1. In this problem we consider a partial equilibrium model of the household sector. The

price of the consumption good is equal to 1, whereas the wage rate is equal to w.

(a) First consider a single household, who derives utility from consumption c and disutility

from labor according to the welfare function

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where y is a positive parameter. There is no other source of income than supplying labor.

Convince yourself that the budget constraint is given by

c≤ wl.

What is the optimal consumption bundle (c, l) for this household?

(b) Now assume that the population consists of a unit interval of households such as

the one from part (a) but with different preference parameters 7. More specifically, two

thirds of the households have y = 1/2 whereas the rest (one third) has y = 2. If all

households maximize their welfare, what is the Gini coefficient of the resulting labor

income distribution?

(c) Finally, assume that the government collects a proportional labor income tax at rate

7 from the households and redistributes the total tax revenue in lump-sum form back to

the households, whereby every household gets the same transfer T. Convince yourself

that the budget constraint of the households is now given by

c≤ (1-7)wl + T.

Assuming again that all households maximize their welfare, what is the relation between

7 and T and how does the Gini coefficient of the net income distribution (i.e., the distri-

bution of incomes after taxes and transfers) depend on 7?