## 1. Maximum Acceptable Leakage, Inequality Aversion, and Ratio of Incomes

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### I. Introduction and purpose
- Public discourse on inequality motivates monetary indicators combining average incomes and their distribution to assess welfare and rank policies.
- The primer revisits Okun’s “leaky bucket” and Atkinson’s “equally-distributed-equivalent income” (EDEI) as tools to elicit social preferences over equality versus efficiency and to summarize societal welfare.
- Empirical illustrations use recent income distribution data for a panel of countries and a calibrated general equilibrium exercise to analyze a hypothetical tax-and-transfer scheme.

### II. Okun’s “Leaky Bucket” (eliciting maximum acceptable leakage)
- Setup (Okun, 1975) example:
  - Five families: one rich with $45,000 and four poor with $5,000 each.
  - Tax $4,000 from the rich; with a leak b (0≤b≤1), each poor family receives (1−b)×$1,000.
- Interpretation of b:
  - b represents administrative costs, potential corruption, and reduced work incentives.
- Two-family general formulation:
  - Initial incomes I_P (poor) and I_R (rich).
  - After a transfer T: I_P + (1−b)T and I_R − T.
  - Observers with welfare weights w(I) prefer the scheme if:
    - w(I_P +(1−b)T) + w(I_R − T) ≥ w(I_P) + w(I_R).
  - For infinitesimal transfers, desirability condition:
    - (1−b) w′(I_P) ≥ w′(I_R).
- Okun’s personal tolerance: maximum acceptable leakage of no more than 60 percent for his five-family example.

### III. Atkinson’s functional form for inequality aversion
- Atkinson (1970) household welfare function:
  - w(I) = I^(1−ε) / (1−ε), where I is disposable income and ε is the coefficient of aversion to inequality.
- Properties:
  - Additive separability and homotheticity imply aversion to inequality depends only on income distribution; proportional increases in every household’s income affect welfare only through mean income.
  - Special case ε = 1 yields w(I) = log(I).
- Relation to Okun:
  - Okun’s indifference condition solved exactly for his five-family numbers yields ε = 0.4338.
  - Marginal-transfer approximation with functional form yields:
    - (I_R / I_P)^ε = 1 / (1 − b).
  - Applied to Okun’s example (I_R / I_P = 9, b = 0.6) gives ε = 0.42 (approximation; Blinder, 1982).

### IV. Inferring ε from choices or inferring acceptable leakage b from assumed ε
- Presenting observers with an assumed leakage b and asking whether transfers are desirable allows inference of their ε.
- Conversely, assuming a value for ε allows inference of the maximum acceptable leakage b that makes a marginal transfer just desirable.

### V. Table of maximum acceptable leakage by income ratio and ε (Table 1) — exact entries
- Table reports maximum acceptable share of the amount transferred from the rich household that leaks out before reaching the poor household. I_R / I_P is initial income ratio. ε is coefficient of aversion to inequality.
- I_R / I_P = 2, 3, 4, 5, 10, 25
  - ε = 0.2 → 0.13, 0.20, 0.24, 0.28, 0.37, 0.47
  - ε = 0.5 → 0.29, 0.42, 0.50, 0.55, 0.68, 0.80
  - ε = 1.5 → 0.65, 0.81, 0.88, 0.91, 0.97, 0.99
  - ε = 2.0 → 0.75, 0.89, 0.94, 0.96, 0.99, 0.998

### VI. Empirical ranges and literature
- Studies typically find ε in the range of 0.2–2.0 (examples: Amiel and Cowell 1999; Pirttila and Uusitalo 2010).
- Pirttila and Uusitalo (2010) surveyed 3,000 Finnish adults (split into four subgroups).

### Leaky-bucket questionnaire results and interpretation (Pirttila and Uusitalo subgroup experiment)
- Each subgroup received a questionnaire asking whether they would support a transfer from the top income decile earners (>€3300 per month) to the bottom decile earners (<€800 per month).
- Each questionnaire had different leakage: b=0.5, 0.75, 0.94, and 0.985, corresponding to ε = 0.5, 1.0, 2.0, and 3.0, respectively.
- Respondent options: “yes”, “no”, or “cannot say”.
- Findings (exact reported observations):
  - Support for the transfer decreased with higher leakage.
  - Even with ε=0.5 (b=50%), less than 30 percent of respondents were in favor.
  - Median inequality aversion of respondents inferred to lie below 0.5.
  - More than 20 percent of respondents favored the transfer for leakage coefficient corresponding to ε as high as 3.

### Comparative evidence from literature — selected findings (exact reported values)
- Harberger (1983):
  - Uses top/bottom decile income ratio nine-to-one and ε = 0.5 to estimate ratio of marginal effects three-to-one from (5).
  - Concludes accepted ε must be less than 0.5 given limited redistribution observed.
- Hendren (2013):
  - Finds additional redistribution from households earning above $400,000 to EITC beneficiaries is desirable for leakage coefficients between 34–56 percent.
- Bargain and others (2014):
  - Measured revealed aversion to inequality for 17 European countries and the United States:
    - Inequality aversion below one for most Southern European countries and the United States.
    - Around one for Continental Europe, the United Kingdom, Ireland and Finland.
    - Between one and three for Belgium and the Scandinavian countries.
- Pirttila and Uusitalo (2010) (alternative method):
  - Presenting respondents with hypothetical income distributions led most to choose the least unequal distribution, yielding a median ε of three or above.

