Distribution of Resources Whether a resource is distributed

Distribution of Resources Whether a resource is distributed evenly among member, of a population is often an important political or economic question How can we measure this? How can we decide if the distribution of wealth in this country is becoming more or less equitable over time? How can we measure which country has the most equitable income distribution? This problem describes a way of making such measurements. Suppose the resource is distributed evenly. Then any 20% of the population will have 20% of the resource. Similarly, any 30% will have 30% of the resource and so on. If. however, the resource is not distributed evenly, the poorest pft of the population (in terms of this resource) will nut have of p% goods. Suppose F(x) represents the fraction of the resources owned by the poorest fraction i of the population. Thus F(0.4) = 0.1 means that the poorest 40% of the population owns 10% ol the resource. What would F be if the resource were distributed evenly? What must be true of any such F ? What must F(0) and F(1) equal? Is F increasing or decreasing? Is the graph of F concave up or concave down? Gini\'s index of inequality, G. is one way to measure how evenly the resource is distributed It is defined by G = 2 integrate 1 0 [x- F(x)] dx. Show graphically what G represents.

Solution

a) If the resource were distributed evenly then F(0.4)=0.4

i.e the poorest 40% of the population would have 40% of the total population

b) F is increasing with increse in x but after reaching 1 it is constant.

F is concave up

F(0)=0

F(1)=1

c)G here represents how the values are distributed