31-10-2016, 12:21 PM
NUMBERS EQUIVALENTS, RELATIVE ENTROPY, AND CONCENTRATION RATIOS: A COMPARISON USING MARKET PERFORMANCE
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INTRODUCTION
Economists who are interested in market
structure search for measures of seller concentration
for two purposes: (1) to classify
individual markets into theoretical boxes--
competitive, monopolistically competitive,
oligopolistic, with or without differentiation
-and; (2) to facilitate cross-section comparisons.
The most frequently used measure,
the four- or eight-firm concentration ratio,
is readily available and easily understood.
It is the share of industry output (or employment
or assets) accounted for by the
largest K firms, and the Bureau of the Census
publishes concentration data for K = 4, 8,
20, and 50 as part of its Census of Manufactures.
Alternative or additional measures
of market structure include the Gini coefficient,
the Herfindahl-Hirschman index(es),
entropy, and various marginal concentration
ratios, but all (save the last) suffer
from a common problem: their availability
is quite limited, and hence their use either
requires some computational estimation or
is impossible.
Even so, entropy (or various manipulations)
seems to be a current front runner as
a rival to concentration ratios. Defined' as
H = -
E!-1 xi logs 1 , where n is the number
of firms and xi is a size variable (assets,
value of shipments, employment) in firm
shares, it presents problems of comprehension
and interpretation not embraced by
concentration ratios. Its counterpart, the
numbers-equivalent2 (F), is more easily
understood. Consider an industry generating
a particular value for H. If all firms were
of equal size, then the definition of H reduces
to
- 'Ex log2 i = - (1/n) 0log2 (1/n)
= - (1/n) [n(log2 1 - log2 n)]
log2 n.
Equating this to the actual value of H and
solving for n (= F): H = log2 F. That is,
given H and n actually existing in an industry,
F suggests the "number of equalsized
firms necessary to generate a level of
concentration comparable to that obtaining
in the industry [where concentration is
measured by H]" [4, 396]. Thus F = 2, 4,
40, 500 suggest respectively duopoly, oligopoly,
monopolistic competition, and perfect
competition.
Another variant of H is relative entropy,
defined as G = H/log2 n, where H and n
pertain to values of an actual market. If all
firms are actually identical in size, G reaches
its maximum of unity (this corresponds to a
straight line diagonal in a Lorenz diagram
and to a straight line in a cumulative concentration
curve from (0, 0) to (n, 100)). If
firms differ in size, H < log2 n and G < 1.
Relative entropy is thus a comparison of the
actual H with what H would have been if all
the actual firms (an actual n of them) had
been of the same size. Thus G suggests "the
* I am indebted to Ira Horowitz and Jon Joyce
for their comments.
SOften H is used to designate the HerfindahlHirschman
index: