Problem Set #4 Answers Date Due: October 25, 2012

Transcription

1 Problem Set #4 Answers Date Due: October 5, The World Bank data library reports the following information on the distribution of income for Mexico in 008. Table 1: Mexico Distribution of Income, 008 Population Quintile Share of Income (%) First (lowest) 4.73 Second 8.65 Third Fourth Fifth (highest) (a) Using these data plot the Lorenz curve. Answer: See Figure 1 (b) Using these data calculate the Gini coefficient, G. Hint: The Lorenz curve and high school geometry are sufficient to find the Gini coefficient. Answer: The Gini Coefficient is the area between the 45 and the Lorenz Curve. There are many alternative ways to find the area. One way is to partition the area beneath the Lorenz curve is into 5 triangles labelled 1 5, and 4 rectangles, labelled 5 9 (Figure 1). Label the area of the Lorenz curve to the x axis area C. The the Gini coefficient is 1/ C (where 1/ is area beneath the 45 line. By my calculations the Gini Coefficient is cumulative Income cumulative population (c) Go to the World Bank web page. What Gini coefficient does the World Bank report for Mexico in 008? Does that equal yours? Assuming both coefficients are calculated correctly, why might your Gini be somewhat different from the World Bank s? Answer: The World Bank Reports the Gini Coefficient to be.48, slightly higher than calculated above. We have calculated the Gini from a Lorenz curve based only on the quintiles. Our Lorenz curve is a set of piecewise linear segments, whereas with a finer partition (say to percentiles) the Lorenz curve would be smooth. Thus, our representa- 1

2 tion of the Lorenz curve is an approximation, so it is not surprising that our Gini does not match exactly the Gini reported by the World Bank.. Go to the World Bank web page and retrieve Gini Coefficients for India, Kenya, Mexico, the United States, and a country of your choice. For each country, please obtain information for the most recent year, most distant year reported by the WB, and for an intermediate year. (The years will vary by country.) Pick as well an intermediate year, such as 1990 (a year when the WB s World Development Report centered on poverty and inequality). Answer: I ve selected Bangladesh as my additional country. It is among the poorest countries in the world. And is home to one of the most extensively studied family planning interventions. During the 1970s and 1980s it suffered drought and famine. Table : Gini Coefficients: Bangladesh, India, Kenya, Mexico, USA Year Bangladesh India Kenya Mexico USA (a) What is the time trend of inequality in these countries? (Is it increasing, decreasing, or stable?) Answer: For Bangladesh and the United States the Gini coefficient has increased over time. For India and Mexico the Gini has been relatively stable. Though only a few Ginis are reported for Kenya the Gini declined and then rebounded, but remains at the last report substantially lower than its initial report in 199. (b) Again, looking across time is there a pattern by the level of development? Answer: The data are scarce, but I think not: Bangladesh and the USA exhibit increasing inequality yet income per capita in the two countries are widely divergent. (c) For the US what substantive economic reasons may explain the time series change in the Gini Coefficient? Answer: Since the early 1990s the earnings distribution in the United States has widen considerably as seen by the trend in the Gini Coefficients. There are many conjectures

3 as to why, but the consensus view is one of increasing returns to skill. That is, the distribution of earnings increased because the earnings of the highest skilled members increased much more than earnings for middle and lower skilled workers. 3. For the same countries as used for the previous question, collect information on poverty measures from the World Bank data bank. Again, please retrieve values (see below) for the most recent, the most distance and an intermediate year. (a) What poverty measures are available in the World Bank s databank? Answer: The Poverty Gap at $1.5 a day (PPP), Poverty Gap at $ a day (PPP), Poverty gap at national poverty line (%), Poverty gap at rural poverty line (%), Poverty gap at urban poverty line (%), Poverty headcount ratios at $1.5 a day, $ a day, as a %of population, Poverty headcount ratio at national poverty line, Poverty headcount ratio at rural poverty line, and Poverty Headcount ratio at urban poverty line. (b) Select poverty measures you think are most informative. Explain why you selected these particular measures. Answer: The Head Count is the most easily measured and consequently the most commonly reported. However, it is susceptible to manipulation by targeting income transfers to individuals just below the poverty line. I selected the Poverty gap at $1.5 a day measured in PPP. And the poverty headcount ratio at $1.5 a day. The $1.5 per day measures are the most conservative and represent the poorest people in the country and in the world. Table 3: Poverty Measures Bangladesh India Kenya Malawi Mexico Year HC Gap HC Gap HC Gap HC Gap HC Gap (c) Briefly describe the trend in poverty across your subset of countries. What patterns appear? Be sure to mention any pattern that you think is surprising. 3

4 Answer: The general pattern across these countries is for poverty rates to decline. Kenya is the exception. (d) Is there a correlation between inequality measures and poverty measures? Overall? Within each time period across countries? Within each country over time? Answer: I didn t calculate the correlation coefficient, but any correlation is weak though may be positive (lower inequality and lower poverty over time). (e) Based on material from Chapter 5 and the Kuznets Curve, should we expect there to be a correlation between inequality and poverty measures? If so why, and is correlation positive or negative. If no correlation ly explain why not. Answer: Hard to say, as the correlation will depend on the stage of economic development. Early in economic development, inequality will increase but poverty is likely to decline. After inequality has peaked and begins to decline, poverty rates should continue to fall through continued economic growth. Hence, as the economy becomes mature I expect the correlation would be positive. 4. Table presents the distribution of income in the United States in calendar year 00 by ventile 1 Income is presented in constant dollars and income in the j th ventile is the mean income among individuals within ventile j. The population in each ventile is 14.6 million. The data are from the World Bank data file wyd for release.dta. Table 4: United States Distribution of Income, 00 Ventile Income (a) Using the information in Table calculate the Theil Index and the Coefficient of Varia- 1 A ventile is 5%. 4

5 tion, and the Gini coefficient. Another formula for the Gini coefficient is: G =1+ 1 n n µ [y 1 +y + + ny n ] (1) with y 1 y y 3... y n. Answer: The calculations are in the Excel spreadsheet e448ps4ans.xlsx. (b) Collapse the income distribution from ventiles to deciles. Calculate the Theil Index, Coefficient of Variation, and Gini Coefficient. How sensitive are the indices to the partition used to report the income distribution? Answer: Again I refer you to the Excel spreadsheet e448ps4ans.xlsx. The is little change in any of the inequality measures aggregating from ventiles to deciles. Yet, aggregating further to quintiles lowers the indices indicating less inequality. The coefficient of variation is most sensitive, at least in this exercise. A. Sen (1996) On Inequality, Expanded Edition, Oxford University Press. p