How to Use VSTACK and LAMBDA to Author Dynamic Kolm Inequality Index Models in Microsoft Excel

Most common inequality measures—including the Gini coefficient, Atkinson index, and Theil entropy metric—are relative inequality indices, meaning they satisfy scale invariance. If every individual’s income doubles, relative inequality remains completely unchanged. However, in welfare economics and wage compensation analysis, absolute inequality measures are critical: if everyone receives an equal flat currency increase, absolute inequality remains unchanged (translation invariance), whereas doubling everyone’s income increases absolute inequality. The Kolm inequality index is the premier mathematical standard for measuring absolute inequality. Modern Microsoft Excel dynamic array functions—specifically VSTACK, LAMBDA, and LET—allow financial analysts to model dynamic Kolm absolute inequality indices cleanly without legacy macros.

Mathematical Foundations of the Kolm Index

Given a distribution vector $y_1, y_2, dots, y_n$ with arithmetic mean $mu$ and inequality aversion parameter $alpha > 0$, the discrete Kolm inequality index $K(alpha)$ is defined as:

K(alpha) = (1 / alpha) * ln((1 / n) * SUM(exp(alpha * (mu - y_i))))

Where:

• n denotes the total count of valid observation units.
• mu represents the arithmetic mean of the distribution vector.
• alpha ($alpha$) represents the policy analyst’s degree of absolute inequality aversion. Higher values place greater priority on reducing absolute disparities at the bottom of the distribution.
• exp(alpha * (mu - y_i)) penalises observations falling below the arithmetic mean exponentially.

Implementing the Kolm Index Formula in Excel

Using the LET function, we can construct the mathematical calculation into readable intermediate steps that handle uncurated columns dynamically:

=LET(
    raw_data, A2:A101,
    alpha, 0.05,
    clean_data, FILTER(raw_data, ISNUMBER(raw_data) * (raw_data > 0)),
    n, COUNT(clean_data),
    mu, AVERAGE(clean_data),
    exp_terms, EXP(alpha * (mu - clean_data)),
    mean_exp, AVERAGE(exp_terms),
    kolm_index, (1 / alpha) * LN(mean_exp),
    kolm_index
)

This dynamic formula removes non-numeric entries, calculates the distribution mean, computes exponential deviations per observation, and derives the exact Kolm absolute inequality index instantly.

Encapsulating the Formula into a Reusable Custom LAMBDA

To deploy this calculation across enterprise workbooks without repeatedly pasting complex formulas, define a reusable custom function named KOLM.INDEX in the Excel Name Manager:

1. Open the Formulas tab and click Name Manager.
2. Click New, enter KOLM.INDEX in the Name field, and set Scope to Workbook.
3. In the Refers to box, input the following formula:

   =LAMBDA(values, alpha_param,
       LET(
           clean, FILTER(values, ISNUMBER(values) * (values > 0)),
           n, COUNT(clean),
           mu, AVERAGE(clean),
           (1 / alpha_param) * LN(AVERAGE(EXP(alpha_param * (mu - clean))))
       )
   )

4. Click OK and close the dialogue.

Building Sensitivity Analysis Tables with VSTACK

Because the Kolm index depends on the choice of inequality aversion parameter $alpha$, financial analysts frequently conduct sensitivity comparisons across multiple values of $alpha$ using VSTACK:

=LET(
    cohort, B2:B100,
    header, HSTACK("Aversion Parameter (Alpha)", "Kolm Absolute Index", "Mean Income"),
    row_1, HSTACK("Alpha = 0.01 (Mild)", KOLM.INDEX(cohort, 0.01), AVERAGE(cohort)),
    row_2, HSTACK("Alpha = 0.05 (Standard)", KOLM.INDEX(cohort, 0.05), AVERAGE(cohort)),
    row_3, HSTACK("Alpha = 0.10 (Severe)", KOLM.INDEX(cohort, 0.10), AVERAGE(cohort)),
    VSTACK(header, row_1, row_2, row_3)
)

This formula dynamically stacks sensitivity test rows into a cohesive comparison table. As underlying salary or revenue figures change, all absolute inequality calculations recalculate instantly, providing robust sensitivity analytics across corporate compensation and public welfare tiers.