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

While the Gini coefficient is the most ubiquitous index for measuring income or wealth inequality, it embodies an implicit normative assumption: it weights transfers near the median of the distribution more heavily than transfers at the extreme tails. In contrast, the Atkinson Inequality Index introduces an explicit parameter ε (ε ≥ 0) representing societal “inequality aversion.” As ε increases, the index becomes progressively more sensitive to disparities at the lower tail of the income distribution. By leveraging modern Excel dynamic array functions—specifically VSTACK, HSTACK, LAMBDA, LET, and REDUCE—analysts can construct dynamic Atkinson sensitivity models in Microsoft Excel without legacy VBA macros.

Mathematical Foundations of the Atkinson Index

For an income distribution of N individuals with incomes y₁, y₂, …, y_N and mean income μ, the Atkinson Index with inequality aversion parameter ε is formulated as:

A(ε) = 1 - (1 / μ) * [ (1 / N) * ∑_{i=1}^N (y_i)^(1 - ε) ]^(1 / (1 - ε))   (for ε ≠ 1)

A(1) = 1 - (1 / μ) * [ ∏_{i=1}^N y_i ]^(1 / N)   (for ε = 1, geometric mean ratio)

The term inside the brackets represents the “equally distributed equivalent income” (y_ede)—the level of income per person which, if distributed equally to everyone, would yield the same total social welfare as the current unequal distribution. The Atkinson score A(ε) represents the percentage of total income that could be sacrificed without reducing social welfare if equal distribution were achieved.

Creating the Reusable LAMBDA Function in Name Manager

In Microsoft Excel (Microsoft 365 or Excel 2024), navigate to Formulas > Name Manager and define a new name titled ATKINSON_INDEX with the following formula:

=LAMBDA(incomes, epsilon,
    LET(
        clean, FILTER(incomes, (incomes > 0) * ISNUMBER(incomes)),
        n, ROWS(clean),
        mu, AVERAGE(clean),
        IF(epsilon = 1,
            LET(
                geo_mean, EXP(AVERAGE(LN(clean))),
                1 - (geo_mean / mu)
            ),
            LET(
                power_sum, AVERAGE(clean ^ (1 - epsilon)),
                y_ede, power_sum ^ (1 / (1 - epsilon)),
                1 - (y_ede / mu)
            )
        )
    )
)

This LAMBDA handles both standard power exponents and the logarithmic geometric mean singularity when ε = 1.

Constructing the Dynamic Sensitivity Matrix with VSTACK

Suppose household survey incomes are stored in cells A2:A20. In cell C2, insert this dynamic reporting formula to evaluate inequality under multiple aversion assumptions (ε = 0.25, 0.5, 1.0, 1.5, 2.0):

=LET(
    inc, A2:A20,
    eps_list, {0.25; 0.5; 1; 1.5; 2},
    mu_val, AVERAGE(FILTER(inc, (inc > 0) * ISNUMBER(inc))),
    
    header, HSTACK("Inequality Aversion (ε)", "Atkinson Index A(ε)", "Equally Distributed Income (y_ede)", "Societal Welfare Sacrifice"),
    
    calc_row, LAMBDA(e,
        LET(
            a_val, ATKINSON_INDEX(inc, e),
            y_ede, mu_val * (1 - a_val),
            HSTACK(e, ROUND(a_val, 4), TEXT(y_ede, "£#,##0"), TEXT(a_val, "0.0%"))
        )
    ),
    
    data_rows, DROP(REDUCE("", eps_list, LAMBDA(acc, e, VSTACK(acc, calc_row(e)))), 1),
    VSTACK(header, data_rows)
)

Interpreting Atkinson Sensitivity Output

1. Welfare Interpretation: An Atkinson score of 0.28 at ε = 1.0 means that if income were distributed completely equally, society could achieve the same level of social welfare with only 72% of the total existing income pool (sacrificing 28%).
2. Parameter Elasticity: If A(ε) surges sharply between ε = 0.5 and ε = 2.0, severe poverty and extreme deficits are concentrated in the bottom decile rather than spread evenly across middle income earners.
3. Zero-Macro Portability: Operates natively in Excel Online, iOS, and desktop without requiring macro permissions or security trust settings.