_1+Y_2\)

In the first stage of the game, the child chooses \(r\) to maximize their utility function which is \(U_1(Y_1+L^*)\). Differentiating the child's utility function with respect to \(r\) and setting it equal to zero yields: \(U_1'(Y_1+L^*)\cdot(Y_1'(r)+\frac{dL^*}{dr})=0\) Since \(U_1'(Y_1+L^*)>0\) (given in the problem statement), we need \(Y_1'(r)+\frac{dL^*}{dr}=0\) to satisfy the above equation. Substituting the expression of \(\frac{dL^*}{dr}\) that we found in step 2 and solving for \(Y_1'(r)\), we have: \(Y_1'(r)+\frac{U_2''(Y_2-L)\cdot(-Y_2'(r))}{\alpha U_1''(Y_1+L)\cdot Y_1'(r)}=0\) This simplifies to: \(Y_1'(r)=-\frac{U_2''(Y_2-L)\cdot(-Y_2'(r))}{\alpha U_1''(Y_1+L)}\) Rearranging the equation and multiplying both sides by \(-\alpha U_1''(Y_1+L)\), we find: \(Y_1'(r)\cdot \alpha U_1''(Y_1+L) = Y_2'(r)\cdot U_2''(Y_2-L)\) Since \(\alpha > 0\), \(U_1''(Y_1+L)>0\) and \(U_2''(Y_2-L)>0\), we have: \(Y_1'(r)=-Y_2'(r)\) Hence, the child chooses the value of \(r\) that maximizes the sum \(Y_1+Y_2\), despite having no altruistic intentions. The Rotten Kid Theorem is proved.

Key concepts

Subgame-Perfect Equilibrium

Understanding subgame-perfect equilibrium is essential in the analysis of games involving sequential moves, such as the scenario described in the Rotten Kid Theorem. This equilibrium concept is applied within the framework of game theory, which helps in deciphering the optimal strategies for players involved in a sequential game.

Subgame-perfect equilibrium is a refinement of the Nash equilibrium, designed specifically for sequential games. It requires that players' strategies not only be in Nash equilibrium overall, but also that they constitute a Nash equilibrium for every subgame of the original game. This implies that the players' strategies are optimal considering the remaining part of the game at every stage, regardless of how the game has progressed up to that point.

The rationale behind its importance in sequential games is that it rules out non-credible threats and promises. A player will not make a move that is irrational for them in a future stage, even if it might lead to a better outcome in the overall game. It ensures that a strategy is consistent across all possible decision points within a game.

Backward Induction

Backward induction stands as a fundamental problem-solving method in game theory, especially for games with perfect information. It is utilized extensively to find subgame-perfect equilibria in sequential games, like the scenario between the 'rotten kid' and their altruistic parent.

To apply backward induction is to reason backward, from the end of the problem to the beginning. In the Rotten Kid Theorem, we start by considering the last move of the game, the parent's choice of the monetary bequest, and what decision will maximize their utility. Then, assumptions made about the parent's best strategy can be used to deduce the child's optimal action in the first step of the game.

Improving students' understanding of backward induction could involve graphical representations of the game, repeatedly illustrating the concept with varied examples, or even engaging in role-playing to simulate sequential decision-making. These methods could solidify the concept as students work through problems.

Implicit Function Rule

The implicit function rule, a concept in calculus, plays a significant role in economic theory, particularly when dealing directly with optimization problems where explicit solutions are hard to find or infeasible.

When variables are related through an equation without an explicit solution, the implicit function rule allows us to find the derivative of one variable with respect to another. This is crucial in the Rotten Kid Theorem scenario, where we can't explicitly solve for the parental bequest, but we certainly can determine how the bequest changes as the kid's behavior varies (dL*/dr). This differentiation is central to determining the child's optimal action in the first stage of the game.

To make the implicit function rule more accessible, one could illustrate the concept using various examples that involve the relationship between different variables, allowing students to practice the mechanics of the rule and apply it to diverse functions. Visualization tools can also be employed to show the impact of changes in one variable on another in a graphical manner, thereby enhancing the students' understanding.