### Atkinson’s Equally-Distributed-Equivalent Income (EDEI): concept and formulas
- Atkinson’s inequality index, A:
  - A = (μ − Ī) / μ = 1 − Ī / μ, where μ is mean income and Ī is EDEI.
  - A varies between 0 (perfect equality) and 1 (most unequal).
- EDEI definition for n households (j=1,..,n):
  - w(Ī) = (1/n) Σ_{j=1}^{n} w(I_j).
- Two-household concavity condition:
  - 2 * w(Ī) = w(I_P) + w(I_R).
- Interpretations:
  - More concave w(I) implies higher inequality aversion.
  - Linear w(I) corresponds to ε = 0 (utilitarian).
  - Rawlsian extreme corresponds to infinite inequality aversion.

### Numerical two-family example and Table 2 — exact numbers
- Two-family example: I_R = $45,000; I_P = $5,000. Mean μ = $25,000; total initial income = $50,000.
- For ε = 0.5:
  - EDEI Ī = $20,000.
  - Total income after equalization = $40,000.
  - Observer would accept a loss of 20 percent in total income to equalize incomes.
  - Tax-and-transfer interpretation: implies a 40 percent leakage — rich taxed $25,000, poor receives $15,000.
- Table 2 entries (exact):
  - ε 0.2 0.5 0.9 1.5 2.0
  - Ī 23,084 20,000 15,931 11,250 9,000
- Leaky-bucket transfer solvability example (exact numbers):
  - With I_R = $45,000, I_P = $5,000, b=0.6, maximum gross transfer T that equalizes incomes solving I_R − T = I_P + (1 − b)T is T = $28,571.
  - Observers would agree to a small transfer ($1–100) if ε ≥ 0.42, and to a gross transfer of $1,000 from the rich if ε ≥ 0.43.
- Relationship: larger transfers require higher ε to be acceptable given fixed leakage (example shown for leakage = 60%).

### Relationship between Okun’s leaky-bucket and Atkinson’s EDEI
- Distinction and linkage:
  - Okun’s exercises: ask observers whether, for a given leakage coefficient, a tax-and-transfer scheme is acceptable.
  - Atkinson’s EDEI: requires only the observer’s ε, not assumptions about leakage.
  - Once EDEI estimated, one can infer the leakage that would be associated with a tax-and-transfer scheme to move from initial situation to perfect equality.
  - If asked whether to equalize incomes, Atkinson’s exercise becomes a special case of Okun’s leaky-bucket.
- Appendix B derives relationships between Atkinson’s index, transfer size required to equalize incomes, leakiness b, and ε.
- Optimal-income-taxation interpretation: in leaky-bucket, dollar value (in public funds) of an extra dollar of consumption equals one for the poor and 1 − b for the rich (Appendix C).

### Empirical application context (lead into panel analysis)
- EDEI estimated for a sample of 139 countries in 2015 using Hellebrandt and Mauro (2016) dataset.
- EDEI measured in international U.S. dollars (purchasing power parity).
- A common ε assumed for all countries in each exercise to facilitate cross-country comparisons.
- Data limitation: most country data do not detail income distribution within households; analysis focuses on household incomes divided by household size.

### Data sources and measurement issues (Box 1 continued)
- Primary survey source: Luxembourg Income Study (LIS) — incomes reported net of income taxes and include monetary transfers; exclude in-kind transfers, subsidized goods and services, or public health or education provision.
- Supplementary source: World Bank database for countries not covered by LIS (largely consumption surveys).
- Measurement gaps:
  - Survey means often lower than per capita GDP or national accounts consumption; example: disposable personal income was 72 percent of GDP in 2015 for the United States.
  - HFCE includes imputed rent from owner-occupied housing, rarely estimated in surveys.
  - Survey errors: under-reporting (especially self-employed), under-sampling of high-income households.
  - Hellebrandt and Mauro (2016) propose adjustments; primer uses unadjusted data (inequality and EDEI would be somewhat stronger with adjustments).

### EDEI versus mean income — empirical findings (exact reported numbers and patterns)
- EDEI computation for charts uses ε = 2.0 to emphasize inequality’s role.
- Strong positive association between EDEI and mean income even with ε = 2.0.
- Rankings based on EDEI often differ substantially from mean-income rankings; rank reversals depend on ε.
- Sen index provides similar results; highest rank correlation between Sen and Atkinson’s EDEI occurs when ε = 1.5.

### Table 3: Equally-Distributed-Equivalent Income for Different ε, 2015 — exact entries (selected countries)
- Note: ε denotes coefficient of aversion to inequality. EDEI equals mean income when ε = 0.
- United States
  - Mean Income: 24,128
  - EDEI ε =0.2: 22,781
  - EDEI ε =0.5: 20,880
  - EDEI ε =1.5: 14,404
  - EDEI ε =2.0: 7,985
- Sweden
  - Mean Income: 18,495
  - EDEI ε =0.2: 18,074
  - EDEI ε =0.5: 17,470
  - EDEI ε =1.5: 15,478
  - EDEI ε =2.0: 14,102
- United Kingdom
  - Mean Income: 18,176
  - EDEI ε =0.2: 17,419
  - EDEI ε =0.5: 16,373
  - EDEI ε =1.5: 13,141
  - EDEI ε =2.0: 10,623
- South Africa
  - Mean Income: 4,366
  - EDEI ε =0.2: 3,679
  - EDEI ε =0.5: 2,848
  - EDEI ε =1.5: 1,323
  - EDEI ε =2.0: 931
- Albania
  - Mean Income: 2,835
  - EDEI ε =0.2: 2,757
  - EDEI ε =0.5: 2,646
  - EDEI ε =1.5: 2,305
  - EDEI ε =2.0: 2,151
- Kyrgyz Republic
  - Mean Income: 1,994
  - EDEI ε =0.2: 1,946
  - EDEI ε =0.5: 1,876
  - EDEI ε =1.5: 1,659
  - EDEI ε =2.0: 1,561

