Economics & Market Microstructure
Prais-Shorrocks Income Mobility Index M(P) Formula: Markov Transition Matrix Trace
Quantify intergenerational or credit-rating transition mobility from a K x K Markov matrix P using the Prais-Shorrocks Trace Index M(P) = (K - trace(P)) / (K - 1) and Mean Class Stay Duration 1 / (1 - p_ii).
Turn a K x K income-bracket or credit-rating transition matrix into a single [0, 1+] Mobility Index!
Two economies can exhibit identical cross-sectional Gini coefficients yet wildly different life chances: in a rigid caste-like economy every diagonal persistence probability p_ii = 1 (trace(P) = K), whereas in an Equal-Opportunity origin-independent society every row equals [1/K, ..., 1/K] (trace(P) = 1)!
Derived by S. J. Prais (1955) and axiomatized by Anthony F. Shorrocks (1978, Econometrica), the Trace Mobility Index M(P) = (K - SUM_{i=1..K} p_ii) / (K - 1) normalizes mobility cleanly: M(I) = 0 under complete immobility, M(P) = 1.000 under origin-independence, and each bracket’s geometric expected spell duration before moving is E[T_i] = 1 / (1 - p_ii) periods!
| 3x3 Tertile Transition Regime | Diagonal Persistence [p_11, p_22, p_33] | Matrix Trace tr(P) | Prais-Shorrocks Mobility M(P) |
|---|---|---|---|
| Complete Immobility (Identity Matrix I) | [1.00, 1.00, 1.00] | 3.00 | 0.0000 |
| Observed Empirical Panel | [0.60, 0.55, 0.70] | 1.85 | 0.5750 |
| Origin-Independent Equal Chance | [0.333, 0.333, 0.333] | 1.00 | 1.0000 |
Prais-Shorrocks Markov Trace Mobility Index M(P) = (K - trace(P)) / (K - 1)
Validates that all 3 rows of transition matrix B2:D4 are non-negative and sum to 1.00 (within 0.001), then computes (3 - (B2 + C3 + D4)) / 2.
=IF(OR(MIN(B2:D4) < 0, ABS(SUM(B2:D2) - 1) > 0.001, ABS(SUM(B3:D3) - 1) > 0.001, ABS(SUM(B4:D4) - 1) > 0.001), "Invalid Row-Stochastic Transition Matrix", ROUND((3 - (B2 + C3 + D4)) / 2, 4))Expected Spell Duration in Origin Bracket i: E[T_i] = 1 / (1 - p_ii)
Computes average consecutive periods spent in Bracket 1 before exiting (guarding absorbing state p_11 = 1).
=IF(OR(B2 < 0, B2 > 1), "Invalid Probability", IF(B2 = 1, "Absorbing State (Infinite Stay)", ROUND(1 / (1 - B2), 2)))Try the example
Use the original values or change them to test your understanding.