Poverty and Income Inequality

This is Unit Fourteen of Principles of Microeconomics: poverty and inequality, how each is measured, and what the measures can and cannot say. Unit thirteen set wages one market at a time. This unit adds them up into a distribution and asks two separate questions of it: how many fall below a line, and how the whole is shared. The two can move apart, and keeping them apart is the unit's first lesson. The tools are a threshold, a trap, a curve and a single number, and the capstone runs one distribution through all of them. By the end you will be able to keep poverty apart from inequality and compute a threshold both ways. To explain and compute the poverty trap. To build a Lorenz curve and define the Gini. To report the attributed causes of rising inequality and the tools against it. And to carry one distribution from fifths to a Gini and through a tax and transfer.

Three entries on poverty: two questions kept apart, where the line comes from, and the arithmetic of a threshold.

Two questions.

MIC 1301 Poverty and Inequality: Two Questions

Poverty asks how many people fall below an income needed for a basic standard of living, the poverty line. Inequality asks how income is shared across the whole distribution, and they can move separately: if every income falls ten percent, poverty rises and inequality is unchanged. A rise at the top raises inequality and leaves poverty untouched, so keep the two measures apart.

Where the line comes from.

MIC 1302 The Poverty Line

The official line is absolute: the nineteen sixty-three cost of an adequate diet times three, since families then spent a third of income on food, price-adjusted since. A relative test calls the bottom of the distribution poor; the European Union's line is sixty percent of median income. The line counts cash only and ignores in-kind aid, and its virtue is consistency over time.

Computing a line.

MIC 1303 Poverty Line Arithmetic

Build a threshold from a food budget by the one-third rule: multiply the cost of an adequate diet by three. Adjust for family size by using that family's food budget, and for a price change by scaling the threshold with prices. Then test a family both ways, absolute against the threshold and relative against a share of median income; the two tests can disagree, and the disagreement is the point. An adequate diet for a family of four costs nine thousand dollars a year, so the threshold is twenty-seven thousand; for a family of three the diet costs seven thousand, threshold twenty-one thousand. Prices rise ten percent and the four-person threshold becomes twenty-nine thousand seven hundred. A family of four earning thirty thousand is above the absolute line, and below a relative line set at sixty percent of a median of seventy thousand, forty-two thousand. The one-third rule is the nineteen sixties' food share, so the multiplier understates a modern budget, and the line stays because a changing definition would break every comparison over time; and a relative line can never be reached, since there is always a bottom fifth.

Three entries on the trap and the net: what aid does to the incentive to work, the trap in numbers, and the programs as designs.

What aid does to the incentive to work.

MIC 1304 The Poverty Trap

If assistance falls a dollar for every dollar earned, working brings no gain: the poverty trap. A fifteen-thousand guarantee makes income flat at fifteen thousand whatever the hours. It loosens when benefits fall by less than a dollar per dollar, or with work requirements and time limits, and a slower phase-out costs more, since benefits reach higher incomes; that is the price.

The trap in numbers.

MIC 1305 The Poverty Trap in Numbers

Given a wage, a guaranteed benefit and a benefit-reduction rate, compute income at several hours of work: earnings plus the guarantee minus the reduction rate times earnings, with the benefit never below zero. Report the net gain per hour, which is the wage times one minus the reduction rate, then rerun with a lower rate and show what changes and what it costs. The wage is ten dollars, the guarantee fifteen thousand, and benefits fall a dollar per dollar. At zero hours income is fifteen thousand; at a thousand hours, earnings of ten thousand cut the benefit to five thousand, still fifteen thousand; at fifteen hundred hours the benefit is gone and income is still fifteen thousand, and only beyond that does work pay. Net gain per hour: zero. Now benefits fall fifty cents per dollar. A thousand hours gives twenty thousand, two thousand hours twenty-five thousand, and the net gain per hour is five dollars, with benefits continuing until earnings reach thirty thousand. The flat stretch is the trap, and its length is the guarantee divided by the wage; the fifty-cent rule pays benefits to families earning up to twice the guarantee, which is why loosening the trap is expensive even though it raises work.

The programs as designs.

