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Chapter 6 What is Economic Inequality? Measurement of Inequality Anonymity, Population, Relative Income, and Dalton Principles The Lorenz Curve Complete Measures: Coefficient of Variation and the Gini Coefficient Economic inequality refers to the distribution of an economic attribute, such as income or wealth, across citizens within a country or across countries themselves. For example, how is the total income in a country distributed across its citizens? What proportion of total wealth is held by the richest? the poorest? Economists study inequality for intrinsic reasons (reducing inequality can be seen as an objective in itself) functional reasons (inequality may affect other indicators of economic performance, such as growth). The first step in understanding economic inequality is to know how to measure it. Suppose there are n individuals in a society, indexed by i = 1,2,3,…,n An income distribution describes how much income yi is received by each individual i: y1 , y2 ,...., yn We are interested in comparing “relative inequality” between two such distributions (over time, or between regions/countries, etc.) 1. The Anonymity Principle Names do not matter, incomes can always be ranked without reference to who is earning it y1 y2 y3 ,..., yn 2. The Population Principle As long as the composition of income classes remain unchanged, changing the size of the population does not matter for inequality What matters are the proportions of the population that earn different levels of income The Relative Income Principle 3. Only relative income matters, and not levels of absolute income Scaling everyone’s income by the same percentage should not affect inequality The Dalton Principle 4. If a transfer is made from a relatively poor to a relatively rich individual, inequality must increase “Regressive” transfers (taking from poor and giving to the rich) must worsen inequality An inequality index is a function of the form I I ( y1 , y2 ,..., yn ) A higher value of this measure I(.) indicates greater inequality The Anonymity Principle: the function I(.) is insensitive to all permutations of the income distribution y1 , y2 ,...., yn among the individuals 1,2,..., n. The Population Principle: For every distribution y1 , y2 ,...., yn , I y1 , y2 ,...., yn I y1 , y2 ,...., yn ; y1 , y2 ,...., yn “cloning” has no effect on inequality The Relative Income Principle: For every positive number , I y1 , y2 ,...., yn I y1 , y2 ,...., yn The Dalton Principle: The function I(.) satisfies the Dalton Principle, if, for every distribution y1 , y2 ,..., yn and every transfer 0, I ( y1 , y2 ,..., yn ) I ( y1 ,..., yi ,..., y j ,..., yn ) wherever yi y j The Lorenz curve illustrates how cumulative shares of income are earned by cumulatively increasing fractions of the population, arranged from the poorest to the richest. A graphical method for measuring inequality If everyone has the same income, then the Lorenz curve is the 450 line The slope of the Lorenz curve is the contribution of the person at that point to the cumulative share of national income The “distance” between the 450 line and the Lorenz curve indicates the amount of inequality in the society The greater is inequality, the further will the Lorenz curve be from the 450 line The previous graph gives us a measure of inequality called the Lorenz Criterion An inequality measure I is Lorenz-consistent if, for every pair of income distributions ( y1 , y2 ,..., yn ) and z1 , z2 ,..., zm , I y1 , y2 ,..., yn I z1 , z2 ,..., zm whenever the Lorenz curve of ( y1 , y2 ,..., yn ) lies to the right of z1 , z2 ,..., zm Can we summarize inequality by a number? Attractive for policymakers and researchers When Lorenz curves cross, we cannot rank inequality across two distributions A numerical measure of inequality helps rank distributions unambiguously Let there be m distinct incomes, divided into j classes In each income class j, the number of individuals earning that income is n j The total population is then given by n m nj j 1 The mean or average of the distribution is given by 1m nj yj n j 1 1. Range 2. Kuznets Ratio 3. Mean Absolute Deviation 4. Coefficient of Variation 5. Gini Coefficient Difference in the incomes of the richest and the poorest individuals, divided by the mean R 1 ym y1 Very crude measure of inequality Does not consider people between the richest and poorest on the income scale Fails to satisfy the Dalton Principle (why?) The ratio of the share of income of the richest x % to the poorest y % where x and y represent population shares Example: share of income of the richest 10% relative to the poorest 60% These ratios are basically “snapshots” of the Lorenz curve Useful when detailed inequality data in not available The sum of all income distances from average income, expressed as a fraction of total income 1 m nj y j M n j 1 The idea: inequality is proportional to distance from mean income May not satisfy the Dalton Principle, if regressive transfers are made between income classes that are all above or below the mean Essentially the standard deviation(sum of squared deviations from the mean), divided by the mean 1 C nj 2 y j j 1 n m Gives greater weight to larger deviations from the mean Lorenz-consistent (satisfies the four principles) Sum of the absolute differences between all pairs of incomes, normalized by (squared) population and mean income 1 mm G 2 n j nk y j yk 2n j 1k 1 Takes the difference between all pairs of income and sums the absolute differences Inequality is the sum of all pair-wise comparisons of two-person inequalities Double summation: first sum over all k’s, holding each j constant. Then, sum over all the j’s. Most commonly used measure of inequality Satisfies all four principles: Lorenz-consistent