Problem Set on Unidimensional Inequality and Poverty Measures

Transcription

1 Problem Set on Unidimensional Inequality and Poverty Measures A) Paper-Based Problems: 1. Compare the following pairs of distributions: X=(1,5,7,2,14) Y=(2,3,2,8,2) M=(2,4,8,6) S=(2,5,8,5) a) Graph the cdf, the Lorenz curves and the GL curves. Is there an FSD or SSD relationship between them? b) For the second pair of distributions, calculate the range, the Coefficient of Variation and the Theil 2 Index (Mean Log Deviation). How do these measures evaluate the two distributions? What problems do you find with the first measure? d) Suppose now two additional distributions obtained from M: L=(7,7,4,2) and P=(3,3,6,8). Calculate the Coefficient of Variation and the Theil 2 Index for these distribution and compare the three. What can you say in terms of the sensitivity to transfers in each of these measures? e) Suppose another distribution T=(3,6,4,7). Calculate the Gini Coefficient for M, L and T, What can you say in terms of the sensitivity to transfers in each of these measure? 2. For these two pairs of distributions: a) x=(2,8) y=(4.10) b) x=(2,4) Y=(8,10) a) Calculate the Gini coefficient: G(x), G(y) and G(x,y). Verify the decomposition formula in each case. What is the value of R in the second pair of distributions? Why? b) Graph the Lorenz Curve for the overall distribution (2,8,4,10), and verify the value of the Gini, calculating it graphically. 3. Given distribution X=(2,3,4,11) and assume z=10 calculate: IGR, FGT0, FGT1, FGT2, the Watt s Index, the CHU indices (with Beta=0.5) and the decomposable version of the CHU Indices (with Beta=-1). a) If the distribution is changed to X =(2,3,3,11), which measures will show an increase in poverty and why? Verify it. 1

2 b) If the distribution is changed to X =(3,3,3,11), which measures will show a decrease in poverty and why? Verify it. c) How much is the contribution of the subgroup of the two people to overall decomposable CHU measure? d) Suppose that there were a fund of money that could be allocated to reduce poverty. How would you do it if the poverty measure were: (i) the headcount ratio, (ii) the income gap ratio (iii)the FGT with alfa=2? B) Computer-Based Problems (Using Stata): First make sure to count with enough memory to work with the data: set memory 256m Open the data set distributed to you, called Half_Sample_Bhutan.dta B.1) Exercises on inequality 1. Which is the mean income? The income variable is pce_real which is the per capita expenditure in real terms. Do it without weights and with weights. (The weights in this survey are the type of importance weights. You need to tell Stata which type of weights your are using). Is there a significative difference between the two? 2. Repeat (1) but distinguishing between urban and rural areas. (Ignore the weights in this case) (You will need to use the option by, and sort the data first by area ): a) Which areas have the higher mean? b) Calculate the range for each area. According to this measure of inequality, which area is more unequal? Would you feel confident about this assertion? What problem does the range have as an inequality measure? c) Which area has higher variance in the income distribution? d) If we used the variance as a measure of inequality, which area would you say that it has higher inequality? Would you feel confident about this assertion? What problem does the variance have as an inequality measure? e) Which measure related to the variance- could be used that is Lorenz consistent? With this measure, which area is more unequal? 2

3 3. Using the command glcurve of the Inequal package, do the following: a) Obtain the Generalized Lorenz (GL) curve for the whole sample. (Use the options pvar and glvar to be able to see the variables that are generated to construct the curve). b) Obtain the GL for rural and urban areas. (You will need to use the by(area) split option). c) What is the value of each of these curves when p=1? Is this coincident with what you analysed in Excercises 1 and 2? (check the content of your glvar variable). d) According to the GL curves, which area has higher inequality? Why can you say this? Is this consistent with your answer in 2d)? Why? What can you conclude in terms of stochastic dominance (cite the corresponding theorem)? e) Using the lorenz option in the glcurve, you can get the Lorenz curves. Do this for rural and urban areas. Comparing this graph with the one of the GL, what can you say about the additional information that the GL curves provide? 4. Obtain the Lorenz and GL curves for the districts of Haa (dcode=15) and Trashiyangste (dcode=26). What can you say about inequality in these two districts? Why? Can you conclude that one of the districts SSD the other? Why? What type of ranking does the Lorenz criterion provide? (Lorenz and GL curves can also be obtained with DASP. You can find the command for this clicking in user/dasp/ curves/ Lorenz concentration: Curves. For more details, check the DASP manual. As an exercise you can try to replicate the previous results with DASP). 5. Using the command ainequal from the Inequal package obtain the following inequality measures for the whole sample and also distinguishing urban and rural areas: Relative Mean Deviation Coefficient of Variation Standard Deviation of Logarithms Gini Coefficient Generalized Entropy Measures for =-1,1,0,2 3

4 Atkinson s Measures for =-1,-0.5,0 (Note that in the command they request you to give the value of, where =1-. Therefore you need to provide the values: =2,1,1.5 correspondngly.) (Type all at the end of the command so that it reports all the measures). a) Which of these measures are Lorenz Consistent and which are not? Which specific axioms do they fail to satisfy? b) How do the Lorenz-consistent measures rank rural and urban areas? Why was this expected (cite the theorem)? 6. Obtain the same measures than in Exercise 5, but for the districts of Haa (dcode=15) and Trashiyangste (dcode=26). How does it measure rank the two districts? Do the rankings coincide? Should they coincide according to the Lorenz criterion? 7. a) With the information you obtained in Excercises 1 and 5, verify the decomposability formula of the Theil 1 Measure, Theil2 Measure, the Squared Coefficient of Variation between rural and urban areas. Recall the weights in each case. How much of total inequality measured by the Theil 2 Index corresponds to inequality within each area and how much corresponds to inequality between areas? b) Verify the decomposition formula for the Gini Coefficient. To simplify calculations, try to write a short code in a dofile to obtain the Gini Between. What is the value of the residual term R? What does this value reflect? C.2) Exercises on poverty 1. Using the poverty command, and considering that the poverty line is Nu , obtain the following poverty measures for rural and urban areas: Headcount Ratio (h) Income Gap Ratio (igr) Poverty Gap Ratio (FGT1) (pgr) FGT2 (fgt3) Watts Index (w) Sen Index (s) CHU with Beta=0.25, 0.50, 0.90 (chu2, chu3, chu5) 4

5 a) According to each measure, where is poverty higher? b) If you had not access to the poverty command, how would you obtain the headcount ratio? (write it in a dofile) You can repeat this exercise for the FGT1 and FGT2. c) Repeat the exercise for the Headcount, the Income Gap Ratio, the Poverty Gap Ratio and the FGT2 (h, igr, pgr and fgt3) for the districts of Haa (dcode=15) and Trashiyangste (dcode=26) and compare: How does each measure compare these two districts? How does the judgement changes as one moves from considering only the incidence, to including the depth and distribution? What is the judgement of the Income Gap Ratio? What problem does this measure have? 2. Using the results from Exercise 1, verify the decomposability of the FGT2 between rural and urban areas. 3. Using the results from Exercise 1 from the whole sample, verify the expressions of the Sen and FGT2 indices when n tends o infinity. 5