AP Microeconomics: Income Distribution and Lorenz Curve / Gini Coefficient

AP Microeconomics – Income Distribution and Lorenz Curve / Gini Coefficient

AP Microeconomics – Income Distribution, Lorenz Curve & Gini Coefficient

What This Is

Income distribution describes how a nation’s total earnings are shared among its residents. The Lorenz Curve graphically shows the cumulative share of income earned by the bottom X?% of the population, while the Gini Coefficient condenses that curve into a single number (0?=?perfect equality, 1?=?perfect inequality). AP students must be able to draw, interpret, and calculate these tools because they appear on multiple?choice items and FRQs that ask you to evaluate the equity effects of taxes, subsidies, or market outcomes (e.g., a government’s progressive income?tax schedule).

Key Terms & Formulas

• Lorenz Curve – A graph with the horizontal axis = cumulative % of households (from poorest to richest) and the vertical axis = cumulative % of total income earned. The 45?degree line (line of perfect equality) runs from (0,0) to (100,100).
• Line of Perfect Equality – The 45° line on a Lorenz diagram; every point on it means the bottom X?% of households earn exactly X?% of total income.
• Gini Coefficient (G) – G = A / (A + B) where A is the area between the line of perfect equality and the Lorenz curve, and B is the area beneath the Lorenz curve.
• Area (A) – The region between the 45° line and the Lorenz curve; larger A-higher inequality.
• Area (B) – The area under the Lorenz curve; equals 0.5 minus A because the total area under the 45° line is 0.5 (a right triangle).
• Progressive Tax – A tax system where the average tax rate rises as income rises; tends to lower the Gini coefficient.
• Regressive Tax – A tax where the average tax rate falls as income rises (e.g., a flat sales tax); tends to raise the Gini coefficient.
• Redistributive Policy – Government action (taxes, transfers, subsidies) aimed at changing the shape of the Lorenz curve to achieve a desired Gini level.
• Cumulative Share of Income – The percentage of total national income earned by the bottom X?% of households; plotted on the vertical axis of the Lorenz curve.
• Percentile – A point on the horizontal axis representing a specific share of the population (e.g., the 20th percentile = bottom 20?%).

Step?by?Step / Process Flow

1. Collect data – Obtain the cumulative income shares for each percentile (or decile) of the population.
2. Plot the Lorenz Curve – On graph paper, mark the horizontal axis (0?% to 100?% of households) and vertical axis (0?% to 100?% of income). Plot each cumulative point and connect them with a smooth, upward?sloping line.
3. Draw the line of perfect equality – Sketch the 45° line from (0,0) to (100,100).
4. Calculate area B – Since the total area under the 45° line is 0.5, compute B = 0.5?A. If you have the coordinates, use the trapezoid rule or the formula for the area of a polygon to find B directly.
5. Compute the Gini Coefficient – Use G = A / (A + B) (or equivalently G = 2A because A?+?B?=?0.5). Insert the measured area to obtain a number between 0 and 1.
6. Interpret the result – Compare the Gini to the national average or to a pre?policy value. Explain whether a proposed tax/subsidy would shift the Lorenz curve toward the 45° line (lower G) or away from it (higher G).

Common Mistakes

• Mistake: Treating a movement along the Lorenz curve as a “shift.”
  Correction: The Lorenz curve itself only moves when the underlying income distribution changes (e.g., a tax reform). A change in a single household’s income moves a point along the existing curve, not the whole curve.

• Mistake: Using the formula G = (A?+?B) / A (or any inversion).
  Correction: The correct formula is G = A / (A?+?B); remember that A is the inequality area, so a larger A yields a larger G.

• Mistake: Forgetting that the total area under the 45° line equals 0.5, leading to mis?calculated B.
  Correction: Always start with the fact that A?+?B = 0.5 (the area of the right?triangle). This shortcut lets you find A directly from the Gini: A = G?×?0.5.

• Mistake: Assuming a “flat” sales tax is neutral for inequality because the tax rate is the same for everyone.
  Correction: A flat sales tax is regressive because lower?income households spend a larger share of their income on taxable goods, raising their effective tax burden and moving the Lorenz curve away from equality.

• Mistake: Mixing up “percentage of households” with “percentage of income” on the axes.
  Correction: Horizontal = cumulative % of households (from poorest upward); vertical = cumulative % of total income earned by those households.

AP Exam Insights

1. Multiple?Choice Focus: 1?point items often ask you to identify whether a given tax (e.g., a progressive income tax) will increase or decrease the Gini coefficient. Look for key words “progressive,” “regressive,” or “proportional.”
2. FRQ Prompt Pattern: You may be given a Lorenz curve before and after a policy and asked to calculate the change in Gini or to explain the shift in terms of redistribution. Remember to state the formula, show the area reasoning, and label the axes.
3. Graphing Requirement: The AP rubric awards points for a correctly drawn Lorenz curve and a clearly labeled line of perfect equality. Even a rough sketch earns credit if the axes and direction (bottom?to?top) are correct.
4. Conceptual Tie?In: The exam frequently links income distribution to market outcomes (e.g., monopoly pricing) and to government intervention (e.g., Earned Income Tax Credit). Be ready to discuss how a market failure can widen the gap between the Lorenz curve and the 45° line.

Quick Check Questions

1. MC: A country’s Lorenz curve lies very close to the 45° line. Which of the following statements is most accurate?
2. A) The Gini coefficient is near 1.
3. B) Income is highly unequal.
4. C) The Gini coefficient is near 0.
5. D) The government must impose a regressive tax.

Answer: C) The Gini coefficient is near 0.
Explanation: Proximity to the line of perfect equality means low inequality, yielding a Gini close to 0.

1. FRQ (short): After implementing a progressive income tax, the area A between the Lorenz curve and the line of perfect equality shrinks from 0.08 to 0.05. What is the change in the Gini coefficient?

Answer: Gini falls from 0.16 (2?×?0.08) to 0.10 (2?×?0.05); the Gini decreases by 0.06.

1. MC: Which policy is most likely to raise a country’s Gini coefficient?
2. A) Expanding a refundable Earned Income Tax Credit.
3. B) Introducing a flat sales tax on all consumer goods.
4. C) Implementing a progressive income?tax schedule.
5. D) Providing universal health care funded by a progressive payroll tax.

Answer: B) Introducing a flat sales tax on all consumer goods.
Explanation: A flat sales tax is regressive, increasing the effective tax burden on low?income households and moving the Lorenz curve away from equality.

Last?Minute Cram Sheet (10 one?liners)

1. Lorenz Curve axes: % of households (horizontal) vs. % of total income (vertical).
2. Line of Perfect Equality = 45° line from (0,0) to (100,100).
3. Gini = 2?×?Area?A (because A?+?B?=?0.5).
4. Gini range: 0?=?perfect equality, 1?=?perfect inequality.
5. Progressive tax-? Gini; Regressive tax-? Gini.
6. Area A = “inequality area” between the 45° line and the Lorenz curve.
7. Area B = 0.5?A (area under the Lorenz curve).
8. A shift of the Lorenz curve toward the 45° line = more equal distribution.
9. “Supply increases” means the curve shifts right – not up. A change in price causes a movement along, not a shift. (applies to Lorenz: a change in a single household’s income moves a point along the curve, not the whole curve.)
10. FRQ tip: Always label the axes, state the Gini formula, and explain the direction of the shift before plugging numbers.