Measure the distribution before explaining how it arose or judging policies.
By the end: Distinguish market income, disposable income, wealth, and the effects of redistribution.
Income is a flow over a period. Wealth is a stock of assets minus liabilities at a date. Market income includes earnings and returns from ownership; disposable income adjusts for taxes and cash transfers under a specified definition. Public services and unpaid work affect living standards even when they do not appear in that cash measure. Household size and needs matter when comparing resources.
The mean divides total income by the number of people. The median is the middle observation, or the average of the two middle observations in an even-sized sample. In our six equal-sized households, incomes of 10, 20, 30, 40, 60, and 140 have a mean of 50 and a median of 35. The top household receives almost 46.7% of the total. None of these summaries describes every household.
A distribution needs a unit of observation, a period, and a resource definition. Annual individual earnings, monthly household disposable income, and household net wealth answer different questions. A person can have low current earnings while living in a household supported by another earner. A household can have high income but little wealth after years of debt repayment. Statistics can be correct within each definition and still appear inconsistent if those definitions are silently exchanged. Start a comparison by asking who is counted together and whether the number measures receipts during a period or assets at a date.
For a paper example, household P receives 3,000 currency units per month and has one member. Household Q receives 4,800 and has four members. Ranking by household income places Q above P. Dividing by members gives 3,000 per person for P and 1,200 for Q. That reverses the ranking but assumes that every additional person requires the same resources as the first. Shared housing and heating can make that assumption too strong. Conversely, disability or care needs can raise required resources. Neither raw household totals nor simple per-person division fully describes living standards.
An equivalence scale is an explicit rule for adjusting resources for household composition. To illustrate rather than prescribe one, give the first adult a weight of one, another adult one half, and each child one quarter. If Q contains two adults and two children, its total weight is two. Dividing 4,800 by two gives adjusted income of 2,400. P's adjusted income remains 3,000. This invented scale produces a smaller gap than dividing by four, because it assumes some shared costs. The arithmetic does not validate the weights. Any reported adjusted distribution needs the actual scale and its interpretation.
Wealth requires another calculation. Suppose P owns assets worth 90,000 and owes 70,000, while Q owns 30,000 and owes 5,000. Net wealth is 20,000 for P and 25,000 for Q. These amounts cannot be added to monthly incomes without specifying a time conversion or an economic question. Nor does Q necessarily have more cash available this month. Its assets might be retirement claims that cannot be withdrawn immediately. Liquidity measures access to usable funds; net wealth measures assets less liabilities. A household may own valuable housing while lacking cash for an urgent bill.
Finally, distinguish absolute changes from shares. If every income in a population doubles while prices stay fixed, each household can afford more, but relative income shares remain unchanged. If incomes and prices both double, purchasing power need not rise at all. A top share can fall because lower incomes rise or because the top income collapses while others stagnate. The same direction of change in an inequality statistic can therefore accompany very different living conditions. Use the statistic with income levels, prices, and the population definition rather than asking it to stand for every aspect of economic security.
Differences in education, health, work hours, ownership, inheritance, discrimination, bargaining, and market power can contribute to unequal incomes and wealth. Returns compound differently when some households can save and absorb risk while others cannot. A high observed return on wealth can reflect changing asset prices rather than new production. One summary statistic cannot identify the relative contribution of these mechanisms.
Policy affects the distribution before taxes as well as afterward. Education, labor rights, competition policy, housing rules, and access to assets influence market incomes and opportunities. Taxes and transfers then redistribute cash resources. Describing the first distribution as purely natural and the second as the only political intervention hides the institutions that shape both.
Saving connects an income flow to a wealth stock. A household that receives more than it spends may add to assets or repay debt. Capital gains change the value of existing assets without necessarily creating current production or arriving as cash. Inheritance transfers claims between people. These mechanisms can operate together, but they require different evidence. A rise in a property's market value is not the same event as constructing another house, receiving rent, or inheriting ownership. Explaining wealth concentration requires identifying which changes in claims and valuations actually occurred.
Consider a paper household beginning with financial wealth of 100,000 currency units. Assume a certain annual return of 5%, no taxes, and no withdrawals. After one year it has 105,000. After a second year it has 110,250, because the second return applies to the first year's interest as well as the original principal. The two-year gain is 10,250. A second household beginning with 10,000 under identical conditions ends with 11,025. Both wealth stocks grow by 10.25%, but the first gains 9,225 more currency units. Equal percentage returns preserve their ten-to-one ratio while widening the absolute difference.
