## wpiea2021252-print-pdf

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---

### Major themes and purpose
- Focus: quantifying the redistributive capacity of taxation (and transfers) and making progressivity and redistribution indices comparable across countries and years.
- Motivation: redistribution has risen to the fore of policy analysis, data availability varies across advanced, emerging market, and low-income countries, and the COVID-19 pandemic heightened the importance of measuring redistribution accurately.
- Key empirical application: personal income tax (PIT) using a dataset of PIT characteristics covering 108 countries over the 2007-2018 period.

### Definitions and core indices
- Pre-tax income distribution for country-year i denoted iX.
- Post-tax distribution notation: iiNX and ()iiNX.
- Reynolds–Smolensky (redistributive power) index:
  - R_i ≡ G_{X_i} − G_{N_i(X_i)}
- Kakwani progressivity index:
  - P_i ≡ G_{T_i} − G_{X_i}
  - where T_i(X_i) ≡ N_i(X_i) − X_i
- Aggregate average tax rate notation: τ_i (ratio of mean tax over mean pre-tax income; also measured as tax revenue over GDP).
- Decomposition (Kakwani):
  - R_i = P_i × (1 − τ_i)

### Issues motivating adjustment for pre-tax distributions
- Standard indices depend on the pre-tax income distribution iX; varying pre-tax distributions mechanically change measured redistributive effects.
- Two correction approaches:
  - Fixed-income method: impose an identical pre-tax distribution across countries (sensitive to choice).
  - Transplant-and-compare method (Dardanoni and Lambert 2002): transplant tax regimes into a common base via iso-elastic transformations so Lorenz-curve-based rankings are robust.
- Endogeneity caveat: paper assumes an inelastic tax base and does not model endogenous responses.

### Simplified transplant-and-compare implementation (proposed)
- Rationale for lognormal specification:
  - Lognormal distributions are iso-elastic transformations of one another.
  - Two parameters (shape σ_i and scale λ_i) can be calibrated on observed Gini and mean.
- Calibration relationships:
  - Gini of lognormal: G_i = Φ(2^{1/2} σ_i) − 1 (as given in source notation)
  - Mean: μ_i = exp(λ_i + 2 σ_i^2 / 2) (as given in source notation)
  - Hence σ_i and λ_i solved from observed Gini and mean.
- Transplantation mechanism:
  - Transplantation function g(x) = e^{x a b} (notation in source); parameters a_i and b_i chosen so ln-incomes map to standard base.
  - Apply same transformation to post-tax incomes to obtain transplanted tax schedule g_i N(.) and transplanted indices.
- Practical advantages:
  - Requires only mean pre-tax income (approximated by GDP per capita) and Gini coefficient.
  - Lower computational complexity, applicable where microdata are unavailable.
- Limitations:
  - Lognormal may poorly fit the very top tail; alternatives (Weibull, Pareto tails, hybrid) could better capture tails but lose iso-elastic simplicity.
  - Assumes inelastic tax base; compliance and informality not modelled.

### Method: transplant-and-compare (simulation-based) — step-by-step
- Step 1 — Calibration:
  - Calibrate lognormal via Gini and mean of pre-tax income (proxied by GDP per capita), solving for σ_i and λ_i.
- Step 2 — Simulation:
  - (a) Generate 10,000 (or more) simulated taxpayers drawn from the calibrated lognormal distribution to obtain X_i.
  - (b) For each simulated taxpayer, compute post-tax income using known tax rules to obtain N_i(X_i).
- Step 3 — Transplantation:
  - (a) Apply iso-elastic transformation g_i^{ab}(x) = a_i x^{b_i} where a_i b_i + λ_i = 0 and b_i σ_i = 1 to X_i and N_i(X_i).
  - (b) Compute transplanted taxes T_i^g = N_i^g − X_i^g.
- Step 4 — Comparison:
  - Compute comparable indices g_i P, g_i τ, and g_i R from transplanted distributions.
- Forward-looking analysis:
  - For policy-change scenarios, hold latest Gini from Step 1 and simulate policy changes in Step 2 under inelastic base assumption.

### Data inputs and implementation caveats
- Gini source: Standardized World Income Inequality Database version 9.1 (Solt 2021); SWIID provides comparable pre-tax income Gini for 198 countries, 1960-2020.
- Study uses 157 countries, 2000-20 for distributional illustrations.
- Exclusion rule: exclude countries-years where two standard error range for Gini exceeds 10 Gini points.
- Mean pre-tax income proxied by GDP per capita reconstructed in original local currency units from IMF’s World Economic Outlook archives.
- PIT design characteristics from EY’s Worldwide Personal Tax Guides and IBFD database.
- PIT taxable income definitions vary; capital income inclusion may bias indices upward where capital income is taxed differently.
- PIT tax unit varies; Ginis follow equivalized individual income on square root scale (LIS practice).
- Compliance and informality not accounted for; method measures redistributive power implied by tax codes.

