31. Suppose that at date 1 the distribution of income is (1, 2, 3), while later, at date 2, it is found to be (2, 3, 4). Verify for yourself that for a poverty line of 2 the reduction in the headcount index of poverty is entirely due to growth in the mean, i.e., that the redistribution component is zero. Now suppose the distribution at date 2 is (1.5, 1.5, 3) (as in question 1). Is the change in poverty now due entirely to redistribution? Discuss how would your answer depends on what measure of poverty you use. 32. Suppose that the initial rural distribution of income is (1, 2, 3, 4) while the initial urban distribution is (3, 4). The poverty line is 2, so the overall poverty rate (headcount index) is 1/3. Now imagine that all of the rural poor move to urban areas and each of them gains 20% in real income. Verify that the overall poverty rate falls to 1/6, yet the urban poverty rate rises from zero to 1/4. 33. What is selection bias? What are some methods for reducing this type of bias? 34. What is meant by the "poverty-inequality" trade off? Has there been such a trade-off in the developing world? Explain 35. What are some explanations that have been given for the rising obesity rates in the United States, particularly among the poor?