Universitat Pompeu Fabra, 1996/1997

Due June 11, 1997.

**1.** Consider the lists of incomes:
x=10,10,20,20,20,30,40,50; and y=15,15,15,20,20,25,50,290.

**1.1**Does the Lorenz curve of one list of incomes dominates the other?**1.2**Does the Generalized Lorenz curve of one list dominates the other?**1.3**Assume a planner has the social welfare function W(x) = ∑U(x_{i}), where U(x_{i})=ln(x_{i}). Which of the above lists of incomes gives a higher welfare level?

**2.** Assume the government taxes individuals by a
progressive tax schedule t; and taxes married couples by the function
r(x)=2t(x/2), where x is the joint income of the couple. Show that a
couple with joint income x pays no more than two individuals with
incomes x_{1},x_{2}, x_{1}+x_{2}=x.

**3.** Assume that a planner can assign to each individual
either a positive tax or a negative one (transfer). Further assume that he
is restricted to collect no more than B by taxes, and has to pay as
transfers all the amount that he collected. What is the optimal
tax-transfer rule if he wants to minimize the Gini coefficient of the
after tax-transfer income list? Apply the rule you found to the list x
in Question 1, and B=20 and B=30.