1 The Rotten Kid Theorem

Transcription

1 ACE 501 Fall, 2007 Comments on Gibbons problem 2.1 The Rotten Kid Theorem Carl H. Nelson 9/7/07 1 The Rotten Kid Theorem The Rotten Kid Theorem that you proved in problem set 2 is an important insight in family economics and incentive theory that was first stated by Becker [1]. Stated in words, if a family has a head who cares sufficiently about all other members to transfer general resources to them, then redistribution of income among members [of the household] would not affect the consumption of any member, as long as the head continues to contribute to all. That is, if such a head exists other family members are motivated to make choices that maximize family income, even if their welfare depends on their own consumption alone. This is what you proved in problem set 2. This is an amazing result, which should become much clearer when we study contracting. Contracting theory is a large literature because it is hard to create incentives for selfish agents to take actions that benefit the group, especially when the agents have unobserved actions or attributes. The Rotten Kid theorem says that selfish kids will take actions that benefit the family if a benevolent head exists. This doesn t even require any form of pre-commitment by the head. The family of such a head will have no free riders or principal-agent problems. This is an important insight in family economics. It is an insight that provides understanding of the distinction between a family and the general economic organizations analyzed in incentive theory. But, it is not a general theorem that holds for all specifications of utility. A more formal analysis of Becker s insight was provided by Bergstrom [2]. I will give you a summary of some of his analysis. A general statement of the theorem is the following. Given a family of n members, where each members consumption of a single consumption good, X, is X i. Let the head s utility be: U(X 1, X 2,..., X n ) 1

2 Let I i be the income of family member i before transfers occur. Then the family budget constraint is: X i = I i i i Then if the head is making transfers to family members the distribution of consumption is the one that maximizes the utility of the head subject to the budget constraint. Given that each of the X i s is a normal good for the head, each X i will be an increasing function of income. Thus any child who has the ability to increase total family income will do so, even if it means reducing his own pretransfer income. Bergstrom begins by providing an example that demonstrates that the Rotten Kid theorem fails to apply when there is asymmetric information. He calls this example the Lazy rotten kid. Let the income of each family member be an increasing function of their time and effort working, Y i, I i = I(Y i ). Let each childs utility depend on consumption, X i, and let leisure be measured by 1 Y i. Suppose that the head can observe income, but not time and effort working. This breaks the incentives created by the transfer from the head. Then a selfish child will not choose effort to maximize family income because because he does not receive the marginal product of effort, but the marginal product times the head s marginal propensity to spend on him. But, it turns out the the Rotten Kid theorem does not hold, even if the head can observe the effort of selfish children. Assume there are two children with utility functions: And assume the head s utility is: U i = X i (1 Y i ) U 0 = U 1/2 1 + U 1/2 2 = X 1/2 1 (1 Y 1 ) 1/2 + X 1/2 2 (1 Y 2 ) 1/2 Assume each child earns income I i = wy i (Note, this means that the head knows Y i if he observes I i.) The head will make transfers to maximize utility subject to the constraint: X 1 + X 2 = I 0 + wy 1 + wy 2 The maximum occurs when: X 1 = (1 Y 1) X 2 (1 Y 2 ) Note, that this means that each child s share of consumption is a decreasing function of his own work effort. Thus each child s incentive to work is pointed in the direction opposite the direction needed to maximize family income. 2

3 Consider another variation of the Rotten Kid theorem that Becker insightfully comments on. There is a husband who wants to read at night, but the light bothers his wife. The husband loves his wife and gives her consumption goods. He is aware that the light bothers his wife, and his love for her causes him to read less than he would otherwise. But he still uses the night-light more than his wife would like. One day when the husband is absent, an electrician offers to disconnect the night-light in such a way that the husband would think it was an accident and would not be able to fix it. Becker claims that even though the wife is selfish and dislikes the night-light, she will not take up the electricians offer. His reasoning is that the husband will be worse off if the electrician disconnects the night-light. This effect is a loss of family income. If the wife s utility is a normal good for the husband, this loss of income will cause the husband to reduce his gifts of consumption goods, so the wife s utility would be lower after the night-light is disconnected. Bergstrom subjects this provocative explanation of the incentive effects of altruism to formal analysis. Let the altruistic husband s utility function be U h (u h (x h, y), u w (x w, y)) where u h (x h, y) is increasing in both arguments, and u w (x w, y) is increasing in x w and decreasing y. Let U h (u h, u w ) = u h u a w, where 0 < a < 1, and u h (w h, y) = x h (1 + y) and u w (x w, y) = x w e y. Thus, the husband chooses x h, x w, y to maximize: U h = x h x a w(1 + y)e ay subject to: x h + x w = I The solution reveals that the husband will distribute income so that x w = ai/(1 + a). So income distribution is independent of night-light, y. Thus the husband will choose y = (1/a) 1 > 0. But the wife would choose y = 0 if she could because her utility is decreasing in y. Thus the wife would take the electrician s offer. This is a case where attaining a Pareto optimal allocation of the public good, y, requires explicit bargaining over the allocation of y. What is the common problem in these examples that causes the incentives of the Rotten Kid theorem to break down? It is the existence of the second commodity. You can see this by considering the utility possibility frontier in the second example. We can solve for this by making x h and x w explicit functions of u h and u w and substituting the solutions into the budget constraint to obtain: u h 1 + y + u we y = I 3

4 We can see that for a given value of y this is linear in u h and u w. Changes in y will change the location and slope of the utility possibility frontier. This frontier will be most favorable to the wife when y = 0. The same is true of the lazy rotten kid. He can commit himself to an action before the head chooses an incentive scheme. The kid will choose inefficient actions that distort the utility possibility frontier in his favor. Bergstrom proves that there must be conditional transferable utility in order for the subgame perfect equilibrium to be Pareto optimal. He further proves that in order for this to happen every agent must have a utility function of the form: u i (m i, a) = A(a)m i + B i (a) This is a form of utility that allows utility to be transferred through m i. So the technical Rotten Kid theorem is much more restrictive than Becker s provocative claims. But Bergstrom acknowledges that the theorem is at least as useful as a heuristic generator of insight as it is as a formal proposition...the richness of insight that Becker has to offer about the economics of the family is only partially captured in formal analysis of the Rotten Kid theorems. Many insights into the economics of the family and the economics of incentives that have escaped formal treatment in this paper are to be found in Becker s discussion and examples in the Treatise on the Family and in his 1974 article. After Bergstrom s analysis, Cornes and Silva [4] claimed to prove that the utility restriction identified by Bergstrom could be relaxed if all children provide a pure public good. Formally, they claim that if all children provide a pure public good, such that their first order conditions have interior solutions, then the firstorder conditions will always satisfy Pareto optimality. Unfortunately, Chiappori and Werning [3] showed that interior solutions can be reached in a very limited set of cases, one of which is the frequently used case of pure symmetry. References [1] Gary S. Becker. A theory of social interactions. Journal of Political Economy, 82(6): , [2] Theodore C. Bergstrom. A fresh look at the rotten kid theorem and other household mysteries. Journal of Political Economy, 97(5): , [3] Pierre-Andre Chiappori and Ivan Werning. Rotten kids, purity, and perfection: Comment. Journal of Political Economy, 110(2):475 80,

5 [4] Richard C. Cornes and Emilson C. D. Silva. Rotten kids, purity, and perfection. Journal of Political Economy, 107(5): ,