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# optimal distribution

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1. Suppose there are only two people, Simon and Charity, who mush split a fixed income of 4100. For Simon, the marginal utility of income is MUs=400-2Is. While for Charity, marginal utility is MUc=400-6Ic, where Ic, Is are the amounts of income to Charity and Simon, respectively.
a). What is the optimal distribution of income if the social welfare function is additive?
b). What is the optimal distribution if society values only the utility of Charity? What if the reverse is true? Comment on your answer.
c). Finally, comment on how your answers change if the marginal utility of income for both Simon and Charity is constant:
MUc=400
MUs=400

2. Consider Eleanor, who qualifies for the Earned Income Tax Credit as depicted in the graph attached. Suppose that Eleanor can earn \$8 per hour. Taking into account EITC and ignoring other aspects of the tax and transfer system:
a). How much do her earnings increase when her labor supply increases from zero to 1,000 hours per year?
b). How much do her earnings increase when her labor supply increases from 1,000 to 1,500 hours per year?
c). How much do her earnings increase when her labor supply increases from 1,500 to 2,000 hours per year?
In each case, compute the incremental amount of earnings associated with the increase in work effort. Relate your answer to the implicit marginal tax rate embodied in the EITC.

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#### Solution Preview

1. Suppose there are only two people, Simon and Charity, who mush split a fixed income of 4100. For Simon, the marginal utility of income is MUs=400-2Is. While for Charity, marginal utility is MUc=400-6Ic, where Ic, Is are the amounts of income to Charity and Simon, respectively.
a). What is the optimal distribution of income if the social welfare function is additive?
If the social welfare is to be

When W = Ic + Is = 4100
We need to consider the goal to maximize the total utility of both persons.
The optimal income level will make MUs = MUc, i.e.,
400-2Is = 400-6Ic
or
Is = 3Ic
Substitute into W = Ic + Is = 4100
Ic + 3Ic = 4100
4Ic = 4100
Ic = 1025
Is = 3 * 1025 = 3075

However, one problem with this result is that their MUs = MUc = 400 - 2*3075 < 0, which means the additional income will ...

#### Solution Summary

The optimal income level is illustrated.

\$2.19