### Computational validation
- Simplified (lognormal-based) vs microdata-based transplant-and-compare:
  - Sample: 23 countries with LIS 2016 microdata.
  - Correlation of simplified versus microdata Reynolds–Smolensky R index: 0.922.
  - Correlation of implied rankings: 0.925.
  - Tests on log pre-tax income in microdata reject normality in all 23 countries at standard significance levels; errors partly related to lognormal approximation.

### Regression evidence on the “Robin Hood” question (pooled regressions, 108 countries, 2007-2018; cluster-robust SEs)
- Column (1) Unadjusted R, pooled:
  - Gini pre-tax income: 0.0914*** (0.0184)
  - Constant: -1.617** (0.811)
  - Observations: 1,364
  - R-squared: 0.091
- Column (2) Transplant-and-compare R, pooled:
  - Gini pre-tax income: 0.0555** (0.0228)
  - Constant: 0.290 (1.042)
  - Observations: 1,364
  - R-squared: 0.030
- Column (3) Unadjusted R, pooled, controlling for GDP per capita:
  - Gini pre-tax income: 0.0774*** (0.0139)
  - GDP per capita: 5.33e-05*** (1.05e-05)
  - Constant: -1.900*** (0.610)
  - Observations: 1,356
  - R-squared: 0.388
- Column (4) Transplant-and-compare R, pooled, controlling for GDP per capita:
  - Gini pre-tax income: 0.0404** (0.0167)
  - GDP per capita: 5.67e-05*** (1.13e-05)
  - Constant: -0.00817 (0.773)
  - Observations: 1,356
  - R-squared: 0.332
- Interpretation:
  - Correcting for pre-tax dispersion reduces magnitude and significance of association between initial inequality and measured redistribution.
  - No evidence of a cross-country “Robin Hood paradox”: higher pre-tax inequality is associated with more redistributive PITs even after adjustment.

### Kakwani decomposition and cross-country patterns
- Kakwani decomposition: R_i = P_i × (1 − τ_i); redistributive power depends on both progressivity P_i and aggregate tax rate τ_i.
- Findings by country groups:
  - AEs and EMEs: progressive capacity P has trended downward; τ slightly increasing or flat; R stagnant (AEs) or decreasing (EMEs).
  - LIDCs: low and declining τ is main constraint on low R; PIT often progressive but low τ yields low R.
  - Multiple P and τ combinations can yield similar R; for R ∈ [2, 4[ there is wide variation in P and τ producing similar R.

