# Problem set 5 of Welfare economics

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(xi), where U(xi)=ln(xi). 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 x1,x2, x1+x2=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.