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

In quantitative economic history and statistical income distribution analysis, various Italian school economists introduced alternatives to the standard Gini coefficient that model inequality from distinct geometric and normative perspectives. The De Vergottini Inequality Index (V) was developed by Mario De Vergottini to assess inequality based on the ratio of income received by the upper versus lower half of the population, weighted by rank distance. By utilizing modern dynamic array formulas—specifically VSTACK, HSTACK, LAMBDA, LET, SORT, and SCAN—analysts can construct dynamic De Vergottini models in Microsoft Excel without legacy VBA macros.

Mathematical Formulation of the De Vergottini Index

For an income distribution of N individuals sorted in ascending order y₁ ≤ y₂ ≤ … ≤ y_N with total income Y = ∑ y_i, let L_i = (1 / Y) * ∑_{j=1}^i y_j represent the cumulative income share of the poorest i individuals, and P_i = i / N represent the cumulative population share.

The De Vergottini Index evaluates the sum of deviations between population shares and income shares weighted by the distance from the median position:

V = ∑_{i=1}^{N-1} (P_i - L_i) / ∑_{i=1}^{N-1} P_i

The denominator ∑ P_i = (N - 1) / 2 represents the theoretical maximum deviation possible under complete concentration. The index ranges between 0.0 (perfect equality where L_i = P_i for all i) and 1.0 (pure monopoly). The De Vergottini index provides a direct bridge between traditional Lorenz curve geometry and discrete percentile gap analysis.

Creating the Reusable LAMBDA Function in Name Manager

In Microsoft Excel (Microsoft 365 or Excel 2024), navigate to Formulas > Name Manager. Click New, enter the name DE_VERGOTTINI_INDEX, and define the formula as follows:

=LAMBDA(incomes,
    LET(
        clean, FILTER(incomes, (incomes > 0) * ISNUMBER(incomes)),
        sorted, SORT(clean, 1, 1),
        n, ROWS(sorted),
        tot_inc, SUM(sorted),
        cum_inc, SCAN(0, sorted, LAMBDA(acc, val, acc + val)),
        p_seq, SEQUENCE(n - 1, 1, 1, 1) / n,
        l_seq, TAKE(cum_inc, n - 1) / tot_inc,
        diff_sum, SUM(p_seq - l_seq),
        max_sum, SUM(p_seq),
        v_index, diff_sum / max_sum,
        v_index
    )
)

This LAMBDA automatically sorts observations in ascending order, calculates cumulative income proportions via SCAN, normalizes population sequences, and computes the ratio against theoretical maximum dispersion.

Constructing the Full Dynamic Scorecard with VSTACK

Suppose household income figures are stored in cells A2:A20. In cell C2, enter this self-contained reporting formula:

=LET(
    inc, A2:A20,
    v_val, DE_VERGOTTINI_INDEX(inc),
    clean_inc, FILTER(inc, (inc > 0) * ISNUMBER(inc)),
    sorted_inc, SORT(clean_inc, 1, 1),
    n, ROWS(sorted_inc),
    tot, SUM(sorted_inc),
    
    // Compute Gini benchmark for comparison
    gini_val, (2 * SUM(SEQUENCE(n, 1, 1, 1) * sorted_inc) / (n * tot)) - ((n + 1) / n),
    
    header, HSTACK("Distribution Metric", "Computed Value", "Analytical Assessment"),
    r1, HSTACK("De Vergottini Index (V)", ROUND(v_val, 4), IF(v_val > 0.45, "Severe Distributional Skew", "Moderate Dispersion")),
    r2, HSTACK("Gini Coefficient (G)", ROUND(gini_val, 4), IF(gini_val > 0.4, "High National Inequality", "Moderate Inequality")),
    r3, HSTACK("Metric Ratio (V / G)", ROUND(v_val / gini_val, 3) & "x", "Relative weighting divergence"),
    r4, HSTACK("Active Population Count", n, "Total valid observations"),
    
    VSTACK(header, r1, r2, r3, r4)
)

Analytical Advantages in Welfare Economics

1. Geometric Transparency: Expresses inequality directly as a normalized fraction of maximum potential Lorenz curve area without complex integrals.
2. Instant Spilled Updates: Modifying or appending sample survey numbers immediately recalculates all cumulative shares and index values across the spreadsheet.
3. Zero-Macro Portability: 100% macro-free formula architecture that functions seamlessly across Excel for Windows, Mac, iOS, and Excel on the Web.