Financial & Accounting
Atkinson Inequality Index & Equally Distributed Equivalent Income (y_EDE) Formula
Measure social-welfare-weighted income inequality with tunable bottom-tail sensitivity epsilon != 1 using Sir Anthony Atkinson’s 1970 formula A(epsilon) = 1 - y_EDE / mu, where y_EDE = (MEAN(y_i^(1 - epsilon)))^(1 / (1 - epsilon)).
Why Anthony Atkinson’s 1970 welfare index reveals normative trade-offs hidden inside a single Gini number!
Whereas the Gini coefficient applies a fixed rank-order weight loudest near the middle of the income distribution, Sir Anthony B. Atkinson (1970, Journal of Economic Theory) grounded inequality measurement directly in social welfare via an explicit inequality-aversion parameter epsilon >= 0! Atkinson defined the Equally Distributed Equivalent income y_EDE as the uniform per-capita income that yields the EXACT same social welfare as the unequal actual distribution: for epsilon != 1, y_EDE = ((1/N) * SUM(y_i^(1 - epsilon)))^(1 / (1 - epsilon)), and Atkinson’s Index is A(epsilon) = 1 - y_EDE / mu!
The economic interpretation is crystal clear: A(epsilon) = 0.1049 means society could achieve the same total welfare with 10.49% LESS aggregate income if distributed equally! Across quartile incomes [$20k, $35k, $60k, $125k] (mean mu = $60,000), mild aversion epsilon = 0.50 gives y_EDE = $53,708.83 and A(0.5) = 0.1049—jumping to A(2.0) = 0.3542 (harmonic-mean EDE = $38,747.88) under strong Rawlsian-leaning bottom-tail aversion epsilon = 2.0!
| Inequality Aversion Parameter epsilon | Equally Distributed Equivalent y_EDE | Atkinson Index A(epsilon) = 1 - y_EDE/mu | Distributional Welfare Interpretation |
|---|---|---|---|
| epsilon = 0.50 (Moderate Aversion) | $53,708.83 (vs $60,000 Mean) | 0.1049 (10.49% Welfare Loss) | Balanced sensitivity across brackets |
| epsilon = 2.00 (High Bottom-Tail Aversion) | $38,747.88 (Harmonic Mean) | 0.3542 (35.42% Welfare Loss) | Heavily weights lowest-income $20k quartile |
| epsilon = 1.00 or Non-Positive Income <= 0 | Requires Geometric Mean / y > 0 | "Requires Positive Incomes & epsilon != 1" | Power-Mean Singularity Guard |
Atkinson Inequality Index A(epsilon) = 1 - ((SUM(y^(1-eps))/N)^(1/(1-eps))) / AVERAGE(y)
Evaluates Atkinson’s index (0 <= A < 1) for 4 income brackets B2:B5 and aversion parameter epsilon != 1 (B6 >= 0).
=IF(OR(MIN(B2:B5) <= 0, B6 < 0, B6 = 1), "Requires Positive Incomes & epsilon != 1", ROUND(1 - (((((B2 ^ (1 - B6)) + (B3 ^ (1 - B6)) + (B4 ^ (1 - B6)) + (B5 ^ (1 - B6))) / 4) ^ (1 / (1 - B6))) / AVERAGE(B2:B5)), 4))Atkinson Equally Distributed Equivalent Income y_EDE ($)
Computes the generalised power mean y_EDE in currency units.
=IF(OR(MIN(B2:B5) <= 0, B6 < 0, B6 = 1), "Requires Positive Incomes & epsilon != 1", ROUND((((B2 ^ (1 - B6)) + (B3 ^ (1 - B6)) + (B4 ^ (1 - B6)) + (B5 ^ (1 - B6))) / 4) ^ (1 / (1 - B6)), 2))Try the example
Use the original values or change them to test your understanding.