Principles of Microeconomics / Unit 14 MIC 1312 review, 2 to 3 minutes
One Distribution, End to End
The idea
One distribution carried from its quintile shares to a single number, and then through a tax and transfer, shows how every measure in this unit connects. Start with the five shares and cumulate them from the bottom: the running totals are the Lorenz curve's five points, and the last must be one hundred. Plot them against twenty through one hundred percent of households and join them to the origin; the diagonal above is perfect equality. Now the Gini. The area under the diagonal is one half. The area under the Lorenz curve can be found as five strips of equal width, each a trapezoid whose sides are consecutive cumulative shares, so the area is one tenth of the sum of each pair of neighboring points. The Gini is one half minus that area, divided by one half. Then apply policy. A progressive tax takes a rising share as income rises, and transfers add to the bottom; recompute the shares after tax and transfer, cumulate, and the Gini falls; the size of the fall is the redistribution's effect on inequality. What the exercise cannot show is poverty: a distribution's shape says nothing about whether the bottom fifth clears a line, and two countries with the same Gini can have very different poverty rates. And the Lorenz curve can cross itself between years, when the Gini alone misleads.
In practice
Worked example: year two's fifths earn four, nine, fifteen, twenty-four and forty-eight percent, cumulating to four, thirteen, twenty-eight, fifty-two and one hundred. The pairs of neighboring points, starting from zero, sum to four, seventeen, forty-one, eighty and one hundred fifty-two, or two hundred ninety-four in percent units; the area is a tenth of that as a fraction, zero point two nine four. The Gini is one half minus zero point two nine four, divided by one half: zero point four one. After a progressive tax and transfers the fifths earn seven, eleven, sixteen, twenty-three and forty-three, cumulating to seven, eighteen, thirty-four, fifty-seven and one hundred; the pair sums total three hundred thirty-two, the area is zero point three three two, and the Gini is zero point three four. Redistribution cut the Gini by about eight hundredths.
Where people slip
Three cautions. The five-strip area is an approximation that treats income as evenly spread within each fifth; the true curve bows within strips too, so the Gini computed this way understates the published figure slightly. A graph of the same distribution can be made to look calm or alarming with true numbers: by stretching an axis, choosing the vertical scale, averaging years, or picking a start date at a peak; every graph is a point of view, and accuracy and honesty are different tests. And the after-tax Gini says redistribution reduced inequality, not whether it was worth its cost in incentives, which is the contested question.
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