### Rank reversals and sensitivity to ε (Table 4) — exact counts and percentages (sample = 139 countries)
- Number of countries with rank reversals (decline in rank by two or more positions when ranked by EDEI versus mean income):
  - ε =0.2: 18 countries
  - ε =0.5: 35 countries
  - ε =1.5: 69 countries
  - ε =2.0: 71 countries
- As percent of sample:
  - ε =0.2: 12.9
  - ε =0.5: 25.2
  - ε =1.5: 49.6
  - ε =2.0: 51.1
- Interpretation: rank reversals by two or more positions increase with higher ε.

### Decomposition of changes in EDEI (2003–15) — exact decomposition formula and empirical finding
- Decomposition equation: ∆ln(Ī) = ∆ln(μ) + ∆ln(1−A), using ε = 2.0.
- Finding: most variation in EDEI over 2003–15 is accounted for by changes in mean incomes even with ε = 2.0.
- Using lower ε yields qualitatively similar results with an even smaller role for changes in inequality.

### Policy assessment — Example: Universal Basic Income (UBI) scenarios (United States, cumulative equivalent to 1 percent of GDP)
- Modeling framework: general equilibrium model applied to households (short- to medium-term horizon up to 5 years); EDEI refers to equally-distributed-equivalent consumption for fair comparison between VAT and PIT financing.
- Financing modalities analyzed:
  - (a) Cuts of spending assumed to be wholly wasteful.
  - (b) Increase in personal income taxation (PIT), assumed to become more progressive with greater reliance.
  - (c) Introduction of a value-added tax (VAT).
- Key model findings (exact reported thresholds and qualitative relations):
  - Gains in EDEI from UBI increase with higher ε.
  - Wasteful spending cuts financing: welfare increases regardless of ε; improvement larger at higher ε. Substituting UBI for inequitable and inefficient programs yields net gains; cutting productive government spending can yield net losses.
  - PIT versus VAT financing:
    - PIT is more desirable for distribution; PIT imposes greater efficiency costs lowering total output relative to VAT financing.
    - PIT financing preferable when observers are highly averse to inequality; VAT financing does comparatively better for lower ε.
    - For low ε, efficiency costs dominate and net EDEI changes can be negative.
    - Thresholds reported:
      - For ε > 1.4, UBI financed by PIT provides net gains in EDEI.
      - UBI financed by VAT requires ε > 2.4 to provide small net gains.

### Modeling details (Box 2 summary)
- Model features (exact framework elements):
  - Households divided by predetermined types (education level); idiosyncratic productivity shocks create within-type heterogeneity.
  - Three sectors: manufacturing, skilled services, low-skilled services.
  - Competitive markets, international trade at exogenous world prices, closed capital markets except government external debt schedule exogenous.
  - Horizon up to 5 years: skill levels fixed; low-skill cannot work in high-skill services.
  - Households borrow/save via a non-state-contingent bond with exogenous borrowing constraints varying by skill level.
  - Stationary equilibrium: households maximize lifetime expected utility; firms maximize profits; markets clear; government budget balances.

### Policy implications and conclusion (exact summary points)
- Monetary indicators combining total income and distribution, such as EDEI, provide intuitive tools summarizing efficiency and equity implications for policymaking.
- Over multi-decade periods, economic growth (mean income) plays a greater role than changes in income distribution in determining EDEI evolution, though distribution remains significant.
- Public policies (taxes and transfers) substantially affect inequality and EDEI; in advanced economies, direct taxes and transfers reduce income inequality on average by about one third, with approximately three-quarters of this fiscal redistribution achieved on the transfer side of the budget (IMF, 2017).
- Policy choices depend importantly on the degree of aversion to inequality; analysis using EDEI and related methods should be more widely applied in policymaking.

