Economics & Market Microstructure
Hannah-Kay Generalized Concentration Index HK(alpha) Formula: Numbers-Equivalent Firms
Unify Shannon Entropy (alpha -> 1), Herfindahl-Hirschman 1/HHI (alpha = 2), and large-firm dominance weighting in a single numbers-equivalent market structure formula: N_HK(alpha) = (SUM(s_i^alpha))^(1 / (1 - alpha)).
One Rényi-entropy parameter alpha unifies Firm Count (alpha=0), Entropy (alpha->1), and 1/HHI (alpha=2)!
Formulated by Leslie Hannah and John A. Kay (1977, Concentration in Modern Industry), the Hannah-Kay Numbers-Equivalent Index N_HK(alpha) = (SUM_{i=1..N} s_i^alpha)^(1 / (1 - alpha)) (for alpha > 0, alpha != 1, with continuous Shannon-entropy limit EXP(-SUM(s_i * LN(s_i))) at alpha = 1) solves a classic antitrust dilemma: how heavily should merger screening weight the top 2 giants versus smaller rivals?
Every alpha yields an exact "Numbers-Equivalent" count of equal-sized firms (and reciprocal concentration ratio HK_conc = 1 / N_HK): at alpha = 0, N_HK(0) = N (raw firm count); as alpha -> 1, N_HK(1) = EXP(Shannon Entropy); at alpha = 2, N_HK(2) = 1 / SUM(s_i^2) = 1 / HHI (Adelman’s numbers-equivalent of Herfindahl!); and at alpha = 3 or 4, N_HK focuses almost exclusively on dominant-leader asymmetry!
| Sensitivity Parameter alpha on Shares [0.50, 0.25, 0.15, 0.10] | Hannah-Kay N_HK(alpha) Equivalent Firms | Concentration Ratio 1 / N_HK(alpha) |
|---|---|---|
| alpha = 0.5 (Emphasizes Fringe Presence) | 3.6505 Equal Firms | 0.2739 |
| alpha = 1.0 (Shannon Entropy Limit EXP(-SUM s*LN s)) | 3.3467 Equal Firms | 0.2988 |
| alpha = 2.0 (Exact Reciprocal of HHI = 0.3450) | 2.8986 Equal Firms | 0.3450 (= HHI!) |
| alpha = 3.0 (Dominant-Firm Sensitivity) | 2.6261 Equal Firms | 0.3808 |
Hannah-Kay Numbers-Equivalent Firm Count N_HK(alpha) (Includes Exact alpha = 1 Shannon Limit)
Computes (SUM(s_i^alpha))^(1 / (1 - alpha)) for positive firm revenues B2:B5 and elasticity parameter alpha >= 0 in D2 (evaluating EXP(-SUM(s_i * LN(s_i))) when alpha = 1).
=IF(OR(MIN(B2:B5) <= 0, D2 < 0), "Requires Positive Firm Revenues & alpha >= 0", IF(D2 = 1, ROUND(EXP(-((B2 / SUM(B2:B5)) * LN(B2 / SUM(B2:B5)) + (B3 / SUM(B2:B5)) * LN(B3 / SUM(B2:B5)) + (B4 / SUM(B2:B5)) * LN(B4 / SUM(B2:B5)) + (B5 / SUM(B2:B5)) * LN(B5 / SUM(B2:B5)))), 4), ROUND((SUMPRODUCT((B2:B5 / SUM(B2:B5)) ^ D2)) ^ (1 / (1 - D2)), 4)))Hannah-Kay Generalized Concentration Ratio 1 / N_HK(alpha)
Returns the 0-to-1 concentration index (matching HHI identically when D2 = 2).
=IF(OR(MIN(B2:B5) <= 0, D2 < 0), "Requires Positive Firm Revenues & alpha >= 0", IF(D2 = 1, ROUND(1 / EXP(-((B2 / SUM(B2:B5)) * LN(B2 / SUM(B2:B5)) + (B3 / SUM(B2:B5)) * LN(B3 / SUM(B2:B5)) + (B4 / SUM(B2:B5)) * LN(B4 / SUM(B2:B5)) + (B5 / SUM(B2:B5)) * LN(B5 / SUM(B2:B5)))), 4), ROUND(1 / ((SUMPRODUCT((B2:B5 / SUM(B2:B5)) ^ D2)) ^ (1 / (1 - D2))), 4)))Try the example
Use the original values or change them to test your understanding.