### Appendix II — Selected PIT 2018 values (Progressive Capacity P, Aggregate Tax Rate τ, Redistributive Capacity R) (values preserved exactly)
- Netherlands NL: P 28.70; τ 25.80; R 9.98
- Austria AT: P 18.34; τ 32.84; R 8.97
- Belgium BE: P 14.85; τ 36.93; R 8.69
- Australia AU: P 18.23; τ 27.29; R 6.84
- Italy IT: P 20.87; τ 24.38; R 6.73
- Israel IL: P 26.56; τ 19.19; R 6.31
- Argentina AR: P 17.67; τ 26.22; R 6.28
- Malta MT: P 20.03; τ 23.70; R 6.22
- France FR: P 22.78; τ 21.29; R 6.16
- United Kingdom GB: P 26.28; τ 18.54; R 5.98
- Luxembourg LU: P 13.70; τ 30.20; R 5.93
- Slovenia SI: P 13.77; τ 30.01; R 5.90
- Ireland IE: P 14.44; τ 28.38; R 5.72
- Germany DE: P 39.08; τ 12.35; R 5.51
- Portugal PT: P 13.13; τ 28.63; R 5.27
- Croatia HR: P 23.15; τ 18.15; R 5.13
- Cyprus CY: P 34.44; τ 12.94; R 5.12
- Finland FI: P 27.55; τ 14.49; R 4.67
- Mexico MX: P 23.48; τ 16.19; R 4.54
- United States US: P 23.76; τ 15.21; R 4.26
- Uruguay UY: P 30.53; τ 11.61; R 4.01
- New Zealand NZ: P 11.68; τ 25.54; R 4.01
- Canada CA: P 15.80; τ 19.32; R 3.78
- Korea KR: P 13.28; τ 22.01; R 3.75
- Sweden SE: P 35.31; τ 9.36; R 3.65
- Poland PL: P 13.16; τ 21.24; R 3.55
- Norway NO: P 20.11; τ 14.62; R 3.44
- China CN: P 34.50; τ 9.07; R 3.44
- Dominican Republic DO: P 37.27; τ 8.30; R 3.37
- Greece GR: P 7.61; τ 28.23; R 2.99
- Turkey TR: P 7.86; τ 27.38; R 2.96
- Malaysia MY: P 34.18; τ 7.97; R 2.96
- Denmark DK: P 16.10; τ 15.45; R 2.94
- Panama PA: P 31.58; τ 8.51; R 2.94
- Estonia EE: P 13.84; τ 17.09; R 2.85
- Philippines PH: P 41.30; τ 6.33; R 2.79
- Nigeria NG: P 22.87; τ 10.82; R 2.77
- Brazil BR: P 40.79; τ 6.32; R 2.75
- Singapore SG: P 25.11; τ 9.76; R 2.71
- Spain ES: P 19.33; τ 11.80; R 2.59
- Moldova MD: P 21.06; τ 10.73; R 2.53
- Slovak Republic SK: P 9.27; τ 21.40; R 2.52
- Vietnam VN: P 43.82; τ 5.26; R 2.43
- Seychelles SC: P 4.53; τ 33.62; R 2.30
- Sierra Leone SL: P 41.79; τ 5.03; R 2.21
- Czech Republic CZ: P 13.59; τ 13.34; R 2.09
- Thailand TH: P 37.45; τ 5.23; R 2.07
- Armenia AM: P 3.87; τ 33.16; R 1.92
- Indonesia ID: P 38.60; τ 4.69; R 1.90
- El Salvador SV: P 41.38; τ 4.30; R 1.86
- Switzerland CH: P 28.60; τ 5.79; R 1.76
- Peru PE: P 39.05; τ 4.25; R 1.73
- Costa Rica CR: P 40.47; τ 3.71; R 1.56
- Lao P.D.R. LA: P 26.21; τ 5.43; R 1.50
- Honduras HN: P 44.79; τ 2.84; R 1.31
- Romania RO: P 39.72; τ 3.02; R 1.24
- Bolivia BO: P 5.04; τ 16.00; R 0.96
- Angola AO: P 4.52; τ 15.18; R 0.81
- Venezuela VE: P 1.00; τ 43.40; R 0.77
- Latvia LV: P 2.03; τ 24.31; R 0.65
- Belarus BY: P 2.41; τ 19.52; R 0.59
- Iran IR: P 2.22; τ 20.41; R 0.57
- Mongolia MN: P 3.55; τ 13.05; R 0.53
- Kazakhstan KZ: P 2.70; τ 14.40; R 0.45
- Colombia CO: P 46.80; τ 0.84; R 0.40
- Paraguay PY: P 46.57; τ 0.76; R 0.36
- Serbia RS: P 47.59; τ 0.70; R 0.33
- Montenegro, Rep. of ME: P 2.48; τ 9.92; R 0.27
- Pakistan PK: P 46.66; τ 0.53; R 0.25
- Ecuador EC: P 47.30; τ 0.50; R 0.24

### Conclusions and policy implications
- The transplant-and-compare simulation enables construction of progressivity and redistributive capacity indices adjusted for pre-tax income distributions using Gini coefficients and GDP per capita.
- Enables international and intertemporal comparisons when microdata are unavailable.
- Major empirical conclusions (108-country, 2007–2018 sample):
  - No evidence of a cross-country “Robin-Hood” paradox: higher pre-tax inequality is associated with more redistributive PITs.
  - In many LIDCs, aggregate tax rate τ is a stronger constraint on redistribution than statutory progressivity P.
- Practical implication: diagnose whether redistributive shortfalls stem from insufficient progressivity P or from limited aggregate tax capacity τ to tailor reform advice.

*Content based solely on "REFERENCES __________________________________________________________________________________________ 23" from the provided PDF content unit.*

### REFERENCES __________________________________________________________________________________________ 23

### wpiea2021252-print-pdf - REFERENCES __________________________________________________________________________________________ 23

### Major themes and purpose
- Focus: quantifying the redistributive capacity of taxation (and transfers) and making progressivity and redistribution indices comparable across countries and years.
- Motivation: redistribution has risen to the fore of policy analysis, data availability varies across advanced, emerging market, and low-income countries, and the COVID-19 pandemic heightened the importance of measuring redistribution accurately.
- Key empirical application described: personal income tax (PIT) using a dataset of PIT characteristics covering 108 countries over the 2007-2018 period.

### Definitions and core indices
- Pre-tax income distribution for country-year i denoted iX.
- Tax schedule mapping and post-tax distribution notation: (.), iiNX and ()iiNX.
- Reynolds–Smolensky (redistributive power) index:
  - ()
    iii
    iXNX
    RG G−≡
- Kakwani progressivity index:
  - ()
    iii
    iTXX
    PGG≡−
  - where ()()iii iiTXX N X= −
- Aggregate average tax rate notation: iτ (ratio of mean tax over mean pre-tax income; also measured as tax revenue over GDP).
- Decomposition property:
  - 1
    i
    i
    i
    RP
    τ
    τ
    =
    −
  - interprets redistribution as product of progressivity and a policy size component 1−τi.