Now add saving from labor income. Suppose the second household saves 2,000 at each year end, while the first makes no additional deposits. After year one, the second has 12,500, its 10,500 after interest plus the new 2,000. After year two, it has 15,125, because 12,500 earns 625 and receives another 2,000. The richer household still has 110,250. Its wealth is now about 7.29 times the second household's, rather than ten times. Relative concentration has fallen in this two-household case even though the absolute wealth difference rises from 90,000 to 95,125. Ratios and differences describe different changes.
These calculations do not forecast actual returns or feasible saving. A family facing an emergency expense may withdraw funds instead. Borrowing rates may exceed deposit returns. Risky assets can lose value, and taxes or fees can reduce receipts. Access to collateral, stable employment, and insurance affects which risks a household can take. The model explains why initial holdings matter under specified accumulation rules; it does not establish that every rich household stays rich or that low wealth reveals impatience. The ability to save depends on resources after necessary expenses as well as preferences.
Market incomes also depend on bargaining and ownership rules before any transfer is paid. If a firm's additional receipts go to wages, workers' market incomes rise; if they go to dividends, the distribution follows share ownership. Which occurs depends on contracts, competition, and institutions, not a national-accounting identity. An observed association between education and earnings likewise does not by itself isolate the causal return to education. Family resources and access to jobs may affect both. A distributional explanation must connect a mechanism to suitable evidence instead of using a correlation or a compounding example as a complete history.
Tax revenue 0; equal rebate 0 each. Total disposable income 300. Mean 50; median 35; top household share 46.67%.
| Measure | Six-household market-income example |
|---|---|
| Total income | 300 |
| Mean income | 50 |
| Median income | 35 |
| Top household share | 46.67% |
The experiment fixes six household identities and market incomes of 10, 20, 30, 40, 60, and 140 currency units per period. Their sum is 300. Selecting a preset changes only the tax and rebate rule, not earnings, prices, or population. The blue market-income bars remain available beside the disposable-income bars so each household's change can be followed. There is no income slider or accumulating history. Returning to No redistribution restores disposable incomes to the fixed market amounts. This is a comparison of accounting scenarios, not a transition through successive policy years.
Select 20% flat tax and equal rebate. The household tax payments are 2, 4, 6, 8, 12, and 28. Adding them gives revenue of 60. Dividing by six gives a rebate of 10 to every household, including those paying more than 10. Disposable income means market income minus that household's tax plus its rebate. The resulting amounts are 18, 26, 34, 42, 58, and 122. For example, H5 keeps 48 after its tax of 12 and then receives 10, ending at 58. Its net loss is 2 even though it receives a rebate.
The flat-tax preset makes the break-even point visible. A household with income 50 would pay 10 and receive 10, leaving it unchanged. Below 50, the rebate exceeds the tax; above 50, tax exceeds the rebate. This is a consequence of the particular balanced rule and sample mean, not a general property of every tax system. The six disposable amounts still sum to 300. Their median is the average of 34 and 42, or 38. The top share is 122 divided by 300, about 40.67%. A constant mean of 50 therefore coexists with a higher median and a lower top share.
Next select 40% tax above 50 and equal rebate. Only the part of an income exceeding 50 is taxable. H5 has 10 above the threshold and pays 4. H6 has 90 above it and pays 36. The other households pay zero, giving total revenue of 40. Dividing by six gives a rebate of 6 and two thirds each. Disposable incomes are approximately 16.67, 26.67, 36.67, 46.67, 62.67, and 110.67. H5 gains about 2.67 in this scenario, unlike its loss of 2 under the flat-tax preset. Rounding displayed amounts does not change the exact balanced budget.
For this second tax rule, the median is about 41.67 and the top share about 36.89%, again with total income 300. Its stated 40% rate is a marginal rate on income above the threshold, not 40% of every household's entire income. H6 pays 36 out of 140, an average tax rate of about 25.71% before its rebate. These distinctions explain why reading the headline rate alone cannot reproduce the diagram. Compare household identities, the tax base, and the revenue budget together. The interface calculates no work response or wealth change, so neither can be inferred from the changed bars.
