A Unified Approach to Measuring Poverty and Inequality
increased because of redistribution. However, distribution x' may be seen as being obtained from distribution x" by merely increasing everyone’s income by the same proportion with balanced growth. As a result, the headcount ratio falls from three-fourths to one-fourth. Hence, the improvement in poverty in this case has resulted from growth rather than redistribution.22
Exercises 1. Consider the following table that enables you to construct a cumulative distribution function (cdf) from income data. Category Income (i) ($ xi) 1 2 3 4 5
12,000 13,000 14,000 15,000 16,000
Number of people (ni)
pi
F(xi)
(pi ĂŤ xi)
10 15 40 20 15
There are five income categories (Xi) in the economy. Each category contains a certain number of people (ni). a. What is the total number of people (n) in the economy? b. Let pi denote the proportion of people in each category. Fill in the column corresponding to pi for each i. The probability mass function is defined as a function that gives the probability of a discrete variable taking the same value. Now draw the probability mass function. Hint: Draw a diagram with x on the horizontal axis and p on the vertical axis. c. Let F(xi) denote the proportion of people who have an income no higher than xi. Fill in the column corresponding to F(xi) for each i. Now draw the cdf. Hint: Draw a diagram with x on the horizontal axis and F(x) on the vertical axis. d. What is the relationship between pi and F(xi)? e. Calculate the proportion of people having an income less than $14,100. What is the proportion of people having an income more than $14,900? f. What is the average income for the economy? g. Fill in the last column, and find the sum of all cells in that column. What does the sum give you?
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