### Appendix B: Relating Atkinson’s Inequality Index to Okun’s Leaky Bucket — exact relations (two-family case)
- Definitions:
  - μ ≡ 1/2 (I_P + I_R)  (B.1)
  - EDEI 퐼̅ satisfies w(퐼̅) = (1/2) [ w(I_P) + w(I_R) ]  (B.2)
  - Deviations: 훾_R ≡ (I_R − 퐼̅) / 퐼̅  and  훾_P ≡ (퐼̅ − I_P) / 퐼̅  (B.3)
- Mean and deviations:
  - μ = 퐼̅ + 1/2 [ (I_R − 퐼̅) − (퐼̅ − I_P) ]  (B.4)
  - μ = 퐼̅ [ 1 + 1/2 ( 훾_R − 훾_P ) ]  (B.5)
- Leaky-bucket specification:
  - 퐼̅ − I_P = (1 − b) (I_R − 퐼̅)  (B.6)
  - 훾_P = (1 − b) 훾_R  (B.7)
- Atkinson index A expressed with leaky-bucket variables:
  - A = 1 − 퐼̅ / μ  (B.8)
  - A = 1 − 1 / [ 1 + 1/2 b 훾_R ]  (B.10)
  - A = 1 − 1 / [ 1 + 1/2 ( b / (1 − b) ) 훾_P ]  (B.11)
- Interpretation: A depends on observer’s inequality aversion (implicit in 훾_R or 훾_P) and leakage b associated with the tax-and-transfer that equalizes incomes at 퐼̅.
- Extension to n households: aggregate below and above 퐼̅ into I_P and I_R respectively and apply two-family analysis.

### Appendix C: Relating Okun’s leaky bucket exercise to optimal income taxation — exact setup and first-order conditions
- Objective: Max w(I_P + S_P) + w(I_R − T_R), subject to R_0 + (1 − b) T_R ≥ S_P.
- Lagrangean:
  - L = w(I_P + S_P) + w(I_R − T_R) + λ [ R_0 + (1 − b) T_R − S_P ].
- First-order conditions:
  - w′(I_P + S_P) = λ.
  - w′(I_R − T_R) = λ (1 − b).
- Implication for optimal taxation:
  - w′(I_R − T_R) = (1 − b) w′(I_P + S_P).
  - With b = 0 government equalizes incomes; with b > 0 the post-tax income of the rich exceeds the poor’s because transfers are lossy; the optimal income gap widens with larger b.
- Interpretation of λ: extra welfare from an additional dollar of R_0 given to government for optimal allocation.
- Dollar-value interpretation (public funds):
  - Extra dollar to poor: dollar value equals one.
  - Extra dollar to rich: dollar value equals 1 − b.
  - Rationale: extra dollar to government would be allocated to poor raising welfare by w′(I_P + S_P); giving it to rich raises welfare by w′(I_R − T_R) = (1 − b) w′(I_P + S_P).

*Source: wp17214 (IMF staff primer excerpts).*

### 1. Maximum Acceptable Leakage, Inequality Aversion, and Ratio of Incomes ........................ 6

### 1. Maximum Acceptable Leakage, Inequality Aversion, and Ratio of Incomes

### I. Introduction and purpose
- Public discourse on inequality has intensified, motivating the use of monetary indicators that combine average incomes and their distribution to assess welfare and rank policies.
- The primer revisits Okun’s “leaky bucket” and Atkinson’s “equally-distributed-equivalent income” (EDEI) as simple, intuitive tools to elicit social preferences over equality versus efficiency and to summarize societal welfare.
- Empirical illustrations use recent income distribution data for a panel of countries and a calibrated general equilibrium exercise to analyze a hypothetical tax-and-transfer scheme.

### II. Okun’s “Leaky Bucket” thought experiment (eliciting maximum acceptable leakage)
- Setup (Okun, 1975): consider five families—one rich with $45,000 and four poor with $5,000 each. Tax $4,000 from the rich and transfer proceeds to the poor; with a leak b (0≤b≤1), each poor family would receive (1−b)×$1,000.
- The leak b represents administrative costs, potential corruption, and reduced economic incentives to work.
- General two-family formulation: initial incomes I_P (poor) and I_R (rich). A transfer T leaves incomes I_P + (1−b)T and I_R − T. Observers with welfare weights w(I) prefer the scheme if:
  - w(I_P +(1−b)T) + w(I_R − T) ≥ w(I_P) + w(I_R).
- For infinitesimal transfers, desirability condition becomes:
  - (1−b) w′(I_P) ≥ w′(I_R).

- Okun’s personal stated tolerance: maximum acceptable leakage of no more than 60 percent for his five-family example.

### III. Atkinson’s functional form for inequality aversion
- Atkinson (1970) proposes the household welfare function:
  - w(I) = I^(1−ε) / (1−ε),
  - where I is disposable income and ε is the coefficient of aversion to inequality.
- Properties:
  - Additive separability and homotheticity imply aversion to inequality depends only on income distribution; proportional increases in every household’s income only affect welfare through mean income.
  - Special case ε = 1 yields w(I) = log(I).
- Relation to Okun’s exercise:
  - Okun’s stated indifference condition (using his five-family numbers and ε unknown) yields ε = 0.4338 when solved exactly.
  - Using the marginal-transfer approximation and the functional form leads to:
    - (I_R / I_P)^ε = 1 / (1 − b).
  - Applied to Okun’s example (I_R / I_P = 9, b = 0.6) gives ε = 0.42 as a good approximation (Blinder, 1982).

### IV. Inferring ε from choices or inferring acceptable leakage b from assumed ε
- If observers are presented with an assumed leakage coefficient b and asked whether transfers are desirable, one can infer their coefficient ε.
- Conversely, assuming a value for ε allows inference of the maximum acceptable leakage b that makes a marginal transfer just desirable.

### V. Table of maximum acceptable leakage by income ratio and ε (Table 1)
- Note: table reports the maximum acceptable share of the amount transferred from the rich household that leaks out before reaching the poor household. I_R / I_P is the initial income of the rich household divided by the initial income of the poor household. The coefficient of aversion to inequality is ε.

