Appendix
population. The curve graphs the area under the quantile function up to each percentile of population, or the height of the Lorenz curve times the mean per capita expenditure. Thus, the height of the generalized Lorenz curve is equal to the mean consumption expenditure when the percentile is one. In other words, the share of the total consumption expenditure spent by the entire population is 100 percent. The curve with the solid line represents the generalized Lorenz curve for urban Georgia in 2003. The generalized Lorenz curve with the dotted line corresponds to urban Georgia in 2006. If a generalized Lorenz curve lies completely above another generalized Lorenz curve, then every lower partial mean of the former distribution is larger than the corresponding lower partial mean of the latter distribution, and the former distribution has a larger Sen mean than the latter distribution. Also, when one generalized Lorenz curve lies above another, the distribution corresponding to the former generalized Lorenz curve is said to second-order stochastically dominate the distribution corresponding to the latter. In this particular example, the distribution of per capita expenditure in 2003 secondorder stochastically dominates the distribution of per capita expenditure in 2006.
General Mean Curve Figure A.3 graphs the general mean curve of urban Georgia’s per capita expenditure for two years. The vertical axis reports per capita expenditure, and the horizontal axis reports parameter α, also known as a society’s degree of aversion toward inequality. A general mean curve plots the value of general means of a distribution corresponding to parameter α. The general mean of a distribution tends toward the maximum and the minimum per capita expenditures in the distribution when α tends to ∞ and – ∞, respectively. Given that the largest per capita expenditure in any distribution is usually several times larger than the minimum per capita expenditure, allowing α to be very large would prevent any meaningful graphic analysis. So we restrict α = 1 to be between –5 and 5, which we consider large enough. The height of the curve at α = 1 denotes the arithmetic mean. Similarly, the heights at α = 0, α = –1, and α = 2 denote the geometric mean, the harmonic mean, and the Euclidean mean, respectively.
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