A Unified Approach to Measuring Poverty and Inequality
7. Consider the distribution x = (3,6,9,12,24,36). a. Divide the distribution into the following two subgroups: x1 = (3,6,9) and x2 = (12,24,36). Calculate the Gini coefficient for x, x1, and x2. Using the traditional additive decomposability formula, check if the Gini coefficient is decomposable in this situation. b. Divide distribution x into the following two subgroups: x3 = (3,24,36) and x4 = (6,9,12). Again, using the traditional additive decomposability formula, check if the Gini coefficient is decomposable in this situation. c. What is the difference between these two circumstances? Explain. d. What is the residual for the Gini coefficient in these two circumstances? 8. For the two distributions x = (2,100; 700; 1,100; 200) and y = (3,410; 620; 2,170; 6,510), do the following: a. Calculate the WGM(.; a) and use it to calculate the Atkinson measure IA(.; a) for a = 0, –1. b. Do you have the same IA(.; a) for both distributions or not? What is going on here? 9. For the income distributions x = (3,3,5,7) and y = (2,4,6,6), do the following: a. Calculate the generalized entropy measure and IGE(x; a) and IGE(y; a) for a = 1,0,1,2,3,4. b. Plot the values of a on the horizontal axis and the values of IGE(x; a) and IGE(x; a) on the vertical axis. c. Join the points, and check if they intersect. If they intersect, then report at what value of a they intersect, and explain why. 10. Are the following statements true, false, or uncertain? a. The second Theil measure is subgroup consistent. b. The arithmetic mean is higher than the harmonic mean but less than the geometric mean. c. The sum of the decomposition weights of the generalized entropy measure is always less than 1. 11. How is the generalized Lorenz curve GL(p) derived from a cdf? Draw this process and explain. What value does the generalized Lorenz curve take at p = 1? 12. Suppose an inequality measure is given by I(x) = (x¯—e(x))/x¯, where e(x) is one of the equally distributed equivalent income functions used by Atkinson (namely, a general mean with a parameter less than one).
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