- I_R / I_P = 2, 3, 4, 5, 10, 25
  - ε = 0.2 → 0.13, 0.20, 0.24, 0.28, 0.37, 0.47
  - ε = 0.5 → 0.29, 0.42, 0.50, 0.55, 0.68, 0.80
  - ε = 1.5 → 0.65, 0.81, 0.88, 0.91, 0.97, 0.99
  - ε = 2.0 → 0.75, 0.89, 0.94, 0.96, 0.99, 0.998

### VI. Empirical ranges and literature
- Several studies (e.g., Amiel and Cowell 1999; Pirttila and Uusitalo 2010) have elicited attitudes toward inequality, usually finding ε in the range of 0.2–2.0.
- Example: Pirttila and Uusitalo (2010) surveyed 3,000 Finnish adults (split into four subgroups) — study cited as evidence of typical ε ranges.

*Source: IMF Staff Primer on welfare measurement — "Maximum Acceptable Leakage, Inequality Aversion, and Ratio of Incomes" (excerpts).*

### 750. Each subgroup received a questionnaire asking respondents whether they would support

### wp17214 - 750. Each subgroup received a questionnaire asking respondents whether they would support

### Leaky-bucket questionnaire results and interpretation
- Each subgroup received a questionnaire asking respondents whether they would support a transfer from the top income decile earners (>€3300 per month) to the bottom decile earners (<€800 per month).
- Each questionnaire had different leakage: b=0.5, 0.75, 0.94, and 0.985, corresponding to inequality aversion ε = 0.5, 1.0, 2.0, and 3.0, respectively—see (5).
- Respondents could answer “yes”, “no”, or “cannot say” to the transfer.
- Findings:
  - Support for the transfer decreased with higher leakage.
  - Even with the lowest aversion to inequality and least leakage (ε=0.5, b=50%), less than 30 percent of respondents were in favor of the transfer.
  - This suggests that the median inequality aversion of the respondents lies below 0.5.
  - More than 20 percent of respondents were in favor of the transfer for a leakage coefficient corresponding to inequality aversion as high as 3.

### Comparative evidence from literature
- Harberger (1983):
  - Combines Okun’s leaky bucket experiment with Atkinson’s social welfare function to argue that aversion to inequality in the U.S. is relatively low.
  - Using the ratio of average incomes of top and bottom deciles (nine-to-one) and setting the coefficient of inequality aversion to 0.5, he estimates the ratio of marginal effects to be three-to-one from (5).
  - Interprets this as society believing an additional dollar is worth three times as much to the poorest as to the richest, yet society stops redistributing at much larger disparities — implying the accepted coefficient of inequality aversion must be less than 0.5.
- Hendren (2013):
  - Analyzes earned income tax credit (EITC) expansions and the top marginal income tax schedule in the U.S.
  - Finds additional redistribution from rich households earning above $400,000 to EITC beneficiaries is desirable for the leakage coefficient ranging between 34–56 percent.
  - Contrasts with Harberger (1983), who thought the leak should be much lower given contemporaneous estimates of marginal dead weight loss near 10 cents per dollar and small administrative/compliance costs.
- Other approaches and cross-country estimates:
  - Pirttila and Uusitalo (2010): presenting respondents with hypothetical income distributions led most to choose the least unequal distribution, yielding a median inequality aversion coefficient of three or above.
  - Bargain and others (2014): measured revealed aversion to inequality for 17 European countries and the United States by analyzing implicit redistribution in tax-and-benefit systems (controlling for labor supply elasticities). Findings:
    - Inequality aversion below one for most Southern European countries and the United States.
    - Around one for Continental Europe, the United Kingdom, Ireland and Finland.
    - Ranging between one and three for Belgium and the Scandinavian countries.
  - Reasons for differing coefficients across approaches may include respondents’ inconsistencies, political systems’ inability to deliver policies aligned with citizens’ preferences, or overly stringent assumptions regarding the shape of the social welfare function.

### Atkinson’s Equally-Distributed-Equivalent Income (EDEI): concept and formulas
- Atkinson (1970) defines the equally-distributed-equivalent income (EDEI), the income that an external observer would consider just as desirable as the existing income distribution.
- Atkinson’s inequality index, A, is defined as:
  - A = (μ − Ī) / μ = 1 − Ī / μ
  - where μ is mean income and Ī is the EDEI.
  - Interpretation: if Ī is 80 percent of mean income, the observer would be willing to give up 20 percent of total societal income in exchange for equal incomes at level Ī. The index varies between 0 (perfect equality) and 1 (most unequal).
- For n households (j=1,..,n), the EDEI is obtained from:
  - w(Ī) = (1/n) Σ_{j=1}^{n} w(I_j)
- Concavity of w(I) implies w(μ) > (1/2)[w(I_P) + w(I_R)] for two-household case; EDEI Ī satisfies:
  - 2 * w(Ī) = w(I_P) + w(I_R)   (equation (6) / (9) in the text)
- Interpretations:
  - The more concave w(I), the higher inequality aversion.
  - If w(I) is linear (utilitarian), inequality aversion is zero.
  - At the opposite extreme (Rawlsian), social well-being assessed solely on the income of the poorer household corresponds to infinite inequality aversion.

