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

Assessing longitudinal income dynamics requires directional mobility indices capable of measuring the welfare loss associated with downward economic churn. The Van Kerm mobility framework adapts generalized Gini rank-dependent mobility to directional transitions, 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 Van Kerm downside mobility models without external statistical packages.

Mathematical Foundations of Van Kerm Downside Mobility

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

$$VK_{down} = sum_{i > j} P_{ij} cdot left(rac{i – j}{n} ight) cdot left[ u cdot left(1 – rac{j}{n} ight)^{ u – 1} ight]$$

When $ u > 1$, downward transitions landing in the lowest income tiers exert an increasingly heavy penalty on the aggregate mobility index, highlighting societal vulnerability.

Authoring the Dynamic LAMBDA Formula

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

=LAMBDA(transition_matrix, nu_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, 0),
        rank_weight, IF(is_downside, nu_param * ((1 - (col_seq / n)) ^ (nu_param - 1)), 0),
        weighted_loss, IF(is_downside, transition_matrix * rel_drop * rank_weight, 0),
        vk_score, SUM(weighted_loss),
        vk_score
    )
)

Stacking Multi-Period Longitudinal Results with VSTACK

Consolidate 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],
    nu, 2.2,
    vk1, VANKERM_DOWNSIDE(cohort1990, nu),
    vk2, VANKERM_DOWNSIDE(cohort2000, nu),
    vk3, VANKERM_DOWNSIDE(cohort2010, nu),
    vk4, VANKERM_DOWNSIDE(cohort2020, nu),
    headers, HSTACK("Observation Window", "Van Kerm Downside Vulnerability Index"),
    r1, HSTACK("1990-2000", vk1),
    r2, HSTACK("2000-2010", vk2),
    r3, HSTACK("2010-2020", vk3),
    r4, HSTACK("2020-2030 (Est)", vk4),
    VSTACK(headers, r1, r2, r3, r4)
)

Sensitivity Testing and Model Verification

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