# Inequality

> AP Microeconomics · Unit 6: Market Failure and the Role of Government
> Source: https://www.owlsprep.com/study/ap-microeconomics-u6-inequality/

This guide covers core concepts of income inequality for AP Microeconomics Unit 6, including measurement via Lorenz curves and Gini coefficients, tax system classification, and the equity-efficiency tradeoff of redistributive policy.

**Prerequisites:** Basic utility theory; Role of government in correcting market failure; Labor market supply and demand

## Learning objectives

- Distinguish between income inequality and wealth inequality
- Construct and interpret a Lorenz curve for income distributions
- Calculate and interpret the Gini coefficient
- Classify tax systems as progressive, proportional, or regressive
- Evaluate the equity-efficiency tradeoff of redistributive policies

## Defining Economic Inequality

Inequality in AP Microeconomics refers to the unequal distribution of economic resources across households, most commonly measured as **income inequality** (annual flow of earnings from work and capital) or **wealth inequality** (stock of accumulated assets like property or stocks). This topic makes up 12-18% of the total AP Micro exam score, appearing in both multiple-choice and free-response sections.

**Income vs Wealth Inequality** — Income measures the annual flow of resources to a household, while wealth measures the total stock of accumulated assets owned by a household. AP Micro focuses primarily on income inequality, and exam questions will test your ability to distinguish the two terms.

*Example:* A recent college graduate may have a moderate annual income but very low accumulated wealth, while a retired person may have low annual income but high accumulated wealth.

## The Lorenz Curve

The Lorenz curve is the standard graphical tool used to visualize income inequality. It plots the cumulative percentage of households (ordered from lowest to highest income on the x-axis, 0% to 100%) against the cumulative percentage of total national income held by that share of households (y-axis, 0% to 100%).

The 45-degree line drawn from (0,0) to (100,100) is called the **line of perfect equality**. If income were perfectly equally distributed, 10% of households would hold 10% of total income, so the Lorenz curve would lie directly on this line. The further the Lorenz curve is below the 45-degree line, the more unequal the distribution of income.

**Worked example:** Five households have annual incomes of \$15,000, \$25,000, \$40,000, \$60,000, and \$160,000. Identify the coordinates of all key points needed to plot the Lorenz curve.

1. First, confirm households are ordered from lowest to highest income, then calculate total income:
2. $$15000 + 25000 + 40000 + 60000 + 160000 = 300000$$
3. Total income is \$300,000, or 300 thousand dollars.
4. Each household represents 20% of total households, so cumulative shares of households are 20% after 1 household, 40% after 2, 60% after 3, 80% after 4, and 100% after 5.
5. Calculate cumulative income shares:
6. - After 1 household: \$15,000 = $\frac{15}{300} = 5\%$ of total income
7. - After 2 households: $\frac{15+25}{300} \approx 13.3\%$
8. - After 3 households: $\frac{80}{300} \approx 26.7\%$
9. - After 4 households: $\frac{140}{300} \approx 46.7\%$
10. - After 5 households: 100% of total income
11. The key points to plot are: $(0,0)$, $(20, 5)$, $(40, 13.3)$, $(60, 26.7)$, $(80, 46.7)$, $(100, 100)$. Connecting these gives the Lorenz curve, which lies below the 45-degree line of perfect equality.

> **Exam tip:** Never order households from highest to lowest income before plotting. AP exam graders will immediately deduct points for a reversed Lorenz curve that lies above the 45-degree line, as this reflects a fundamental misunderstanding.

## The Gini Coefficient

The Gini coefficient is a numerical summary statistic that quantifies the level of inequality from a Lorenz curve. It is defined as the ratio of the area between the line of perfect equality and the Lorenz curve (area A) to the total area under the line of perfect equality (A + B, where B is the area under the Lorenz curve).

$$G = \frac{A}{A+B}$$

Since the graph is a 1x1 unit square, the total area under the 45-degree line is 0.5, so we can simplify the formula to $G = 2A$. The Gini coefficient ranges from 0 (perfect equality) to 1 (perfect inequality): higher values always mean greater income inequality. For AP exams, you will usually use the trapezoid rule to calculate Gini for discrete distributions.