### Numerical example and Table 2 (preserving exact numbers)
- Two-family example: I_R = $45,000 (rich), I_P = $5,000 (poor). Mean income μ = $25,000; total initial income = $50,000.
- Assuming constant relative inequality aversion functional form and ε=0.5:
  - EDEI Ī = $20,000.
  - Total income after equalization = $40,000.
  - From the external observer’s perspective, a loss of 20 percent in total income would be acceptable to equalize incomes.
  - If accomplished through a tax-and-transfer with Okun’s terminology, this implies a 40 percent leakage: the rich family is taxed $25,000, the poor family receives $15,000.
- Table 2. EDEI and Coefficient of Inequality Aversion (exact entries):
  - ε 0.2 0.5 0.9 1.5 2.0
  - Ī 23,084 20,000 15,931 11,250 9,000
  - Note: Assuming initial incomes of $45,000 for the rich and $5,000 for the poor.
- Leaky-bucket transfer solvability example (exact numbers preserved):
  - With I_R = $45,000, I_P = $5,000, and leakage coefficient b=0.6, the maximum gross transfer T that equalizes incomes (solving I_R − T = I_P + (1 − b)T) is T = $28,571.
  - Observers would agree to a small transfer ($1–100) if ε ≥ 0.42, and to a (gross) transfer of $1,000 from the rich if ε ≥ 0.43. Larger transfers require larger ε.
- Relationship depicted: larger transfers require higher inequality aversion ε to be acceptable, given fixed leakage (example shown for leakage = 60%).

### Relationship between Okun’s leaky-bucket exercise and Atkinson’s EDEI
- Distinction and linkage:
  - Okun’s exercises: ask observers whether, for a given leakage coefficient, a tax-and-transfer scheme (often not equalizing incomes fully) is acceptable.
  - Atkinson’s EDEI exercise: requires only the external observer’s coefficient of aversion to inequality, not assumptions about leakage coefficients.
  - Once EDEI is estimated, one can infer the leakage that would be associated with a tax-and-transfer scheme to move from initial situation to perfect equality.
  - If the observer were asked whether to proceed with a tax-and-transfer to equalize incomes, Atkinson’s exercise becomes a special case of Okun’s leaky-bucket exercise.
- Appendix B (as referenced) derives the relationship between Atkinson’s index, the size of the transfer required to equalize incomes, the leakiness of the bucket, and the observer’s aversion to inequality ε.
- Optimal-income-taxation interpretation:
  - In the leaky-bucket setup, the dollar value (in terms of public funds) of an extra dollar of consumption is equal to one for the poor, and to 1 − b for the rich (Appendix C).

### Empirical application context (lead into panel analysis)
- The concepts introduced are applied to:
  - (a) assessing economic welfare in a panel of countries and
  - (b) assessing policies such as a tax-and-transfer scheme.
- Example of empirical scope:
  - EDEI is estimated for a sample of 139 countries in 2015 using the dataset of Hellebrandt and Mauro (2016).
  - EDEI is measured in international U.S. dollars (purchasing power parity).
  - A common coefficient of aversion to inequality is assumed for all countries in each exercise to facilitate cross-country comparisons.
  - Note on data limitation: available internationally-comparable data for most countries do not provide detail regarding income distribution within households; analysis focuses on household incomes divided by the number of people in the household.

*Source: wp17214 (excerpt provided).*

### Box 1: Income Distribution – Data from Household Surveys (continued)

### wp17214 - Box 1: Income Distribution – Data from Household Surveys (continued)

### Data sources and measurement issues
- Primary survey source: Luxembourg Income Study (LIS) — harmonized income survey data for many middle- and high-income countries; incomes reported net of income taxes and include monetary transfers (but not in-kind transfers, subsidized goods and services, or public health or education provision).
- Supplementary source: World Bank database for countries not covered by LIS (largely consumption surveys); mixing of consumption and income surveys is used despite imperfections.
- Key measurement gaps and sources of divergence:
  - Mean incomes or consumption from household surveys often differ, usually lower than, per capita GDP or consumption from national accounts (gap varies across countries).
  - GDP includes depreciation, retained earnings of corporations, and government revenue not redistributed as cash transfers; example: disposable personal income in the national accounts was 72 percent of GDP in 2015 for the United States.
  - Aggregate household final consumption expenditure (HFCE) includes imputed rent from owner-occupied housing, rarely estimated in surveys.
  - Survey measurement errors: under-reporting (especially self-employed fearing self-incrimination), under-sampling of high-income households (lower response rates).
  - Hellebrandt and Mauro (2016) propose adjustments for under-reporting and under-sampling, but data in this primer are unadjusted; inequality and its role in EDEI would be somewhat stronger with adjusted data.

### EDEI (Equally-Distributed-Equivalent Income) versus mean income — empirical findings
- EDEI computation for expository charts uses a high coefficient of inequality aversion, ε=2.0, to emphasize inequality’s role and improve legibility.
- Strong positive association between EDEI and mean income even with ε=2.0.
- Rankings based on EDEI often differ substantially from rankings based on mean income; rank reversals depend on the inequality aversion coefficient.
- Sen index provides similar results; highest rank correlation between Sen index and Atkinson’s EDEI occurs when EDEI is computed with ε=1.5 (sample considered).