### Issues motivating adjustment for pre-tax distributions
- Standard indices depend on the pre-tax income distribution iX; varying pre-tax distributions mechanically change measured redistributive effects.
- Comparing intrinsic redistributive capacities requires correcting for cross-country and over-time differences in pre-tax distributions.
- Two main correction approaches in the literature:
  - Fixed-income method: impose an identical pre-tax distribution across countries (sensitive to choice of fixed distribution).
  - Transplant-and-compare method (Dardanoni and Lambert 2002): transplant tax regimes into a common base via iso-elastic transformations so rankings of Lorenz-curve-based indices are robust to original pre-tax distributions.
- Endogeneity caveat: tax policy may influence pre-tax distributions (labor supply, general equilibrium effects); this paper assumes an inelastic tax base and does not model endogenous responses.

### Simplified transplant-and-compare implementation (proposed)
- Goal: enable transplant-and-compare comparisons where household microdata are unavailable by using low-dimensional statistics (mean and Gini) and parametric pre-tax distributions.
- Rationale for lognormal specification:
  - All lognormal distributions are iso-elastic transformations of one another, facilitating transplantation via simple transformations.
  - Lognormal is a long-standing income-distribution workhorse and is two-parameter (shape λ and scale σ), allowing calibration on observed Gini and mean.
  - Unlike a bounded uniform, lognormal is unbounded above and better captures high-income features than a uniform.
- Calibration relationships (as given in the source):
  - Gini of a lognormal with shape parameter iiσ and scale parameter iiλ: 2   (/   2)   1
    ii
    Gσ=Φ−
    (where Φ(.) is the cumulative of the standard normal distribution).
  - Mean: 2
    exp()
    2
    i
    i
    i
    μ
    σ
    λ
    =+
  - Hence iiσ and iiλ can be solved for a given Gini and mean.
- Transplantation mechanism (iso-elastic transformation):
  - Define transplantation function g(x) = exab (notation in the source), with parameters a and b chosen so that ln-transformed incomes map to a standard base (e.g., standard normal for ln).
  - Apply the same transformation to post-tax incomes to obtain the transplanted tax schedule (.)giN.
- Practical advantages:
  - Requires only mean pre-tax income (approximated by GDP per capita) and Gini coefficient—data that are more widely available than household microdata.
  - Lower computational complexity and easier applicability across many country-years.
- Limitations and caveats:
  - Lognormal may poorly fit the very top tail (highest incomes); if interest centers on the highest income tranches, parametric lognormal approach could be suboptimal.
  - Alternatives suggested: Weibull, Pareto tails, or hybrid/mixture specifications, but they may sacrifice the simplicity of iso-elastic transformations.
  - The approach assumes inelastic tax base (no behavioral responses).

### Validation and empirical findings (summary of approach and main reported results)
- Validation: compare simplified simulated-data indices against indices computed with household microdata for the subset of countries-years where microdata exist.
  - Reported finding: the two sets of indices are strongly correlated, lending confidence in the simulated-data approach when microdata are missing.
- Main empirical questions addressed:
  - Is there evidence of a “Robin-Hood” paradox (countries with low pre-tax inequality redistributing more, high-inequality countries redistributing less)?
    - Finding reported: no evidence of a Robin-Hood paradox even after applying transplant-and-compare adjusted indices.
  - Cross-country/time patterns: examination of PIT redistributive capacity across country groups (LIDCs, EMEs, AEs), its evolution over time, and decomposition into progressivity and policy size components.

### Methodological recap (simplified step-by-step intent)
- For each country-year i:
  - Obtain mean pre-tax income (approximated by GDP per capita) and pre-tax Gini coefficient.
  - Calibrate a lognormal distribution (shape iiσ and scale iiλ) to match mean and Gini.
  - Define iso-elastic transplantation function g(x)=exab that maps country i’s log income to the common standardized base (standard lognormal/lognormal with standard normal ln).
  - Apply g to the tax schedule and post-tax incomes to compute transplanted progressivity and redistribution indices in the common base.
  - Compare transplanted indices across countries-years to isolate intrinsic redistributive capacities.

