chp 6 inequality.pdf

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 distribu...

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 3. The Relative Income Principle § Only relative income matters, and not levels of absolute income § Scaling everyone’s income by the same percentage should not affect inequality 4. The Dalton Principle § 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 l, I ( y1 , y2 ,...., yn ) = I ( l y1 , l y2 ,...., l yn ) ¡ The Dalton Principle: The function I(.) satisfies the Dalton Principle, if, for every distribution ( y1 , y2 ,..., yn ) and every transfer d > 0, I ( y1 , y2 ,..., yn ) < I ( y1 ,..., yi - d ,..., y j + d ,..., 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 ) ³ 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 m n = å nj j =1 ¡ The mean or average of the distribution is given by 1m µ = å 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 1 R= ( ym - y1 ) µ § 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 M= å nj yj - µ µn 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 nj å (y j - µ ) m C= 2 µ j =1 n § 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 å å n j nk y j - yk 2n µ 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