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 27 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?