MIC 1306 The Safety Net's Designs

Cash assistance is a block grant with work requirements and a five-year limit. The earned income credit rises with earnings, plateaus, then phases out. Food aid, in kind, falls thirty dollars per hundred earned, and a food card frees the cash a family would have spent on food. In-kind aid is mostly income, and a coverage cliff at a job threshold is another trap.

Three entries on measuring inequality: quintiles and the Lorenz curve, two years compared, and the single number.

Measuring the whole distribution.

MIC 1307 Quintiles and the Lorenz Curve

Cut ranked households into fifths and record each fifth's income share. The Lorenz curve plots cumulative household share against cumulative income share; the diagonal is perfect equality, and the more the curve bows below it, the more unequal. The bottom two fifths at eleven percent is the point forty, eleven. Plot cumulative shares, not each fifth's own, and the last point is one hundred, one hundred.

Drawing two years.

MIC 1308 Drawing Two Lorenz Curves

From each year's quintile shares, build cumulative shares by adding each fifth to the total before it; the fifth entry must be one hundred, which checks the arithmetic. Plot each year's five points against twenty through one hundred percent of households, with the origin, and compare: the curve lying further below the diagonal is the more unequal year, and say which fifths gained share and which lost. Year one's fifths earn six, eleven, sixteen, twenty-four and forty-three percent; year two's earn four, nine, fifteen, twenty-four and forty-eight. Year one cumulates to six, seventeen, thirty-three, fifty-seven and one hundred; year two to four, thirteen, twenty-eight, fifty-two and one hundred. At every point but the last year two's curve lies lower, so year two is more unequal; the bottom three fifths each lost share, the fourth held, and the top fifth gained five points. Cumulate in order from the bottom, since a curve built from the top down is a different object, and two curves can cross, in which case neither year is unambiguously more unequal and a single number like the Gini hides the crossing.

The single number.

MIC 1309 The Gini Coefficient

The Gini coefficient is the area between the diagonal and the Lorenz curve, divided by the area under the diagonal: zero for perfect equality, one for one household holding everything. The Census reported point four seven for two thousand six and point four eight five for twenty twenty-three. The anchor never defines it; the definition is from the witness, the numbers from the Census, dated.

Three entries to close: why inequality rose, what governments do about it, and the unit's capstone, one distribution end to end.

Why inequality rose.

MIC 1310 Sources of Rising Inequality

Two attributed explanations. Household composition: two-earner couples, like marrying like, more single parents. Wages: graduates' supply rose greatly, which alone would lower the skill premium, yet it rose, so demand shifted right by more, as technology and global markets raised what skilled workers produce. Winner-take-all markets push top pay beyond education's reach, and income is a flow while wealth is a more unequal stock.

What governments do about it.

MIC 1311 Tools and the Trade-off

Governments reduce inequality three ways: redistribution funded by a progressive income tax, which takes a rising percentage as income rises; a ladder of opportunity, preschool to apprenticeships; and inheritance taxes. Public education redistributes and builds skills at once. Whether equality costs output is contested: one attributed view sees a steep trade-off with incentives, another a flat region where similar economies differ.

One distribution from fifths to Gini and back.

MIC 1312 One Distribution, End to End

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, with the diagonal above as 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, and 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, cumulate, and the Gini falls, and the size of the fall is the redistribution's effect on inequality. What the exercise cannot show is poverty, since a distribution's shape says nothing about whether the bottom fifth clears a line, and the Lorenz curve can cross itself between years, when the Gini alone misleads. 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 two hundred ninety-four in percent units, so the area is zero point two nine four and the Gini is 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. Three cautions. The five-strip area treats income as evenly spread within each fifth, 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, 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.

Look at what you can now do. You can ask poverty and inequality as two separate questions, say that the official line is an absolute one built from a nineteen sixty-three food budget, and compute a threshold by the one-third rule and test a family both ways. You can explain the poverty trap as a benefit that falls a dollar for a dollar, compute the flat stretch as the guarantee over the wage, and say why a slower phase-out raises work and costs more. You can cumulate quintile shares from the bottom into a Lorenz curve, compare two years by which lies lower, and define the Gini as the bowed area over the half. You can report the two attributed explanations of rising inequality and the three tools governments use, with the incentive trade-off left contested. And you can carry one distribution from five shares through five strips to a Gini, run it through a tax and transfer, and say what that number cannot tell you about poverty. Unit fifteen asks what happens when one side of a trade knows more.