### Table 3: Equally-Distributed-Equivalent Income for Different Inequality Aversion, 2015 — reported values
- Note: ε denotes the coefficient of aversion to inequality. EDEI equals mean income when ε=0.
- United States
  - Mean Income: 24,128
  - EDEI ε =0.2: 22,781
  - EDEI ε =0.5: 20,880
  - EDEI ε =1.5: 14,404
  - EDEI ε =2.0: 7,985
- Sweden
  - Mean Income: 18,495
  - EDEI ε =0.2: 18,074
  - EDEI ε =0.5: 17,470
  - EDEI ε =1.5: 15,478
  - EDEI ε =2.0: 14,102
- United Kingdom
  - Mean Income: 18,176
  - EDEI ε =0.2: 17,419
  - EDEI ε =0.5: 16,373
  - EDEI ε =1.5: 13,141
  - EDEI ε =2.0: 10,623
- South Africa
  - Mean Income: 4,366
  - EDEI ε =0.2: 3,679
  - EDEI ε =0.5: 2,848
  - EDEI ε =1.5: 1,323
  - EDEI ε =2.0: 931
- Albania
  - Mean Income: 2,835
  - EDEI ε =0.2: 2,757
  - EDEI ε =0.5: 2,646
  - EDEI ε =1.5: 2,305
  - EDEI ε =2.0: 2,151
- Kyrgyz Republic
  - Mean Income: 1,994
  - EDEI ε =0.2: 1,946
  - EDEI ε =0.5: 1,876
  - EDEI ε =1.5: 1,659
  - EDEI ε =2.0: 1,561

### Rank reversals and sensitivity to inequality aversion (Table 4)
- Sample size: 139 countries.
- Number of countries with rank reversals (decline in rank by two or more positions when ranked by EDEI versus mean income):
  - ε =0.2: 18 countries
  - ε =0.5: 35 countries
  - ε =1.5: 69 countries
  - ε =2.0: 71 countries
- As percent of all countries in the sample:
  - ε =0.2: 12.9
  - ε =0.5: 25.2
  - ε =1.5: 49.6
  - ε =2.0: 51.1
- Interpretation: rank reversals by two or more positions tend to increase with the degree of inequality aversion.

### Decomposition of changes in EDEI (2003–15)
- Decomposition equation: ∆ln(Ī) = ∆ln(μ) + ∆ln(1−A), where Ī is EDEI, μ is mean income, and A is the Atkinson index; analysis uses ε=2.0.
- Finding: most variation in EDEI over 2003–15 is accounted for by changes in mean incomes, even when using a high inequality aversion coefficient (2.0).
- Using lower inequality aversion coefficients yields qualitatively similar results with an even smaller role for changes in inequality.
- Corroborates Dollar and others (2015): mean income changes dominate social welfare measures’ variation over time.

### Policy assessment — Example: Universal Basic Income (UBI) scenarios (United States, cumulative equivalent to 1 percent of GDP)
- Modeling framework: general equilibrium model applied to households (short- to medium-term horizon up to 5 years); EDEI here refers to equally-distributed-equivalent consumption for fair comparison between VAT and PIT financing.
- Three financing modalities analyzed:
  - (a) Cuts of spending assumed to be wholly wasteful.
  - (b) Increase in personal income taxation (PIT), assumed to become more progressive with greater reliance.
  - (c) Introduction of a value-added tax (VAT).
- Key model findings:
  - All gains in EDEI from UBI are upward sloping in inequality aversion: benefits of UBI increase with higher ε.
  - Wasteful spending cuts financing: welfare increases regardless of ε; improvement larger at higher ε. If UBI substitutes for inequitable and inefficient programs, net gains occur; cutting productive government spending could yield net losses.
  - PIT versus VAT financing:
    - PIT is more desirable from the perspective of income distribution; PIT imposes greater efficiency costs, lowering total output relative to VAT financing.
    - PIT financing preferable when external observers are highly averse to inequality; VAT financing does comparatively better for lower levels of inequality aversion.
    - For low aversion to inequality, efficiency costs dominate and net EDEI changes can be negative.
    - Thresholds reported by the model:
      - For inequality aversion above 1.4, UBI financed by PIT provides net gains in EDEI.
      - UBI financed by VAT would require inequality aversion in excess of 2.4 to provide small net gains.

### Modeling details (Box 2 summary)
- Model features:
  - Households divided by predetermined types (education level), idiosyncratic productivity shocks create within-type heterogeneity.
  - Three industrial sectors: manufacturing, skilled services, low-skilled services.
  - Competitive markets, international trade at exogenous world prices, closed capital markets except government external debt schedule assumed exogenous.
  - Short-to-medium-term horizon (up to 5 years): skill levels fixed; low-skill cannot work in high-skill services.
  - Households borrow/save only via a non-state-contingent bond with exogenous borrowing constraints varying by skill level.
  - Stationary equilibrium: households maximize lifetime expected utility, firms maximize profits, markets clear, government budget balances.

### Policy implications and conclusion
- Monetary indicators that combine total income and distribution, such as EDEI, provide intuitive tools for policymaking by summarizing both efficiency and equity implications.
- Over multi-decade periods, economic growth (mean income) plays a greater role than changes in income distribution in determining EDEI evolution, but income distribution remains a significant factor.
- Public policies (taxes and transfers) have sizable impacts on inequality and thus on EDEI; in advanced economies, direct taxes and transfers reduce income inequality on average by about one third, with approximately three-quarters of this fiscal redistribution achieved on the transfer side of the budget (IMF, 2017).
- Policy choices depend importantly on the degree of aversion to inequality; analysis using EDEI and related methods should be more widely applied in policymaking.

