Table 1.
Proposed concentration indices based on probabilities (proportions, market shares) p1≥p2≥⋯≥pn and their lacking properties (LP, in Section 2.1).
Fig 1.
The value of CK in (17) corresponds to the area above the step function formed by the cumulative pi‘s and the reciprocal rank 1/i where the cumulative proportions (CP) = 0 for 0<1/i<1/n; CP = pn for 1/n≤1/i<1/(n−1); CP = pn+pn−1 for 1/(n−1)≤1/i<1/(n−2),…; CP = pn+pn−1+⋯+p2 for 1/2≤1/i<1; and CP = 1 for 1/i = 1.
This exemplary graph is for the distribution P5 = (0.40, 0.30, 0.15, 0.10, 0.05) with CK = 0.635.
Fig 2.
Comparison of values of CR4 and CK for 1000 randomly generated market-share distributions Pn = (p1,…,pn) with the number of firms n varying as a random integer between 5 and 100.
Fig 3.
Comparison of values of HHI and CK for 1000 randomly generated market-share distributions Pn = (p1,…,pn) with the number of firms n varying as a random integer between 5 and 100.
Table 2.
Values of CK in (17), HHI defined in Table 1, and CKH in (25) for in (9) with varying λ and n.
Table 3.
Values of CK in (17), in (21) and (26), d* in (26), HHI and CR4 defined in Table 1, and CKH in (25) for randomly generated Pn = (p1,…,pn) and 2≤n≤30.
Table 4.
Values of CK in (17), in (21) and (26), d* in (26), HHI defined in Table 1, and CKH in (25) for a sample of real market-share data.
Fig 4.
Scatter diagram of HHI versus CK from the data in Table 2 (dots), Table 3 (crosses) and Table 4 (circles).
The curve represents the fitted model in (25).
Table 5.
Values of CK in (17), in (21), d* in (26), HHI defined in Table 1, and CKH in (25) for randomly generated Pn = (p1,…,pn) and 2≤n≤10.