How to Use VSTACK and LAMBDA to Author Dynamic Shorrocks Asymmetric Decile Downside Aaberge Mobility Models in Microsoft Excel

In quantitative welfare economics, assessing social mobility requires frameworks that integrate rank-dependent inequality measures into directional transition dynamics. The Aaberge mobility framework adapts Lorenz-based Gini principles to transitional matrices, allowing analysts to penalise downward transitions based on social aversion to bottom-tier stagnation. By implementing dynamic LAMBDA and VSTACK algorithms in Microsoft Excel, quantitative researchers can construct real-time Aaberge downside mobility models without external statistical packages.

Mathematical Foundations of Aaberge Downside Mobility

The Aaberge downside framework incorporates a rank-weighting distribution parameter $k ge 1$. For an $n imes n$ decile transition matrix $P$ where $P_{ij}$ represents the probability of moving from decile $i$ to decile $j$:

$$A_{ab, down} = sum_{i > j} P_{ij} cdot left[1 – left(1 – rac{i – j}{n – 1} ight)^k ight] cdot left(rac{n – i + 1}{n} ight)$$

When $k > 1$, downward transitions from lower-income tiers exert an increasingly heavy penalty on the aggregate mobility index, highlighting societal vulnerability.

Authoring the Dynamic LAMBDA Formula

Create a named formula AABERGE_DOWNSIDE inside Excel’s Name Manager:

=LAMBDA(transition_matrix, k_param,
    LET(
        n, ROWS(transition_matrix),
        row_seq, SEQUENCE(n, 1, 1, 1),
        col_seq, SEQUENCE(1, n, 1, 1),
        is_downside, row_seq > col_seq,
        rel_drop, IF(is_downside, (row_seq - col_seq) / (n - 1), 0),
        weight_factor, IF(is_downside, 1 - ((1 - rel_drop) ^ k_param), 0),
        origin_weight, (n - row_seq + 1) / n,
        weighted_loss, IF(is_downside, transition_matrix * weight_factor * origin_weight, 0),
        aaberge_score, SUM(weighted_loss),
        aaberge_score
    )
)

Stacking Multi-Period Longitudinal Results with VSTACK

Aggregate multi-decade panel survey data into an automated summary dashboard using VSTACK:

=LET(
    cohort1990, Panel1990[#Data],
    cohort2000, Panel2000[#Data],
    cohort2010, Panel2010[#Data],
    cohort2020, Panel2020[#Data],
    k, 2.0,
    a1, AABERGE_DOWNSIDE(cohort1990, k),
    a2, AABERGE_DOWNSIDE(cohort2000, k),
    a3, AABERGE_DOWNSIDE(cohort2010, k),
    a4, AABERGE_DOWNSIDE(cohort2020, k),
    headers, HSTACK("Observation Window", "Aaberge Downside Vulnerability Index"),
    r1, HSTACK("1990-2000", a1),
    r2, HSTACK("2000-2010", a2),
    r3, HSTACK("2010-2020", a3),
    r4, HSTACK("2020-2030 (Est)", a4),
    VSTACK(headers, r1, r2, r3, r4)
)

Sensitivity Testing and Model Verification

Ensure that all transition matrix rows sum to 100%. Adjusting $k$ in linked cells allows researchers to stress-test differing degrees of social risk aversion across all stacked cohort comparisons simultaneously.