G. Bruckmann
A Generalized
Concept
of Concentration
and its Measurement
"Our ultimate subject is not a collection of objects (establishments, firms), but a relationship among them, like gravity or electricity, which can be "seen" only by its effects". M.A. Adelman, {2}, p. 370 I. The
problem
The concepts
of concentration
are several,
not one.
I) First and foremost, one is likely to speak of an industry or a sector to be highly concentrated when a high percentage of total turnover (or labor force, production, etc. ) is accounted for by a small number of (large) firms, or of a small percentage of the firms; or, in more general terms, when the size distribution is strongly dispersed. 2) Secondly, concentration may arise from the fact that many of the firms of a certain sector are interlinked in some way or other. On first sight, the size distribution within a given branch might not be too strongly dispersed; but some (many, or all) of the firms might have combined their marketing actions. They might have exchanged part of their stock. Some might be under common management or might be owned by one holding company. 3) Furthermore, "concentration" on the domestic market may stem from the influence of foreign firms who might have acquired a substantial share of domestic sales in their respective branch. 4) Yet another aspect of concentration is given by the fact that large enterprises often are engaged in a w{de variety of activities, even within a single plant. The "monopoly power" of such firms, therefore, is much stronger than the share of one of its activities shows. Furthermore, efforts to group companies in order to measure concentration by product or by industry or by industrial segment encounter much uncertainty as to the proper location of the boundaries of the group. In measurements through time, the difficulty is compounded by frequent changes in the activities that companies carry on.
204
In t h e v a s t l i t e r a t u r on c o n c e n t r a t i o n , a l l of t h o s e a s p e c t s h a v e b e e n a n a l y s e d a n d d i s c u s s e d t h o r o u g h l y in a q u a l i t a t i v e m a n n e r . A n y q u a n t i t a t i v e i n v e s t i g a t i o n , h o w e v e r , a n d , in p a r t i c u l a r , a l l i n d i c e s of c o n c e n tration developed so far have restricted themselves to the first aspect of c o n c e n t r a t i o n o n l y . T h e c u s t o m a r y a p p r o a c h t o t h e c o n c e p t of c o n c e n tration has been the following: N e n t i t i e s ( i n d i v i d u a l s , f i r m s ) T . (i = 1, 2 . . . . . N) a r e g i v e n . . A v a l u e p. ( s h a r e of t o t a l i n c o m e , s h a r e of ~ o t a l t u r n o v e r a n d t h e l i k e ) i s a s s i g n e ~ t o e a c h of t h e s e e n t i t i e s , w i t h N
7 pi = 1 i= 1 T w o t y p e s of i n d i c e s c a n b e d i s t i n g u i s h e d : i n d i c e s of r e l a t i v e c o n c e n t r a t i o n ( i n e q u a l i t y ) o r i n d i c e s of a b s o l u t e c o n c e n t r a t i o n ( c o n c e n t r a t i o n ) . I n a r e c e n t p a p e r {3}, c e r t a i n c r i t e r i a w e r e e s t a b l i s h e d t h a t a n i n d e x m u s t m e e t to q u a l i f y e i t h e r a s a n i n d e x of r e l a t i v e c o n c e n t r a t i o n o r a s a n i n d e x of a b s o l u t e c o n c e n t r a t i o n . T h e m o s t i m p o r t a n t d i s t i n c t i o n b e t w e e n t h e m i s t h e f o l l o w i n g : If a r a n d o m s a m p l e of n e n t i t i e s (0 < n < N) i s t a k e n , a c o e f f i c i e n t of r e l a t i v e c o n c e n t r a t i o n c o m p u t e d f r o m t h e s a m p l e of s i z e n s h o u l d b e i n d e p e n d e n t of n, m o r e p r e c i s e l y , i t s e x p e c t e d v a l u e s h o u l d e q u a l t h e p o p u l a t i o n c o e f f i c i e n t of r e l a t i v e c o n c e n t r a t i o n . T h e v a l u e of a c o e f f i c i e n t of a b s o l u t e c o n c e n t r a t i o n , on t h e o t