*Source: wp17214 - Box 1: Income Distribution – Data from Household Surveys (continued), IMF staff compilation from the supplied content.*

### Appendix B. Relating Atkinson’s Inequality Index to Okun’s Leaky Bucket

### Appendix B. Relating Atkinson’s Inequality Index to Okun’s Leaky Bucket

### Two-family case: definitions and setup
- Consider two families with incomes I_R and I_P.
- Define mean initial income, μ, as:
  - 휇 ≡ 1/2 (I_P + I_R)  (B.1)
- The equally-distributed-equivalent income (EDEI), 퐼̅, is the level that would make the observer indifferent to the initial distribution; 퐼̅ is lower than μ.
- The observer’s inequality aversion determines 퐼̅. Under the constant-inequality-aversion functional form, 퐼̅ satisfies:
  - w(퐼̅) = (1/2) [ w(I_P) + w(I_R) ]  (B.2)
- Define percentage deviations of I_R and I_P from 퐼̅:
  - 훾_R ≡ (I_R − 퐼̅) / 퐼̅  and  훾_P ≡ (퐼̅ − I_P) / 퐼̅  (B.3)

### Mean income and transfers (leaky bucket)
- Relationship between μ and 퐼̅:
  - μ = 퐼̅ + 1/2 [ (I_R − 퐼̅) − (퐼̅ − I_P) ]  (B.4)
  - μ = 퐼̅ [ 1 + 1/2 ( 훾_R − 훾_P ) ]  (B.5)
- Leaky-bucket transfer specification:
  - Net transfer received by the poor: 퐼̅ − I_P = (1 − b) (I_R − 퐼̅)  (B.6)
    - b is the leakage coefficient; gross transfer from rich = I_R − 퐼̅.
  - Equivalent relation in deviations:
    - 훾_P = (1 − b) 훾_R  (B.7)

### Atkinson’s inequality index expressed with leaky-bucket variables
- Atkinson’s inequality index, A, defined as percentage loss in total income that makes observer indifferent between existing situation and perfect equality:
  - A = 1 − 퐼̅ / μ  (B.8)
- Substituting μ expression:
  - A = 1 − 퐼̅ / [ 퐼̅ ( 1 + 1/2 ( 훾_R − 훾_P ) ) ]  (B.9)
- Two equivalent expressions linking A to leakage and proportional distances:
  - A = 1 − 1 / [ 1 + 1/2 b 훾_R ]  (B.10)
  - A = 1 − 1 / [ 1 + 1/2 ( b / (1 − b) ) 훾_P ]  (B.11)
- Interpretation:
  - A depends on observer’s inequality aversion (implicit in 훾_R or 훾_P) and the leakage coefficient b associated with the tax-and-transfer scheme that equalizes incomes at 퐼̅.
  - The gap 훾_R − 훾_P is larger when initial inequality is large and when the observer is highly averse to inequality (so 퐼̅ is positioned closer to I_P and further from I_R).

### Extension to n households
- For n households (i = 1,..,n) ranked by income, with m the household earning 퐼̅:
  - Set I_P = (1/m) ∑_{i=1}^m I_i  and  I_R = (1/(n−m)) ∑_{i=m}^n I_i
- The two-family analysis applies by these aggregations (i.e., replace I_P and I_R with the group means below and above 퐼̅).

### Appendix C: Relating Okun’s leaky bucket exercise to optimal income taxation
- Framing Okun’s leak within optimal-income-taxation logic:
  - Objective: Max w(I_P + S_P) + w(I_R − T_R), subject to R_0 + (1 − b) T_R ≥ S_P
    - I_P: initial income of the poor
    - S_P: subsidy to the poor (negative tax)
    - I_R: initial income of the rich
    - T_R: tax on the rich
    - b: leak (fraction lost in transfer)
    - R_0: exogenous revenues (e.g., foreign aid or resource revenues) in excess of necessary spending
- Lagrangean:
  - L = w(I_P + S_P) + w(I_R − T_R) + λ [ R_0 + (1 − b) T_R − S_P ]
- First-order conditions (differentiate w.r.t. S_P and T_R):
  - w′(I_P + S_P) = λ
  - w′(I_R − T_R) = λ (1 − b)
- Implication for optimal taxation:
  - w′(I_R − T_R) = (1 − b) w′(I_P + S_P)
  - With b = 0 (no leak) government would equalize incomes of the two individuals; with b > 0 the government will permit the rich’s post-tax income to exceed the poor’s because transfers are lossy.
  - The income gap between rich and poor at the optimum widens with larger b.
- Interpretation of λ:
  - λ is the extra welfare from an additional dollar of R_0 given to the government for optimal allocation.
- Dollar-value interpretation (in terms of public funds) of an extra dollar of consumption:
  - For the poor: the dollar value equals one.
  - For the rich: the dollar value equals 1 − b.
  - Rationale: an extra dollar to government (R_0) would be allocated to the poor raising welfare by w′(I_P + S_P); giving it instead to the rich raises welfare by w′(I_R − T_R) = (1 − b) w′(I_P + S_P).

*Source: IMF staff appendix text.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2017/wp17214.pdf_