Chapter 2: Income Standards, Inequality, and Poverty
h. Use the cdf to calculate the area to the left of the cdf bounded by x = 0 and F(x) = 1. What do you get? i. Calculate the median, the 95th percentile, and the 20th percentile using the cdf that you drew in 1c. 2. The Gini coefficient is probably the most commonly used index of relative inequality. What are some of the advantages and disadvantages of this measure? 3. The variance of logarithm (VL) is an inequality measure that is computed as VL (x) =
1 N ∑[ln x n − WL (x)]2, N n =1
where WL(x) is the mean of the logarithm of elements in x as defined in the chapter. a. Verify that the variance of logarithms satisfies scale invariance. What property of the variance of logarithms ensures scale invariance? b. Graph the Lorenz curves for the two distributions x = (1,1,1,1,41) and y = (1,1,1,21,21). Can the curves be ranked? c. Find the variance of logarithms of the two distributions. What is wrong here? d. Find the mean log deviation (the second Theil measure) of the two distributions. What is correct here? 4. Construct an inequality measure that violates replication invariance. 5. Are the following statements true, false, or uncertain? In each case, support your answer with a brief but precise explanation. a. The Kuznets ratios satisfy the Pigou-Dalton transfer principle. b. Distribution y = (1,2,3,2,41) is more unequal than distribution x = (1,8,4,1,36) in terms of the Lorenz criterion. c. The four basic properties of inequality measurement are enough to compare any two income distributions in terms of relative inequality. d. If everyone’s income increases by a constant dollar amount, inequality must fall. 6. Consider the distribution x = (1,3,6). a. Draw the Lorenz curve, and calculate the area between the 45-degree line and the curve. b. Calculate the Gini coefficient for x. What is the relationship between the Gini coefficient and the calculated area?
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