*Italic source attribution: Content based solely on "REFERENCES __________________________________________________________________________________________ 23" from the provided PDF content unit.*

### 1. Calibration.  Calibrate a lognormal parametric distribution based on the Gini coefficient and

### 1. Calibration. Calibrate a lognormal parametric distribution based on the Gini coefficient and

### Method: transplant-and-compare (simulation-based)
- Step 1 — Calibration
  - Calibrate a lognormal parametric distribution based on the Gini coefficient and mean of pre-tax income (proxied by GDP per capita), solving for the shape parameter σ_i and scale parameter λ_i (footnote 15).
- Step 2 — Simulation
  - (a) Generate 10,000 (or more) simulated taxpayers each with pre-tax income randomly drawn from the calibrated lognormal distribution and obtain a simulated X_i.
  - (b) For each simulated taxpayer, compute post-tax income based on known tax rules and obtain a simulated N_i(X_i). The simulated tax regime N_i(.) is available.
- Step 3 — Transplantation
  - (a) Transplant N_i(.) into the (standardized) common base by applying the iso-elastic transformation g_i^ab(x) = a_i x^{b_i} to X_i and N_i(X_i), where a_i and b_i satisfy a_i b_i + λ_i = 0 and b_i σ_i = 1.
  - (b) Compute transplanted taxes as T_i^g = N_i^g − X_i^g.
- Step 4 — Comparison
  - Using the 3 transplanted distributions g_i X, g_i N(X) and g_i T, compute the comparable indices g_i P, g_i τ, and g_i R.
- Forward-looking analysis note
  - For policy-change scenarios, keep the assumption of an inelastic tax base and hold the latest Gini information from Step 1 unless additional information is available; simulate policy changes in Step 2.

### Implementation caveats embedded in method
- Lognormal approximation is assumed acceptable for cross-country comparison of PIT progressive and redistributive capacity; microdata-based rankings are strongly consistent with the lognormal-based approach in many cases but differences exist where microdata are available.
- PIT taxable income definitions vary across countries; capital income elements (interest, dividends, capital gains) included in Gini and GDP per capita data may not be taxed like ordinary income, biasing progressivity and redistribution indices upward where capital income faces lower rates.
- The PIT tax unit notion (household vs individual) varies; Gini coefficients used follow equivalized individual income on a square root scale (LIS practice).
- Compliance and informality are not accounted for; approach measures redistributive power as intended in tax codes (can be applied to compliance-adjusted post-tax distributions if such data exist).

### Data inputs described
- Gini coefficients of pre-tax income come from the Standardized World Income Inequality Database version 9.1 (Solt 2021); SWIID provides comparable pre-tax income Gini for 198 countries over the 1960-2020 period.
- The study uses 157 countries, 2000-20 for distributional illustrations.
- Exclusion rule: countries-years where the two standard error range for the Gini exceeds 10 Gini points are excluded.
- Mean pre-tax income proxied by GDP per capita reconstructed in original local currency units from IMF’s World Economic Outlook archives to address currency redenominations.
- PIT design characteristics (rates, thresholds, deductions, allowances, credits) tabulated from EY’s historical Worldwide Personal Tax Guides and the IBFD database.
- Dataset structure by country group reported for years 2007–2018 with yearly totals (example: 2007 total 108; 2018 total 153).

### Key empirical features of PIT included in dataset
- Liability threshold reported in multiples of GDP per capita, with historical patterns differing across country groups; in LIDCs historically high liability thresholds often excluded middle-high incomes from taxation.
- Lowest and top statutory PIT marginal rates and top-rate thresholds tracked over time and across country groups.

### Computational validation
- Simplified (lognormal-based) versus microdata-based transplant-and-compare comparison:
  - Sample: 23 countries with LIS 2016 microdata.
  - Correlation of simplified versus microdata Reynolds–Smolensky R index: 0.922.
  - Correlation of implied rankings: 0.925.
  - Tests on the logarithm of pre-tax income in microdata reject normality in all 23 countries at standard levels of significance; errors between index sets are partly related to the lognormal approximation.

### Regression evidence on the “Robin Hood” question
- Pooled regressions over 108 countries, 2007-2018 (cluster-robust standard errors), dependent variable R (redistribution index).
- Key results (coefficients and standard errors preserved exactly):
  - Column (1) Unadjusted R, pooled:
    - Gini pre-tax income: 0.0914*** (0.0184)
    - Constant: -1.617** (0.811)
    - Observations: 1,364
    - R-squared: 0.091
  - Column (2) Transplant-and-compare R, pooled:
    - Gini pre-tax income: 0.0555** (0.0228)
    - Constant: 0.290 (1.042)
    - Observations: 1,364
    - R-squared: 0.030
  - Column (3) Unadjusted R, pooled, controlling for GDP per capita:
    - Gini pre-tax income: 0.0774*** (0.0139)
    - GDP per capita: 5.33e-05*** (1.05e-05)
    - Constant: -1.900*** (0.610)
    - Observations: 1,356
    - R-squared: 0.388
  - Column (4) Transplant-and-compare R, pooled, controlling for GDP per capita:
    - Gini pre-tax income: 0.0404** (0.0167)
    - GDP per capita: 5.67e-05*** (1.13e-05)
    - Constant: -0.00817 (0.773)
    - Observations: 1,356
    - R-squared: 0.332
- Interpretation: correcting for differences in pre-tax dispersion reduces the magnitude and statistical significance of the association between initial inequality and measured redistribution; nevertheless, no evidence of a cross-country “Robin Hood paradox” emerges—countries with more unequal pre-tax income tend to implement more redistributive PITs even after adjustment.

