Income, Wealth, and Inequality

Prerequisites: Surplus, Taxes, and Subsidies · Labor Markets, Wages, and Employment

The average monthly income in both locations is 30. Is the standard of living for an ordinary household the same? If a household’s income remains unchanged but housing prices rise, does it become wealthier? The first question requires analyzing the distribution, while the second requires distinguishing between period income and wealth at a specific point in time.

This article constructs a ledger using five households of equal size, then interprets the Lorenz curve and the Gini coefficient. All amounts are in teaching units; households are equally weighted, and incomes are non-negative. For real-world comparisons, one must verify taxes, household size, prices, and statistical coverage.

Place income and wealth on the correct timeline

Income is a flow, such as receiving a salary of 30 in a month; assets and liabilities are stocks, such as having assets of 200 and liabilities of 80 at the end of the month, resulting in a net wealth of 120. One cannot directly call assets of 200 the household's net wealth, nor can one add the month-end balance to the monthly salary and call it "monthly income."

Assume beginning net wealth is 120. After deducting consumption and related expenses from this month's income, savings amount to 5. Existing assets appreciate by 10, with no other changes. Thus, ending net wealth is 135. The increase of 15 consists of savings and revaluation; only 5 of that comes from current accumulation.

Book value appreciation does not guarantee an equivalent amount of available cash immediately. Homeowners whose primary residence appreciates in value still need to live there; upon selling, they may face more expensive replacement housing. The ease of borrowing and long-term improvements in actual living standards are separate issues. To evaluate a household's situation, one should look separately at disposable income, net assets, debt servicing, and liquidity.

Same averages, but who actually gets what?

Both groups have a mean and median of 30. However, the lowest household in A earns only 10, while in B it is 20; the lowest two households in A earn a combined 30, accounting for 20% of total income, whereas the lowest two in B earn a combined 45, accounting for 30%. The average does not reveal these differences.

If only "average income of 30" is published, a reader might mistakenly assume most households are close to 30. The median answers how much the person in the middle earns; quantiles and income shares describe the distribution at both ends. No single number can replace the complete distribution.

Reading a Lorenz curve by hand

After ranking by income, accumulate from the lowest-income household. The horizontal axis represents the cumulative proportion of households, and the vertical axis represents the cumulative proportion of income received by these households:

Both means are 30; B’s cumulative income distribution is closer to equality.

The point (40%, 20%) on curve A does not mean a single household earns 20, nor does it mean the richest 40% receive 20%; it means the lowest-income 40% of households collectively receive 20% of total income. If everyone's income were exactly the same, the lowest 40% would receive exactly 40%, forming the diagonal line of perfect equality.

In this dataset, B's curve is higher than A's at intermediate positions, closer to equal distribution. If the curves of two locations cross, they may provide different rankings between the lowest and middle groups; one cannot declare one location more equal overall based on a single point on the graph.

What the Gini coefficient compresses and what it loses

For $n$ equally weighted households with non-negative incomes and a positive mean income, the Gini coefficient can be written as the sum of the absolute differences in income between all ordered pairs of households, divided by 2 × n² × mean income. "Ordered" means both (Household X, Household Y) and (Household Y, Household X) are counted; if only unordered pairs are enumerated once, the denominator must be adjusted accordingly.

In this example, the result for A is 4/15 ≈ 0.267, and for B it is 2/15 ≈ 0.133. With both axes on a 0–1 scale, it is twice the area between the line of equality and the Lorenz curve, or that area divided by the area under the equality line; a larger value indicates greater relative disparity under this metric. Reports may use 0–1 or multiply by 100; check the scale before comparing.

Different distributions can yield the same Gini coefficient, and wealth including negative values can produce results outside the simple non-negative case. The Gini coefficient does not directly explain whether disparities stem from education, age, inheritance, market power, discrimination, or other mechanisms; describing the distribution and identifying causes are two separate tasks.

Improvement for every household is not the same as narrowing the gap

Double every household's income in A, resulting in 20, 40, 60, 80, 100. Every household's nominal income increases, and the income shares and Gini coefficient remain unchanged; however, the absolute difference between the highest and lowest increases from 40 to 80. Proportionally equal growth maintains relative differences but does not maintain absolute monetary differences.

Consider another policy: add 10 to every household, resulting in 20, 30, 40, 50, 60, with a total income of 200. The lowest household's share rises from approximately 6.67% to 10%, and the Gini coefficient becomes 0.20, indicating a decrease in relative inequality, but the absolute difference between the highest and lowest remains 40.

In a third scenario, only the lowest household increases from 10 to 12, and the highest from 50 to 100, with the middle unchanged. The lowest household improves while relative disparity expands. Therefore, "bottom income rising," "relative inequality decreasing," and "absolute poverty reduction" should be stated separately and cannot be substituted for one another.

These are nominal amounts. If prices rise faster than income, purchasing power may still decline; the next set of macroeconomic measurements will address price adjustments. If the poverty line is defined by actual living needs, one cannot simply compare unadjusted income growth rates.

Supplement household and institutional information before comparing

A one-person household with a monthly income of 30 and a five-person household with a monthly income of 30 do not necessarily have the same standard of living. Pre-tax income, post-tax and transfer disposable income, and income adjusted for household size answer different questions; public services such as free education and healthcare may not directly enter cash income statistics.

Comparing households of different ages in the same year cannot directly infer the opportunity differences experienced by the same cohort over their lifetime. One needs to track income mobility, health, education, region, and family structure to distinguish between lifecycle changes and persistent barriers. Conversely, the existence of mobility does not necessarily imply equal opportunities.

Policy evaluation should clarify its goals: raising the standard of living at the bottom, reducing risk, altering relative distribution, or expanding total output may require different indicators. Statistical indicators help check goals but do not automatically decide which distribution is most desirable.

Answering with ledgers and shares

A household's net wealth increases from 120 to 150, with current savings of 5 and no debt relief or other transfers. Can we say it earned 30 this month?

In a certain location, the average income rises, but the income share of the lowest 40% falls. Did the actual income of the lowest 40% definitely fall?

Sources and Further Reading

These references support concepts and statistical definitions; the numerical cases and diagrams are original synthetic teaching examples.

CORE · Institutions, Power, and Inequality