Problem set 4 of Welfare economics

Universitat Pompeu Fabra, 1996/1997

Due June 4, 1997.

1. Consider the list of incomes: 10,10,20,20,20,30,40,50.

1.1 What is the mean income? What is the Schutz coefficient of inequality of this society?
1.2 Draw the Lorenz curve that corresponds to this list, and draw the Schutz coefficient as well.
1.3 Find the discrete density function that corresponds the above list of incomes.

2. Let p:R+→[0,1] be a discrete density function that has positive values at points in S, S={x1,...,xk}, and 0≤x1<...<xk.

2.1 Define the values of the Lorenz curve L(p(x1)), L(p(x1)+p(x2)) and so on, i.e., L(∑i≤rp(xi)) for all 1≤r≤k.
2.2 Draw the Lorenz curve that corresponds to the density function p(10)=0.4, p(20)=0.6, and p(x)=0 for all x≠10, x≠20

3

3.1 Write a formula for the Gini coefficient of a Lorenz curve generated from a discrete density function. (Note that the triangulars do not sum up "nicely").
3.2 Compute the Gini coefficient of the Lorenz curve that corresponds to the density function that is mentioned in 2.2.
3.3 Compute the Gini coefficient of the Lorenz curve that corresponds to the density function that you found in 1.3.