**Worked example:** Using the 5-household income distribution from the Lorenz example above, calculate the Gini coefficient.

1. Convert all percentages to decimals for area calculation: each x interval between consecutive points is 0.2 (20% of households).
2. Calculate area B (under the Lorenz curve) as the sum of trapezoid areas:
3. $$\text{Trapezoid 1: } \frac{0 + 0.05}{2} \times 0.2 = 0.005$$
4. $$\text{Trapezoid 2: } \frac{0.05 + 0.133}{2} \times 0.2 \approx 0.0183$$
5. $$\text{Trapezoid 3: } \frac{0.133 + 0.267}{2} \times 0.2 = 0.04$$
6. $$\text{Trapezoid 4: } \frac{0.267 + 0.467}{2} \times 0.2 \approx 0.0734$$
7. $$\text{Trapezoid 5: } \frac{0.467 + 1}{2} \times 0.2 \approx 0.1467$$
8. Sum all trapezoid areas: $B \approx 0.005 + 0.0183 + 0.04 + 0.0734 + 0.1467 \approx 0.2834$
9. Calculate $A = 0.5 - B \approx 0.2166$, so $G = 2A \approx 0.433$. The Gini coefficient is approximately 0.43.

> **Exam tip:** If asked to shade area A on an FRQ, always shade the gap between the 45-degree line and the Lorenz curve. A common mistake is shading the area under the Lorenz curve (area B), which will lose you the point.

## Tax Classification and Redistributive Policy

Governments use progressive taxation paired with transfer payments to reduce income inequality. Tax systems are classified by their **average tax rate (ATR)**, the share of total income paid in taxes:

- **Progressive**: ATR increases as income increases (higher income households pay a larger share of income in tax)
- **Proportional**: ATR is constant across all income levels (also called a flat tax)
- **Regressive**: ATR decreases as income increases (higher income households pay a smaller share of income in tax)

Only progressive taxation reduces income inequality when combined with transfer payments to low-income households (such as welfare, unemployment insurance, or refundable tax credits).

**Worked example:** A country has a progressive income tax with brackets: 10% on income up to \$30,000, 20% on income between \$30,000 and \$100,000, and 40% on income above \$100,000. Calculate the average tax rate for a low-income household earning \$40,000 and a high-income household earning \$200,000, then explain how this system reduces inequality.

1. Calculate total tax for the low-income household:
2. Tax on first \$30,000: $0.10 \times 30000 = \$3000$. Tax on remaining \$10,000: $0.20 \times 10000 = \$2000$. Total tax = \$5000.
3. Average tax rate for low-income household: $\frac{5000}{40000} = 0.125 = 12.5\%$.
4. Calculate total tax for the high-income household:
5. Tax on first \$30,000 = \$3000, tax on next \$70,000 = $0.20 \times 70000 = \$14000$, tax on remaining \$100,000 = $0.40 \times 100000 = \$40000$. Total tax = 3000 + 14000 + 40000 = \$57000.
6. Average tax rate for high-income household: $\frac{57000}{200000} = 0.285 = 28.5\%$.
7. Explanation: The high-income household pays a higher average tax rate. Tax revenue funds transfers to low-income households, so after-tax income is much more equal than pre-tax income, reducing overall inequality.

## The Equity-Efficiency Tradeoff

The core concept for evaluating redistributive policy in AP Microeconomics is the **equity-efficiency tradeoff**: while redistribution increases equity (a more equal distribution of income), it can reduce economic efficiency by distorting incentives.

For example, high marginal tax rates on high earners can reduce the incentive to work or invest, leading to lower total output and deadweight loss. However, some redistributive policies can increase *both* equity and efficiency, such as public education that corrects credit market failures preventing low-income households from investing in human capital.

**Check your understanding**

Test your understanding of Gini interpretation:

1. The Gini coefficient for Country A is 0.38, and the Gini coefficient for Country B is 0.48. Which statement is correct?

