ACE 501 Fall, 2008

Comments on Gibbons problem 2.1 — The Rotten Kid Theorem

Carl H. Nelson 9/15/08

1 The Rotten Kid TheoremThe Rotten Kid Theorem that you proved in problem set 2 is an important in-sight in family economics and incentive theory that was first stated by Becker[?]. Stated in words, if a family has a head who “cares sufficiently about allother members to transfer general resources to them, then redistribution of in-come among members [of the household] would not affect the consumption ofany member, as long as the head continues to contribute to all.” That is, if sucha head exists other family members are motivated to make choices that maximizefamily 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 studycontracting. Contracting theory is a large literature because it is hard to createincentives for selfish agents to take actions that benefit the group, especially whenthe agents have unobserved actions or attributes. The Rotten Kid theorem saysthat 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 familyof such a head will have no free riders or principal-agent problems. This is an im-portant insight in family economics. It is an insight that provides understanding ofthe distinction between a family and the general economic organizations analyzedin incentive theory. But, it is not a general theorem that holds for all specificationsof utility.

A more formal analysis of Becker’s insight was provided by Bergstrom [?].I will give you a summary of some of his analysis. A general statement of thetheorem is the following. Given a family of n members, where each members’consumption of a single consumption good, X , is Xi. Let the head’s utility be:

U(X1, X2, . . . , Xn)

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Let Ii be the income of family member i before transfers occur. Then the familybudget constraint is: ∑

i

Xi =∑

i

Ii

Then if the head is making transfers to family members the distribution of con-sumption is the one that maximizes the utility of the head subject to the budgetconstraint. Given that each of the Xi’s is a normal good for the head, each Xi

will be an increasing function of income. Thus any child who has the ability toincrease total family income will do so, even if it means reducing his own pre-transfer income.

Bergstrom begins by providing an example that demonstrates that the RottenKid theorem fails to apply when there is asymmetric information. He calls thisexample the “Lazy rotten kid”. Let the income of each family member be anincreasing function of their time and effort working, Yi, Ii = I(Yi). Let eachchilds’ utility depend on consumption, Xi, and let leisure be measured by 1− Yi.Suppose that the head can observe income, but not time and effort working. Thisbreaks the incentives created by the transfer from the head. Then a selfish childwill not choose effort to maximize family income because because he does notreceive the marginal product of effort, but the marginal product times the head’smarginal propensity to spend on him. But, it turns out the the Rotten Kid theoremdoes not hold, even if the head can observe the effort of selfish children. Assumethere are two children with utility functions:

Ui = Xi(1− Yi)

And assume the head’s utility is:

U0 = U1/21 + U

1/22 = X

1/21 (1− Y1)

1/2 + X1/22 (1− Y2)

1/2

Assume each child earns income Ii = wYi (Note, this means that the head knowsYi if he observes Ii.) The head will make transfers to maximize utility subject tothe constraint:

X1 + X2 = I0 + wY1 + wY2

The maximum occurs when:X1

X2

=(1− Y1)

(1− Y2)

Note, that this means that each child’s share of consumption is a decreasingfunction of his own work effort. Thus each child’s incentive to work is pointed inthe direction opposite the direction needed to maximize family income.

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Consider another variation of the Rotten Kid theorem that Becker insightfullycomments on. There is a husband who wants to read at night, but the light bothershis wife. The husband loves his wife and gives her consumption goods. He isaware that the light bothers his wife, and his love for her causes him to read lessthan he would otherwise. But he still uses the night-light more than his wife wouldlike. One day when the husband is absent, an electrician offers to disconnect thenight-light in such a way that the husband would think it was an accident andwould not be able to fix it. Becker claims that even though the wife is selfish anddislikes the night-light, she will not take up the electricians offer. His reasoningis 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 forthe husband, this loss of income will cause the husband to reduce his gifts ofconsumption goods, so the wife’s utility would be lower after the night-light isdisconnected.

Bergstrom subjects this provocative explanation of the incentive effects of al-truism to formal analysis. Let the altruistic husband’s utility function be

Uh(uh(xh, y), uw(xw, y))

where uh(xh, y) is increasing in both arguments, and uw(xw, y) is increasingin xw and decreasing y. Let Uh(uh, uw) = uhu

aw, where 0 < a < 1, and

uh(wh, y) = xh(1 + y) and uw(xw, y) = xwe−y. Thus, the husband choosesxh, xw, y to maximize:

Uh = xhxaw(1 + y)e−ay

subject to:xh + xw = I

The solution reveals that the husband will distribute income so that xw = aI/(1+a). So income distribution is independent of night-light, y. Thus the husband willchoose y = (1/a)− 1 > 0. But the wife would choose y = 0 if she could becauseher utility is decreasing in y. Thus the wife would take the electrician’s offer. Thisis a case where attaining a Pareto optimal allocation of the public good, y, requiresexplicit bargaining over the allocation of y.

What is the common problem in these examples that causes the incentives ofthe Rotten Kid theorem to break down? It is the existence of the second commod-ity. You can see this by considering the utility possibility frontier in the secondexample. We can solve for this by making xh and xw explicit functions of uh anduw and substituting the solutions into the budget constraint to obtain:

uh

1 + y+ uwey = I

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We can see that for a given value of y this is linear in uh and uw. Changes in ywill change the location and slope of the utility possibility frontier. This frontierwill be most favorable to the wife when y = 0. The same is true of the lazy rottenkid. He can commit himself to an action before the head chooses an incentivescheme. The kid will choose inefficient actions that distort the utility possibilityfrontier in his favor.

Bergstrom proves that there must be conditional transferable utility in orderfor the subgame perfect equilibrium to be Pareto optimal. He further proves thatin order for this to happen every agent must have a utility function of the form:

ui(mi, a) = A(a)mi + Bi(a)

This is a form of utility that allows utility to be transferred through mi. So thetechnical Rotten Kid theorem is much more restrictive than Becker’s provocativeclaims. But Bergstrom acknowledges that the theorem is “at least as useful asa heuristic generator of insight as it is as a formal proposition...The richness ofinsight that Becker has to offer about the economics of the family is only partiallycaptured in formal analysis of the Rotten Kid theorems. Many insights into theeconomics of the family and the economics of incentives that have escaped formaltreatment in this paper are to be found in Becker’s discussion and examples in theTreatise on the Family and in his 1974 article.”

After Bergstrom’s analysis, Cornes and Silva [?] claimed to prove that theutility restriction identified by Bergstrom could be relaxed if all children providea pure public good. Formally, they claim that if all children provide a pure publicgood, such that their first order conditions have interior solutions, then the first-order conditions will always satisfy Pareto optimality. Unfortunately, Chiapporiand Werning [?] showed that interior solutions can be reached in a very limitedset of cases, one of which is the frequently used case of pure symmetry.

References

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