### Kakwani decomposition and cross-country patterns
- Kakwani decomposition used to examine interplay of progressive capacity (P) and aggregate tax rate (τ) as drivers of redistributive capacity (R).
- Findings:
  - Multiple combinations of P and τ can yield similar R; in the range R ∈ [2, 4[ there is wide variation in P and τ leading to similar R values.
  - For Advanced Economies (AEs) and Emerging Market Economies (EMEs), progressive capacity has trended downward in recent years while aggregate tax rate is slightly increasing or flat; redistributive capacity is stagnant (AEs) or decreasing (EMEs).
  - In Low-Income Developing Countries (LIDCs), low and declining aggregate tax rate τ is the key driver of relatively low redistributive capacity R; PIT is often progressive (levied on a small formal high-income group) but with low average tax rates producing low overall R.
  - OECD series show progressive capacity declining while aggregate tax rate is almost flat, yielding slightly diminishing redistributive capacity over time.

### Conclusions and policy implications
- The transplant-and-compare simulation technique allows construction of progressive and redistributive capacity indices adjusted for pre-tax income distributions using commonly available aggregate information (Gini coefficients and GDP per capita).
- The approach enables international and intertemporal comparisons where microdata are unavailable.
- Major empirical conclusions from the 108-country, 2007–2018 sample:
  - No evidence of a cross-country “Robin-Hood” paradox: higher pre-tax inequality is associated with more redistributive PITs.
  - In many LIDCs, the aggregate tax rate (size of PIT revenue relative to GDP) is a stronger constraint on redistribution than the progressivity of statutory rates and schedules.
- Practical implication for policy analysts: diagnose whether redistributive shortfalls stem from insufficient progressivity or from limited aggregate tax capacity (τ) to tailor reform advice.

*Authors’ computations and figures as provided in the source document.*

### REFERENCES

### REFERENCES

### Appendix I. The Kakwani Decomposition
- As shown by Kakwani (1977), iR can be decomposed into its size and progressivity components. Noting the means of pre-tax income and taxes as iXμ and iTμ, and the aggregate tax rate as i i T i X μ τ μ ≡, the Kakwani decomposition reads as:

  1 i i i i RP τ τ = −

- This expression shows that the global redistributive power of tax policy depends not only on progressivity iP but also on a multiplicative term 1 i i τ τ − which is increasing in the aggregate tax rate iτ. That term is the amplitude, or size, of tax policy. The Kakwani decomposition captures the simple fact that the redistributive power of taxation depends not only on the progressivity of tax schedules and rates, but also on total tax collection as a proportion of aggregate income.
- This decomposition assumes no income reranking, which is appropriate for any tax policy where the marginal tax rate is below unity (which is empirically always the case, excluding rare pathological situations), but may not hold for transfers. Extending the Kakwani decomposition to include a reranking term is straightforward (Kakwani 1984; Vellutini 2021).

