Exploring Lognormal Income Distributions 11 Oct, 2014 1C 1C 2014 NNN2 1 2014 NNN2 2 Exploring Log-Normal Distributions Lognormal Incomes A Log-Normal distribution is generated from a normal with Milo Schield mu = Ln(Median) and sigma = Sqrt[2*Ln(Mean/Median)]. Augsburg College The lognormal is always positive and right-skewed. Editor: www.StatLit.org Examples: US Rep: International Statistical Literacy Project • Incomes (bottom 97%), assets, size of cities 11 October 2014 • Weight and blood pressure of humans (by gender) National Numeracy Network Benefit: www.StatLit.org/pdf/2014-Schield-NNN2-Slides.pdf • calculate the share of total income held by the top X% www.StatLit.org/Excel/Create-LogNormal-Incomes-Excel2013.xlsx • calculate Gini Coefficient, • explore effects of change in mean-median ratio. 1C 1C 2014 NNN2 3 2014 NNN2 4 Log-Normal Distributions Lognormal and Excel Use Excel to focus on the model and the results. “In many ways, it [the Log-Normal] has remained the Cinderella of distributions, the interest of writers in the Excel has two Log-Normal functions: learned journals being curiously sporadic and that of the Standard: =LOGNORM.DIST(X, mu, sigma, k) authors of statistical test-books but faintly aroused.” k=0 for PDF; k=1 for CDF. Inverse: =LOGNORM.INV(X, mu, sigma) “We … state our belief that the lognormal is as fundamental a Use Standard to calculate/graph the PDF and CDF. distribution in statistics as is the normal, despite the stigma of Use Inverse to find cutoffs: quartiles, to 1%, etc. the derivative nature of its name.” Use Excel to create graphs that show comparisons. Aitchison and Brown (1957). P 1. 1C 1C 2014 NNN2 5 2014 NNN2 6 Log-Normal Bibliography Distribution of Units . . Theoretical Distribution of Units by Income Mode: 20K 100% Cumulative Distribution Function (CDF): Percentage of Units with Incomes below price 75% 50% Units can be individuals, households or families 25% Probability Distribution Function (PDF): as a percentage of the Modal PDF 0% 0 50 100 150 200 250 300 350 400 450 500 Incomes ($1,000) LogNormal Dist of Units Median=50K; Mean=80K 2014-Schield-NNN2-Slides.pdf 1
Exploring Lognormal Income Distributions 11 Oct, 2014 1C 1C 2014 NNN2 7 2014 NNN2 8 Distribution of Households Paired Distributions and Total Income by Income For anything that is distributed by X, there are Suppose the distribution of households by income always two distributions: is log-normal with normal parameters mu# and sigma#. 1. Distribution of subjects by X 2. Distribution of total X by X. Then the distribution of total income by amount has Sometime we ignore the 2 nd : height or weight. a log-normal distribution with these parameters: mu$ = mu# + sigma#^2; sigma$ = sigma#. Sometimes we care about the 2 nd : income or assets. Surprise: If the 1 st is lognormal, so is the 2 nd . See Aitchison and Brown (1963) p. 158. Special thanks to Mohammod Irfan (Denver University) for his help on this topic. 1C 1C 2014 NNN2 9 2014 NNN2 10 Distribution of Distribution of Total Income Households and Total Income Distribution of Total Income . Distribution of Households by Income; by Income per Household Distribution of Total Income by Amount Mode: 50K 100% 100% Distribution of Total Income by Median: Amount of Income Percentage of Maximum 128K 75% Mode: $50K 75% Cumulative Distribution Function (CDF): Median: $128K Percentage of Total Income below price Ave $205K 50% 50% Probability Distribution Function (PDF): 25% Households by Income 25% as a percentage of the Modal PDF Mode: $20K; Median: $50K Mean=$80K 0% 0% 0 50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 Income ($1,000) Unit Incomes ($1,000) Log Normal Distribution of Households by Income Income/House: Mean=80K; Median=50K LogNormal Dist of Units