Exercise 22.34. Consider an economy consisting of three groups, a fraction p poor agents each with income

Exercise 22.34. Consider an economy consisting of three groups, a fraction θp poor agents each with income yp, a fraction θm middle-class agents with income yr > ym, and the remaining fraction θr = 1 − θp − θm rich agents with income yr > yp. Suppose that both θp and θr are less than 1/2, so that the individual with the median income (the “median voter”) is a middle-class individual. (1) Construct a change in incomes that leaves mean income in the society unchanged and increases the gap between the mean and the median but does not constitute a mean preserving spread of the distribution. (2) Construct a mean preserving spread of the distribution such that the gap between the mean and the median narrows. [Hint: increase ym and reduce yp, holding yr constant].