### Appendix II. Progressive and Redistributive Capacities of the PIT (2018, in percent)
- Netherlands NL: Progressive Capacity (P) 28.70; Aggregate Tax Rate (τ) 25.80; Redistributive Capacity (R) 9.98
- Austria AT: Progressive Capacity (P) 18.34; Aggregate Tax Rate (τ) 32.84; Redistributive Capacity (R) 8.97
- Belgium BE: Progressive Capacity (P) 14.85; Aggregate Tax Rate (τ) 36.93; Redistributive Capacity (R) 8.69
- Australia AU: Progressive Capacity (P) 18.23; Aggregate Tax Rate (τ) 27.29; Redistributive Capacity (R) 6.84
- Italy IT: Progressive Capacity (P) 20.87; Aggregate Tax Rate (τ) 24.38; Redistributive Capacity (R) 6.73
- Israel IL: Progressive Capacity (P) 26.56; Aggregate Tax Rate (τ) 19.19; Redistributive Capacity (R) 6.31
- Argentina AR: Progressive Capacity (P) 17.67; Aggregate Tax Rate (τ) 26.22; Redistributive Capacity (R) 6.28
- Malta MT: Progressive Capacity (P) 20.03; Aggregate Tax Rate (τ) 23.70; Redistributive Capacity (R) 6.22
- France FR: Progressive Capacity (P) 22.78; Aggregate Tax Rate (τ) 21.29; Redistributive Capacity (R) 6.16
- United Kingdom GB: Progressive Capacity (P) 26.28; Aggregate Tax Rate (τ) 18.54; Redistributive Capacity (R) 5.98
- Luxembourg LU: Progressive Capacity (P) 13.70; Aggregate Tax Rate (τ) 30.20; Redistributive Capacity (R) 5.93
- Slovenia SI: Progressive Capacity (P) 13.77; Aggregate Tax Rate (τ) 30.01; Redistributive Capacity (R) 5.90
- Ireland IE: Progressive Capacity (P) 14.44; Aggregate Tax Rate (τ) 28.38; Redistributive Capacity (R) 5.72
- Germany DE: Progressive Capacity (P) 39.08; Aggregate Tax Rate (τ) 12.35; Redistributive Capacity (R) 5.51
- Portugal PT: Progressive Capacity (P) 13.13; Aggregate Tax Rate (τ) 28.63; Redistributive Capacity (R) 5.27
- Croatia HR: Progressive Capacity (P) 23.15; Aggregate Tax Rate (τ) 18.15; Redistributive Capacity (R) 5.13
- Cyprus CY: Progressive Capacity (P) 34.44; Aggregate Tax Rate (τ) 12.94; Redistributive Capacity (R) 5.12
- Finland FI: Progressive Capacity (P) 27.55; Aggregate Tax Rate (τ) 14.49; Redistributive Capacity (R) 4.67
- Mexico MX: Progressive Capacity (P) 23.48; Aggregate Tax Rate (τ) 16.19; Redistributive Capacity (R) 4.54
- United States US: Progressive Capacity (P) 23.76; Aggregate Tax Rate (τ) 15.21; Redistributive Capacity (R) 4.26
- Uruguay UY: Progressive Capacity (P) 30.53; Aggregate Tax Rate (τ) 11.61; Redistributive Capacity (R) 4.01
- New Zealand NZ: Progressive Capacity (P) 11.68; Aggregate Tax Rate (τ) 25.54; Redistributive Capacity (R) 4.01
- Canada CA: Progressive Capacity (P) 15.80; Aggregate Tax Rate (τ) 19.32; Redistributive Capacity (R) 3.78
- Korea KR: Progressive Capacity (P) 13.28; Aggregate Tax Rate (τ) 22.01; Redistributive Capacity (R) 3.75
- Sweden SE: Progressive Capacity (P) 35.31; Aggregate Tax Rate (τ) 9.36; Redistributive Capacity (R) 3.65
- Poland PL: Progressive Capacity (P) 13.16; Aggregate Tax Rate (τ) 21.24; Redistributive Capacity (R) 3.55
- Norway NO: Progressive Capacity (P) 20.11; Aggregate Tax Rate (τ) 14.62; Redistributive Capacity (R) 3.44
- China CN: Progressive Capacity (P) 34.50; Aggregate Tax Rate (τ) 9.07; Redistributive Capacity (R) 3.44
- Dominican Republic DO: Progressive Capacity (P) 37.27; Aggregate Tax Rate (τ) 8.30; Redistributive Capacity (R) 3.37
- Greece GR: Progressive Capacity (P) 7.61; Aggregate Tax Rate (τ) 28.23; Redistributive Capacity (R) 2.99
- Turkey TR: Progressive Capacity (P) 7.86; Aggregate Tax Rate (τ) 27.38; Redistributive Capacity (R) 2.96
- Malaysia MY: Progressive Capacity (P) 34.18; Aggregate Tax Rate (τ) 7.97; Redistributive Capacity (R) 2.96
- Denmark DK: Progressive Capacity (P) 16.10; Aggregate Tax Rate (τ) 15.45; Redistributive Capacity (R) 2.94
- Panama PA: Progressive Capacity (P) 31.58; Aggregate Tax Rate (τ) 8.51; Redistributive Capacity (R) 2.94
- Estonia EE: Progressive Capacity (P) 13.84; Aggregate Tax Rate (τ) 17.09; Redistributive Capacity (R) 2.85