by Income Median=50K; Mean=80K 1C 1C 2014 NNN2 11 2014 NNN2 12 Lorenz Curve and Champagne-Glass Gini Coefficient Distribution Pctg of Households vs. Pctg of Income Pctg of Income vs. Pctg. of Households . The Gini coefficient 100% 100% is determined by the Top 50% (above $50k): 83% of total Income Bottom ‐ Up Top 10% (above $175k: 38% of total Income 80% 80% Mean#/Median# ratio. Top 1% (above $475k): 8.7% of total Income Gini = 0.507 Percentage of Income Percentage of Households Top 0.1% (above $1M): 1.7% of total Income 60% 60% The bigger this ratio 40% 40% Gini Coefficient: the bigger the Gini Top 50% (above $50k) have 83% of total Income 0.507 Top 10% (above $175k) have 38% of total Income 20% Bigger means 20% Top 1% (above $475k) have 8.7% of total Income coefficient and the Top 0.1% (above $1M) have 1.7% of total Income more unequal 0% greater the economic 0% 0% 20% 40% 60% 80% 100% 0% 20% 40% 60% 80% 100% inequality. Percentage of Income Percentage of Households Log Normal Distribution of Households by Income Income/House: Mean=80K; Median=50K Log Normal Distribution of Households by Income Income/House: Mean=80K; Median=50K 2014-Schield-NNN2-Slides.pdf 2
Exploring Lognormal Income Distributions 11 Oct, 2014 1C 1C 2014 NNN2 13 2014 NNN2 14 As Mean-Median Ratio As Mean-Median ratio rises, Rich get Richer (or vice-versa) Modal Income may decrease! Log-normal distribution. Median HH income: $50K. Median fixed at $50K Top 5% Households Median Ratio Mean# Mode# Min$ %Income Gini Top 5% Top 1% 50 1.2 60 35 135 15% 0.33 Mean# Min$ %Income Min$ %Income Gini 50 1.3 65 30 165 18% 0.39 55 103 11% 138 2.9% 0.24 50 1.4 70 26 193 20% 0.44 60 135 15% 204 4.2% 0.33 50 1.5 75 22 220 23% 0.48 65 165 18% 270 5.5% 0.39 50 1.6 80 20 246 25% 0.51 70 193 20% 337 6.6% 0.44 50 1.7 85 17 272 27% 0.53 75 220 23% 406 7.7% 0.48 50 1.8 90 15 298 29% 0.56 80 246 25% 477 8.7% 0.51 85 272 27% 549 9.7% 0.53 Does this mean the poor get poorer as the rich get richer 90 298 29% 623 10.7% 0.56 when median Income stays constant? 1C 1C 2014 NNN2 15 2014 NNN2 16 As Mean-Median ratio & Share of Top 10%, Bottom 40% Median , Mode may increase and their Palma Ratio Palma ratio: [Share of top10%] / [Share of bottom 40%]. ----- Top 5% ----- Cobham and Sumner (2014) argue that the Palma ratio is a Median Ratio Mean# Mode# Min$ %Income Gini more understandable measure of inequality than the Gini. 40 1.2 48 28 108 15% 0.33 ---- Top 10% ---- -- Bottom 40% -- 50 1.3 65 30 165 18% 0.39 Mean# Min$ %Income Max$ %Income Palma Gini 60 1.4 84 31 231 20% 0.44 55 87 20% 45 25% 0.8 0.24 70 1.5 105 31 308 23% 0.48 60 108 25% 43 20% 1.3 0.33 80 1.6 128 31 394 25% 0.51 65 127 29% 42 16% 1.8 0.39 90 1.7 153 31 490 27% 0.53 70 143 32% 41 14% 2.3 0.44 100 1.8 180 31 595 29% 0.56 75 159 35% 40 12% 2.8 0.48 80 173 38% 39 11% 3.4 0.51 What does this mean? 85 187 40% 39 10% 4.0 0.53 Median Income: $50K 1C 1C 2014 NNN2 17 2014 NNN2 18 Share of Top 10%, Bottom 40% Minimum Income Minimum Income for Top 5% and top 1% and their Palma Ratio versus Mean Income 900 Minimum Income ($,1000) 800 y = 5.4 x 700 Palma and Gini are independent of the Median Income 600 when the Mean-Median Income ratio is constant. . 500 ---- Top 10% ---- -- Bottom 40% -- 400 Median Ratio Mean# Min$ %Income Max$ %Income Palma Gini 300 y = 2.93 x 40 1.5 60 127 35% 32 12% 2.83 0.48 200 50 1.5 75 159 35% 40 12% 2.83 0.48 100 60 1.5 90 190 35% 48 12% 2.83 0.48 0 70 1.5 105 222 35% 56 12% 2.83 0.48 60 70 80 90 100 110 120 130 140 150 80 1.5 120 254 35% 64 12% 2.83 0.48 Mean Income ($,1000) 90 1.5 135 285 35% 72 12% 2.83 0.48 Log Normal Distribution of Households by Income Median Income: 50K 100 1.5 150 317 35% 80 12% 2.83 0.48 Constant Mean-Median Ratio 2014-Schield-NNN2-Slides.pdf 3
Recommend
More recommend