   - Country A has a more equal distribution of income than Country B.
   - Country B has a more equal distribution of income than Country A.
   - The Lorenz curve for Country A lies further from the line of perfect equality than Country B's.
   - Total income in Country A is lower than total income in Country B.

   *Why:* Gini ranges from 0 (perfect equality) to 1 (perfect inequality), so lower Gini means more equal income distribution. Gini measures distribution, not total income.

> **Exam tip:** When asked to explain the equity-efficiency tradeoff, always explicitly mention both sides: higher equity from redistribution, and potential efficiency losses from distorted incentives. You will not get full credit for only discussing one side.

## Common pitfalls

- **Wrong:** Calling a proportional tax progressive because high-income households pay more total tax than low-income households.
  - Why it fails: Students confuse total tax paid with average tax rate, which is the correct metric for classifying tax systems.
  - Correct: Always use the average tax rate (share of income paid in tax) to classify a tax system as progressive, proportional, or regressive.
- **Wrong:** Interpreting a higher Gini coefficient as meaning a more equal income distribution.
  - Why it fails: Students mix up the definition of the Gini coefficient and the area A calculation.
  - Correct: Memorize that Gini = 0 is perfect equality, Gini = 1 is perfect inequality, so Gini increases as inequality increases.
- **Wrong:** Confusing income inequality with wealth inequality on the exam.
  - Why it fails: The terms are used interchangeably in casual discussion, but they have distinct definitions tested on the AP exam.
  - Correct: Remember income is a flow of annual earnings, while wealth is a stock of accumulated assets; always use the term the question asks for.
- **Wrong:** Claiming all redistribution policies always reduce economic efficiency.
  - Why it fails: Students overgeneralize the equity-efficiency tradeoff, ignoring market failures that redistribution can correct.
  - Correct: Acknowledge that while high marginal taxes can reduce efficiency, policies like public education or the EITC can increase efficiency by correcting market failures.
- **Wrong:** Forgetting to order households from lowest to highest income when calculating cumulative shares for the Lorenz curve.
  - Why it fails: Students skip the sorting step when working with unsorted data, leading to an incorrect curve.
  - Correct: Always sort household incomes in ascending order before calculating any cumulative percentages.

## Cheatsheet

| Category | Definition / Formula | Key Notes |
| --- | --- | --- |
| Gini Coefficient | $G = \frac{A}{A+B} = 2A$ | A = area between 45° line and Lorenz curve; ranges 0 (perfect equality) to 1 (perfect inequality) |
| Lorenz Curve Axes | X: Cumulative % of households (lowest → highest income) Y: Cumulative % of total income | Always sort incomes ascending before plotting |
| Line of Perfect Equality | 45-degree line from (0,0) to (100,100) | Represents perfectly equal distribution of income |
| Average Tax Rate | $\text{ATR} = \frac{\text{Total Tax Paid}}{\text{Total Income}}$ | Used to classify all tax systems |
| Progressive Tax | ATR increases with income | Reduces inequality when paired with transfers to low-income households |
| Proportional Tax | ATR is constant across all income levels | Also called a flat tax; no net effect on income inequality |
| Regressive Tax | ATR decreases with income | Increases overall income inequality |
| Equity-Efficiency Tradeoff | Redistribution increases equity but can reduce efficiency via distorted incentives | Some policies correct market failures and increase both equity and efficiency |

## What's next

Inequality is the capstone topic in AP Microeconomics Unit 6: Market Failure and the Role of Government, which accounts for 12-18% of your total exam score. Mastery of inequality measurement and the equity-efficiency tradeoff prepares you to answer full-length FRQs that ask you to evaluate government intervention beyond correcting standard efficiency market failures. This topic builds on your earlier understanding of factor markets, where you learned how income is distributed across different factors of production. After completing this guide, you should review all Unit 6 topics to prepare for practice exams, focusing on connecting inequality concepts to other market failure topics. Solid understanding of this sub-topic will help you earn full credit on both multiple-choice and free-response questions on exam day.

- [Unit 6: Market Failure and the Role of Government Overview](https://www.owlsprep.com/study/ap-microeconomics-u6-overview/)

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