The lab levies a specified tax on six households and rebates all revenue equally. Total disposable income is unchanged by construction; mean income therefore stays 50. Individual amounts, the median, and the top share change. The diagrams keep market and disposable income visible together so a total cannot conceal who gains or loses.
This is a first-round accounting exercise. It omits responses in work, saving, prices, tax avoidance, migration, administration, and future growth. Those responses need evidence rather than an assumption that they are always zero or always overwhelming. Distributional judgments also require explicit values: concern for poverty, opportunity, security, equality, or rights may lead to different evaluations of the same measured outcome.
The experiment's conservation of total income follows from rebating every unit collected while holding market incomes fixed. Summing household taxes subtracts revenue from households, and summing their rebates adds exactly the same amount back. That equality is an accounting identity within the selected boundary. Whether a real reform leaves production, taxable income, or prices unchanged is a different question. A forecast needs a comparison of behavior with and without the policy. It cannot borrow the experiment's fixed-income assumption as empirical proof that no response exists, or assume that any response must eliminate redistribution.
A separate paper budget shows what administration changes. Keep the six market incomes and the flat tax's revenue of 60, but suppose operating the program costs 6 during the period. If the remaining 54 is rebated equally, each household receives 9. Their disposable incomes become 17, 25, 33, 41, 57, and 121. These sum to 294, with mean 49. The missing 6 has not vanished from every account in the economy. It paid for administration outside this household cash-income boundary. Whether those administrative resources are worth using requires comparing the service and alternative uses, not simply labeling all program spending a household transfer.
Cash measures also omit services and compulsory expenses. Imagine two households each with disposable cash income of 100 per week. One must pay 25 for medicine that the other receives through a public service. After that bill, the first has 75 for other purchases and the second has 100. Equal cash income has not produced equal command over remaining goods. Valuing the service at its budget cost could support a broader resource measure, but cost need not equal the recipient's benefit. A service's availability, quality, and suitability matter. State which measure is being compared before concluding that resources are equal.
Price changes complicate interpretation further. A cash transfer of 10 raises a household's nominal resources by 10, but its gain in purchasing power depends on what it can buy. Suppose, purely on paper, a household initially spends 40 on rent and 60 on other necessities. After receiving 10, it faces rent of 48 with other prices unchanged. Only 2 of the transfer remains after buying its old bundle. This arithmetic does not establish that the transfer caused the rent increase. Establishing that would require evidence on demand, housing supply, contracts, and the counterfactual rent. Legal receipt of a payment and ultimate economic incidence are separate questions.
Normative evaluation comes after these definitions and mechanisms. One evaluator may prioritize lifting resources below a specified poverty threshold. Another may emphasize insurance against disability or protection of established property claims. These criteria can rank the same distribution differently without disagreeing about the sums. They still cannot justify inaccurate arithmetic or unsupported causal claims. To assess an actual reform, specify who pays, who receives cash or services, how those entries affect the government budget, and which behavioral effects have evidence. Then state the value judgment being applied rather than presenting a chosen inequality statistic as an automatic policy verdict.
Incomes are 20, 30, and 100. The top income rises to 160 while the others stay fixed. What happens to mean and median?
Recompute the total and identify the middle observation.
| Measure | Before | After |
|---|---|---|
| Total | 150 | 210 |
| Mean | 50 | 70 |
| Median | 30 | 30 |
The mean rises from 50 to 70. The median stays 30. A rise in average income does not show that the typical middle observation became better off, though it does show a larger total in this sample.
Two equal-sized households earn 40 and 160. A tax of 20 on the second is transferred to the first. Find both disposable incomes and the mean. Is this a complete forecast of the policy?
Separate the immediate entries from behavioral responses.
| Household | Market income | Net transfer | Disposable income |
|---|---|---|---|
| First | 40 | +20 | 60 |
| Second | 160 | −20 | 140 |
Disposable incomes become 60 and 140. The mean remains 100 because total income stays 200. This describes the stated first-round transfer only. A forecast needs assumptions and evidence about behavioral and economic responses.
The lessons and answers stand alone. These optional references identify underlying models, evidence, and perspectives. All numerical examples here are invented unless explicitly attributed; none is a forecast or a personal investment recommendation.
Wealth, earnings, disposable income, redistribution, and policies shaping market outcomes. Our six-household sample is invented and equal-sized.