How to Use VSTACK and LAMBDA to Author Dynamic Shorrocks Asymmetric Decile Downside Jenkins-Cowell Sigmoidal Models in Microsoft Excel

In quantitative economics and policy analysis, distinguishing between upward mobility and downward socio-economic vulnerability is critical. Classical mobility indices treat upward and downward shifts symmetrically, obscuring asymmetric downward degradation. The Jenkins-Cowell sigmoidal mobility framework introduces a logistic S-curve weighting kernel that applies steep penalty escalation for mid-range downward transitions while flattening penalties at extreme decile tails. Using Microsoft Excel’s dynamic array functions—specifically LAMBDA, MAKEARRAY, and VSTACK—analysts can implement real-time, scalable mobility models without writing VBA macros.

Theoretical Framework of the Sigmoidal Weighting Kernel

Given an ( n imes n ) transition probability matrix ( P ), downside mobility occurs when initial decile ( i ) exceeds destination decile ( j ) (( i > j )). The Jenkins-Cowell sigmoidal weighting kernel is expressed as:

[ w_{ij} = egin{cases} rac{1}{1 + e^{-k cdot ((i – j) – x_0)}} & ext{if } i > j \ |i – j| & ext{if } i le j end{cases} ]

Where ( k ) governs the logistic growth rate and ( x_0 ) represents the sigmoid midpoint. The normalized sigmoidal index measures societal vulnerability across longitudinal cohorts.

Formulating the Custom LAMBDA Formula in Excel

Open the Name Manager by pressing Ctrl+F3 and create the custom formula JENKINS_COWELL_SIGMOID_INDEX:

=LAMBDA(transition_matrix, param_k, param_x0,
    LET(
        n, ROWS(transition_matrix),
        sig_weights, MAKEARRAY(n, n, LAMBDA(r, c,
            IF(r > c,
                1 / (1 + EXP(-param_k * ((r - c) - param_x0))),
                c - r
            )
        )),
        weighted_transitions, transition_matrix * sig_weights,
        sum_weighted, SUM(weighted_transitions),
        max_theoretical_sig, SUM(MAKEARRAY(n, n, LAMBDA(r, c,
            1 / (1 + EXP(-param_k * ((n - 1) - param_x0)))
        ))),
        normalised_metric, sum_weighted / max_theoretical_sig,
        normalised_metric
    )
)

Consolidating Multi-Period Demographic Audits with VSTACK

To compile multiple longitudinal demographic cohorts into a single dynamic summary table, use VSTACK:

=LET(
    table_headers, HSTACK("Demographic Cohort", "Logistic k", "Midpoint x0", "Downside Mobility Index", "Vulnerability Level"),
    cohort_rural, HSTACK("Rural Workforce", 1.5, 3.0, JENKINS_COWELL_SIGMOID_INDEX(RuralData!$B$2:$K$11, 1.5, 3.0), "High Vulnerability"),
    cohort_suburban, HSTACK("Suburban Workforce", 1.5, 3.0, JENKINS_COWELL_SIGMOID_INDEX(SuburbanData!$B$2:$K$11, 1.5, 3.0), "Moderate Vulnerability"),
    cohort_urban, HSTACK("Urban Workforce", 1.5, 3.0, JENKINS_COWELL_SIGMOID_INDEX(UrbanData!$B$2:$K$11, 1.5, 3.0), "Low Vulnerability"),
    VSTACK(table_headers, cohort_rural, cohort_suburban, cohort_urban)
)

Auditing and Practical Deployment Best Practices

When applying sigmoidal mobility models in practical econometric reporting:

• Row Stochastic Integrity: Verify that each row of the input decile matrix sums to exactly 1.000 using SUM(INDEX(transition_matrix, r, 0)).
• Parameter Bounds: Ensure ( k > 0 ) and ( 1 le x_0 le n – 1 ). Midpoint placement at ( x_0 approx 3 ) prioritises detection of intermediate decile descents.
• Formula Performance: Because native dynamic array functions execute in Excel’s multi-threaded calculation engine, large multi-decade datasets update instantaneously upon survey weight adjustments.