Economics & Market Microstructure
Kolm-Pollak Absolute Inequality Index K(kappa) & Translation-Invariant EDE Income
Measure absolute dollar income dispersion using Serge-Christophe Kolm’s (1976) Leftist Inequality Index K(kappa) = (1 / kappa) * LN((1 / N) * SUM(EXP(kappa * (mu - y_i)))) and Equally Distributed Equivalent income mu - K(kappa).
Why Gini shrinks when every household gets +$10,000 — while Kolm’s Absolute Index stays constant!
Scale-invariant relative indices (Gini, Atkinson, Theil) judge a 2x doubling of all incomes [$20k, $80k] -> [$40k, $160k] as "unchanged inequality" even though the dollar gap tripled from $60k to $120k! Conversely, Serge-Christophe Kolm (1976) and Robert Pollak axiomatized Absolute ("Leftist") Inequality Aversion: adding a uniform lump-sum transfer +$c to every person’s income y_i + c leaves absolute dollar gaps unchanged and therefore leaves K(kappa) = (1 / kappa) * LN((1 / N) * SUM_{i=1..N} EXP(kappa * (mu - y_i))) identically invariant!
Here kappa > 0 (in inverse income units, e.g., per $1,000) sets absolute inequality aversion: K(kappa) >= 0 is measured in the same currency units as income, representing the per-capita welfare cost of dispersion, so the social Equally Distributed Equivalent income is simply y_EDE = mu - K(kappa)!
| Income Vector ($k) | Mean Income mu ($k) | Kolm Index K(kappa = 0.05) | Social EDE Income mu - K ($k) |
|---|---|---|---|
| Baseline [20, 40, 60, 80] | $50.0000k | $11.0779k | $38.9221k |
| Uniform +$100k Transfer [120, 140, 160, 180] | $150.0000k | $11.0779k (Exact Translation Invariance!) | $138.9221k |
| Complete Equality [50, 50, 50, 50] | $50.0000k | $0.0000k | $50.0000k |
Kolm-Pollak Absolute Inequality Index K(kappa) (in Currency Units)
Evaluates (1 / kappa) * LN((1 / 4) * SUM_{i=2..5} EXP(kappa * (AVERAGE(B2:B5) - B_i))) for incomes B2:B5 and aversion parameter kappa > 0 in D2.
=IF(D2 <= 0, "Requires Positive Aversion Parameter kappa > 0", ROUND((1 / D2) * LN((EXP(D2 * (AVERAGE(B2:B5) - B2)) + EXP(D2 * (AVERAGE(B2:B5) - B3)) + EXP(D2 * (AVERAGE(B2:B5) - B4)) + EXP(D2 * (AVERAGE(B2:B5) - B5))) / 4), 4))Kolm Equally Distributed Equivalent Income y_EDE = mu - K(kappa)
Subtracts the absolute inequality welfare loss K(kappa) from arithmetic mean income AVERAGE(B2:B5).
=IF(D2 <= 0, "Requires Positive Aversion Parameter kappa > 0", ROUND(AVERAGE(B2:B5) - (1 / D2) * LN((EXP(D2 * (AVERAGE(B2:B5) - B2)) + EXP(D2 * (AVERAGE(B2:B5) - B3)) + EXP(D2 * (AVERAGE(B2:B5) - B4)) + EXP(D2 * (AVERAGE(B2:B5) - B5))) / 4), 4))Try the example
Use the original values or change them to test your understanding.