- Philippines PH: Progressive Capacity (P) 41.30; Aggregate Tax Rate (τ) 6.33; Redistributive Capacity (R) 2.79
- Nigeria NG: Progressive Capacity (P) 22.87; Aggregate Tax Rate (τ) 10.82; Redistributive Capacity (R) 2.77
- Brazil BR: Progressive Capacity (P) 40.79; Aggregate Tax Rate (τ) 6.32; Redistributive Capacity (R) 2.75
- Singapore SG: Progressive Capacity (P) 25.11; Aggregate Tax Rate (τ) 9.76; Redistributive Capacity (R) 2.71
- Spain ES: Progressive Capacity (P) 19.33; Aggregate Tax Rate (τ) 11.80; Redistributive Capacity (R) 2.59
- Moldova MD: Progressive Capacity (P) 21.06; Aggregate Tax Rate (τ) 10.73; Redistributive Capacity (R) 2.53
- Slovak Republic SK: Progressive Capacity (P) 9.27; Aggregate Tax Rate (τ) 21.40; Redistributive Capacity (R) 2.52
- Vietnam VN: Progressive Capacity (P) 43.82; Aggregate Tax Rate (τ) 5.26; Redistributive Capacity (R) 2.43
- Seychelles SC: Progressive Capacity (P) 4.53; Aggregate Tax Rate (τ) 33.62; Redistributive Capacity (R) 2.30
- Sierra Leone SL: Progressive Capacity (P) 41.79; Aggregate Tax Rate (τ) 5.03; Redistributive Capacity (R) 2.21
- Czech Republic CZ: Progressive Capacity (P) 13.59; Aggregate Tax Rate (τ) 13.34; Redistributive Capacity (R) 2.09
- Thailand TH: Progressive Capacity (P) 37.45; Aggregate Tax Rate (τ) 5.23; Redistributive Capacity (R) 2.07
- Armenia AM: Progressive Capacity (P) 3.87; Aggregate Tax Rate (τ) 33.16; Redistributive Capacity (R) 1.92
- Indonesia ID: Progressive Capacity (P) 38.60; Aggregate Tax Rate (τ) 4.69; Redistributive Capacity (R) 1.90
- El Salvador SV: Progressive Capacity (P) 41.38; Aggregate Tax Rate (τ) 4.30; Redistributive Capacity (R) 1.86
- Switzerland CH: Progressive Capacity (P) 28.60; Aggregate Tax Rate (τ) 5.79; Redistributive Capacity (R) 1.76
- Peru PE: Progressive Capacity (P) 39.05; Aggregate Tax Rate (τ) 4.25; Redistributive Capacity (R) 1.73
- Costa Rica CR: Progressive Capacity (P) 40.47; Aggregate Tax Rate (τ) 3.71; Redistributive Capacity (R) 1.56
- Lao P.D.R. LA: Progressive Capacity (P) 26.21; Aggregate Tax Rate (τ) 5.43; Redistributive Capacity (R) 1.50
- Honduras HN: Progressive Capacity (P) 44.79; Aggregate Tax Rate (τ) 2.84; Redistributive Capacity (R) 1.31
- Romania RO: Progressive Capacity (P) 39.72; Aggregate Tax Rate (τ) 3.02; Redistributive Capacity (R) 1.24
- Bolivia BO: Progressive Capacity (P) 5.04; Aggregate Tax Rate (τ) 16.00; Redistributive Capacity (R) 0.96
- Angola AO: Progressive Capacity (P) 4.52; Aggregate Tax Rate (τ) 15.18; Redistributive Capacity (R) 0.81
- Venezuela VE: Progressive Capacity (P) 1.00; Aggregate Tax Rate (τ) 43.40; Redistributive Capacity (R) 0.77
- Latvia LV: Progressive Capacity (P) 2.03; Aggregate Tax Rate (τ) 24.31; Redistributive Capacity (R) 0.65
- Belarus BY: Progressive Capacity (P) 2.41; Aggregate Tax Rate (τ) 19.52; Redistributive Capacity (R) 0.59
- Iran IR: Progressive Capacity (P) 2.22; Aggregate Tax Rate (τ) 20.41; Redistributive Capacity (R) 0.57
- Mongolia MN: Progressive Capacity (P) 3.55; Aggregate Tax Rate (τ) 13.05; Redistributive Capacity (R) 0.53
- Kazakhstan KZ: Progressive Capacity (P) 2.70; Aggregate Tax Rate (τ) 14.40; Redistributive Capacity (R) 0.45
- Colombia CO: Progressive Capacity (P) 46.80; Aggregate Tax Rate (τ) 0.84; Redistributive Capacity (R) 0.40
- Paraguay PY: Progressive Capacity (P) 46.57; Aggregate Tax Rate (τ) 0.76; Redistributive Capacity (R) 0.36
- Serbia RS: Progressive Capacity (P) 47.59; Aggregate Tax Rate (τ) 0.70; Redistributive Capacity (R) 0.33
- Montenegro, Rep. of ME: Progressive Capacity (P) 2.48; Aggregate Tax Rate (τ) 9.92; Redistributive Capacity (R) 0.27
- Pakistan PK: Progressive Capacity (P) 46.66; Aggregate Tax Rate (τ) 0.53; Redistributive Capacity (R) 0.25
- Ecuador EC: Progressive Capacity (P) 47.30; Aggregate Tax Rate (τ) 0.50; Redistributive Capacity (R) 0.24

*References and appendices as listed in the source PDF.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021252-print-pdf.pdf_