h e r h a n d , c l e a r l y d e p e n d s on n. N e i t h e r i n d e x i s s u p e r i o r t o t h e o t h e r ; in s o m e p r o b l e m s , a c o e f f i c i e n t of r e l a t i v e c o n c e n t r a t i o n , i n o t h e r p r o b l e m s a c o e f f i c i e n t of a b s o l u t e concentration gives the appropriate answer. It w a s s h o w n t h a t t h e G I N I - c o e f f i c i e n t ( t h e r a t i o of t h e a r e a b e t w e e n L o r e n z - c u r v e a n d d i a g o n a l to t h e a r e a of t h e t r i a n g l e ) q u a l i f i e s a s a n i n d e x of r e l a t i v e c o n c e n t r a t i o n , w h e r e a s a n e n t i r e c l a s s of c o e f f i c i e n t s w a s s h o w n to q u a l i f y a s i n d i c e s of a b s o l u t e c o n c e n t r a t i o n ( c f r . {3}). O n e s p e c i a l c a s e of t h e l a t t e r c l a s s of c o e f f i c i e n t s i s t h e w e l l - k n o w n H e r f i n d a h l - i n d e x , ( a l s o a t t r i b u t e d to H i r s c h m a n ) : 1
N 2 H=E Pi i= I A n o t h e r s p e c i a l c a s e of t h i s c l a s s i s t h e e x p o n e n t i a l of t h e e n t r o p y - m e a sure 7~Pilog Pi'
]7 Pi Pi 1
F o r a r e c e n t p a p e r on tt~e H - i n d e x ,
c f r . Ill .
2 05
Of the second, third and fourth aspect of concentration mentioned in the introductory remarks, it may well never be possible to express the fourth one quantitatively, at least not in the form of one coefficient. In this paper, however, a methodological solution will be offered to express quantitatively, in one coefficient, concentration arising both from the first and the second aspect, viz. from a dispersed size distribution and from relations between firms. This index, defined in section IV under (5), is applicable whenever a) it is possible to assess, at least roughly, the shares of individual firms and the strength of the relation between any two of them; b) these relations can, to a sufficient degree, be considered symmetric. On first sight, a relation between two firms will rarely be symmetric. If firm T. (with a share p.), e. g. , owns the majority of the stock of firm I 1 . . T. (with a share p.) the relation r.. between T. and T. is one of dominantj z c~. For the purpose of measuring concentration in th~at Branch, however, it might often be irrelevant whether T. dominates T. or vice versa: the fact remains that the two firms appea# on the market not in a separated, but rather in a combined way. In section II and III this concept of "relation" will be formalized; in section IV the index will be defined; section V gives a brief comment on the third aspect of concentration (influence of foreign firms), sections VI through IX show properties of the index and give some applications.
II. Relations
amongst
entities
The concept of a possible interrelation between lined above can be formalized as follows: Between defined, i)
any pair of entities T., T. (i, j -- i, 2 ..... N) a relation r.. is with the followzng properties: lj
0 <
r..
zj
:
1
(1)
The more interdependent rely independent, r.. = zj
2)
r.. lJ
entities (firms) as out.-
=
r.. Jt
T. and T., the larger n..; if T. and T. are entiL . I 0; tf T. aJnd T. are totafl~ znterdepen~ent, r.. = i. t
j
zj
(2)
T h e v a l u e of r i j d o e s not e x p r e s s t h e d i r e c t i o n of d e p e n d e n c e . W h e t h e r f i r m T i d e t e r m i n e s t h e b u s i n e s s p o l i c y of T j , o r v i c e v e r s a , d o e s not affect r... lJ 3)
r.. ZZ
206
=
1
(3)
F o r f o r m a l r e a s o n s , it i s n e c e s s a r y to s e t t h e v a l u e of t h e r e l a t i o n of a n e n t i t y w i t h i t s e l f e q u a l to 1 ( t o t a l d e p e n d e n c e ) . 4)
r i ]• -> r i k
(i, j , k
rjk
= 1, 2, . . - , N )
(4)
This condition, which is essential for the following, requires justificat i o n . S u p p o s e b o t h T i and T j a r e t o t a l l y d e p e n d e n t on T k ; ( f o r e x a m p l e , company T k is the holding company for both T i and Tj). We then have r i k = 1 a n d r j k = 1. It w o u l d , a p p a r e n t l y , not m a k e s e n s e to a l l o w r... l j = 0; i t w o u l d not e v e n m a k e s e n s e t o a l l o w r,.. i j < 1, s i n c e t h e t o t a l d e p e n d e n c e of T i a n d T j on T k i m p l i e s t o t a l d e p e n d e n c e a l s o b e t w e e n T . and T.. 1 ] Tk
r.. 1]
z
J
N o w l e t u s s u p p o s e r i k = . 5 a n d r:,_ J~ = . 5 ( w h a t e v e r t h e e c o n o m i c m e a n i n g of t h e s e v a l u e s m a y be). A g a i n it w o u l d not b e j u s t i f i e d t o l e t r i j = 0; on t h e o t h e r h a n d , it m a k e s s e n s e t o a l l o w r i j < . 5 . 2 C o n d i t i o n (4), a s s t a t e d a b o v e , p r o v i d e s a u s e f u l t o o l f o r d e t e r m i n i n g t h e v a l u e of a n i n d u c e d r e l a t i o n r . . t h a t r e s u l t s f r o m t h e e x i s t e n c e of r e l a t i o n s r i k a n d r~ k. I n o t h e r wol~ds: r i k a n d r j k i n d u c e t o g e t h e r a v a l u e rij = rik . rjk, unless there exists an autonomous~e rij > rik . rjk. Examples:
rik = r jk = 0 - - r . .1] --> 0 r.. may, in fact, have any value 0 ~- r.. ~- I. a)
1]
1]
b)
rik
=
c)
rik
= r jk = " 5 -~
Of c o u r s e , d)
rik
r
jk
=
1
--
r..
z]
=
1
r .l] . > .25
rij may have an autonomous value rij >. 25. =
.3,
2 These properties of r.. > 0
rjk
=
.7
--
rij ~- . 21.
are not implausible,
d.. = - l o g r . . z] z] r e p r e s e n t s a n e c a r t (a p s e u d o m e t r i c ) I) d.. = 0 11 2) d i j ~ d j i 3) dij dik + dkj '
insofar as the negative
logarithm
zj
on t h e s e t T , f u l f i l l i n g
207
If, rij the rij as
in this example, rij had an autonomous value of rij < . 21 (for example: = . 15), the induced value outweighs (and t h e r e f o r e : substitutes) autonomous value. If, however, the autonomous value is, e . g . , = . 8, this would - in turn - have an effect upon rik, inasmuch rik -~ rij . rjk ->- . 8 " . 7 = .56 .
III. M a t r i x of r e l a t i o n s Considering the four p r o p e r t i e s stated above, we can now construct the m a t r i x of r e l a t i o n s
"rll r12 r21 R
r22
...
rlN
...
r2N
-. .
•
•
°°
•
° 6 , * o . * ,
oo
•
m , * o
rNl
rN2
°
o,
mo
° . o * , , o
...
rNN
F i r s t step: Insertion of autonomous relations We i n s e r t any given autonomous r e l a t i o n s , including the elements in the main diagonal ( r i i -- i , i = 1,2 . . . . . N). As an example, let us cons i d e r the case N = 6 with r12 = . 3 , r34 = . 3 , r35 = .5, r45 = .9, r46 = . 3 , r56 = . 5 . The (preliminary) matrix R then looks as follows:
R(prel.)
=
3 0 0
I 0 0
0 i .3
0 .3 1
0 .5 .9
0 0
0 0
.5 0
.9 .3
1 .5
00 .3 .
F o r the examples in section VIII and IX, a different (equivalent) algorithm - developed by Erich W e r n e r - w a s used that lends itself better for computer application. T h e need to compute the value of an induced relation m a y also arise f r o m the fact that s o m e autonomous relation rij is unknown. In this ease, the induced value rij (rev) = rik • r"k serves as a substitute (in fact: the smallest possible substitute) for {he true, but u n k n o w n rij.
208
Second step: Revision of the (autonomous) relations A l l e l e m e n t s 0 -< r i j < 1 m u s t b e c h e c k e d to s e e w h e t h e r t h e y s a t i s f y c o n d i t i o n (4) w i t h r e s p e c t to a n y T k . W h e n e v e r t h e r e i s a ( p r e l i m i n a r y ) a u t o n o m o u s r i j < r i k . r .Jk ' s u b s t i t u t e r i j = r i k . r .Jk . • 3
A n e a s y a l g o r i t h m f o r d e t e c t i n g a n d s u b s t i t u t i n g a l l r i j < r i k . r j k is
:
Compute all products r.- r.-(i#k,j+k,r~,.> 0, r ~ t . > 0) a n d o r d e r t h e m IK JK ~ 1" b y s i z e . C h e c k in d e s c e n d i n g o r d e r , w h e t h e r c o n d i t i o n (4) i s s a t i s f i e d . W h e n e v e r it t u r n s out t h a t r i k , r j k > r i j , s u b s t i t u t e t h e v a l u e of t h e "
product rik . rjk
4 r,.
=
1] ( r e v . )
rik " rjk
I m m e d i a t e l y a f t e r t h i s s u b s t i t u t i o n , r e c a l c u l a t e a l l p r o d u c t s in w h i c h treiJd i s a f a c t o r ( i f w e had f o r m e r l y r i. = 0, t h e s e p r o d u c t s a r e c a l c u l a f o r t h e f i r s t t i m e ) a n d i n s e r t t h e ]new v a l u e s i n t h e d e s c e n d i n g o r d e r of a l l p r o d u c t s i n t h e i r p r o p e r p l a c e s b e f o r e p r o c e e d i n g t o t h e n e x t (smaller) product. By this procedure, any newly created induced r V > 0 w i l l , in t u r n , p a r t i c i p a t e i n t h e ( p o s s i b l e ) c r e a t i o n of ( s m a l l e r ) i n d u c e d elements rlm. In o u r e x a m p l e ,
w e find r35 .r45
= .45 > r34
--
r34(rev.)
= .45
r45 .r56
= .45 > r46
-- r46(rev.)
= .45
r35 .r56
= .25 > r36
-* r 3 6 ( r e v . )
= .25
The final matr~
R =
R,
in o u r e x a m p l e ,
0
is t h e r e f o r e
3
1
0
0
0 0 0 0
0 1 .45 0 .45 1 0 .5 .9 0 .25 .45
•5
O0
.25 . 9 .45 1 . .5
w e h a v e t w o g r o u p s of e n t i t i e s : T 1 a n d T 2 on one s i d e , T 3, T 4 , T 5 and T 6 on t h e o t h e r s i d e . T h e r e a r e r e l a t i o n s r i j > 0 w i t h i n e a c h g r o u p , but n o n e b e t w e e n t h e t w o g r o u p s .
As
o n e c a n see,
T o g i v e a s e c o n d e x a m p l e , l e t u s s e e how R w o u l d b e a l t e r e d if j u s t one r e l a t i o n b e t w e e n t h e t w o g r o u p s i s a d d e d , e. g, r23 = . 5 :
( p r e l . ) --
I
.3
3 0 0 0 0
1 .5 0 0 0
0
0
0
0
.5 I •3 .5 0
0 .3 1 .9 .3
0 .5 .9 1 .5
0 0 .3 .5 1
209
This one a d d i t i o n a l r e l a t i o n r 2 3
r14 r26
= .0675, r15 = .075, = .125. This yields "1.0000 .3000 .1500 .0675 .0750 .0375
R =
r16
.3000 1.0000 .5000 .2250 .2500 .1250
= . 5 i n d u c e s t h e r e l a t i o n s r13 = . 1 5 , = . 0 3 7 5 , r 2 4 = . 2 2 5 , r25 = . 2 5 ,
.1500 .5000 1.0000 .4500 .5000 •2500
.0675 .2250 .4500 1•0000 .9000 .4500
.0750 •2500 .5000 .9000 1.0000 .5000
.0375 .1250 .2500 .4500 .5000 1.0000
B y i n t r o d u c i n g t h e a u t o n o m o u s r23 = . 5 e a c h s i n g l e e n t i t y i s now l i n k e d t o e a c h o t h e r one, a t l e a s t to s o m e s m a l l d e g r e e . O b v i o u s l y , a n y m a t r i x R constructed by the above procedure fulfills the four initial conditions.
IV. A g e n e r a l i z e d c o e f f i c i e n t of c o n c e n t r a t i o n I n the i n t r o d u c t i o n , it w a s s t a t e d t h a t " c o n c e n t r a t i o n " m a y b e a t t r i b u t e d t o two p h e n o m e n a , n a m e l y
amongst entities
a) to a n u n e q u a l d i s t r i b u t i o n of s h a r e s a m o n g s t e n t i t i e s , b) to ( s t r o n g e r o r w e a k e r ) r e l a t i o n s a m o n g s t e n t i t i e s . T h e H - i n d e x m e a s u r e s t h e f i r s t , t h e m a t r i x of r e l a t i o n s R m e a s u r e s second• L e t u s w r i t e t h e v e c t o r of s h a r e s
the
Pl
p
=
P'
(Pl
=
. . . . .
PN )
Pb A coefficient of concentration that combines both p h e n o m e n a defined as the quadratic f o r m N N G
= p' R p =
Z
Y.
can n o w be
rijpip j
(5)
i=1 j = l Let us proceed to investigate the properties of this coefficient.
a)
If r . . = 0 f o r a l l i ~ j, 13 G = p'[ p = p'p = H
T h e H - i n d e x , t h e r e f o r e , c a n b e c o n s i d e r e d a s p e c i a l c a s e of G, t h e c a s e in w h i c h a l l T. a r e i n d e p e n d e n t .
1
210
b) M a x i m u m c o n c e n t r a t i o n is a t t a i n e d w h e n all r . . = 1 (i, j = 1,2 . . . . . N), r e g a r d l e s s of t h e siZe d i s t r i b u t i ~ Pi" In t h i s c a s e all e n t i t i e s a r e c o m p l e t e l y d e p e n d e n t and
piPj = Z PiZ pj = 1
G =yZ
Obviously, m a x i m u m concentration is also attained if there is only one entityT 1 with Pl = 1 and P2 = "'' = PN = 0. In this case G = r l l p2
c) If r . . 13
= 1
1, it f o l l o w s f r o m c o n d i t i o n (4) t h a t rik
m r. k = rjk - r i .]3
rjk > rijrik If rij
1,
rik
and
= rik,
hence:
> r j k (k = 1 , 2 . . . . .
N)
(6)
L e t u s now c o n s i d e r a s u b s e t of n . e n t i t i e s (1 < n < N); w i t h o u t l o s s of g e n e r a l i t y it m a y be a s s u m e d t h a t this s u b s e t c o n s i s t s of the l a s t n e n t i t i e s . F o r a n y p a i r T. and T. b e l o n g i n g to the s u b s e t , let r.. = 1 . 1 3 13 We then have G =
+
N N N-n N-n N-n Z Z rijPiP j = Z Z rijPiP j + Z i=l j=l i=l j=l i= 1
N N-n N N )~ ~ ri.pip.]j+ ~ F, r..p.p. i=N-n+l j=l i=N_n+ 1 j=N_n+113 I j N - n N-n 7. ~ i=l j=l N-n j=l
We set
N Z r..p.p. j=N_n+lIj I ]
N rijPiPj +
=
N )~ p. j=N-n~l
7. p. j=N-n~l
=
=
N Pi ~ ri, N_n+IP i + Z i=l i=N-n+l
N-n
N rN_n+ I jpj + 7: Pi ' i=N-n+l
N )~ Pi i=N-n+l
+
N 7. pj j=N-n+l
* PN-n+I
and
Pi
=
Pi*
(i = 1,2,
....
N-n)
We t h e n h a v e
211
G =
N-n N-n S Z rijpi~P~ + i:l
j=l
N-n+1
N-n+1
Y. i=l
j=l
~
N-n
2PN_n+ 1
i=Si ri' N'n+IPi
~
~ + (PN-n+I
rijP[P;
)2
(7)
I n o t h e r w o r d s : If t h e r e i s a s u b s e t of n e n t i t i e s w h i c h a r e t o t a l l y d e p e n d e n t , G y i e l d s e x a c t l y t h e s a m e v a l u e a s if t h e s u b s e t of n e n t i t i e s w e r e t a k e n a s one e n t i t y .
d) W e c o m p a r e t w o s t a t e s of c o n c e n t r a t i o n , Z a n d Z ~, c h a r a c t e r i z e d b y p a n d R, p* a n d R ~ r e s p e c t i v e l y • L e t p = p ~ , but f o r one o r m o r e e l e m e n t s of R r ~ - > r ; ~ . It f o l l o w s i m m e d i a t e l y f r o m (5) t h a t G ~ > G. In o t h e r w o r d s : ~ n y i'~J n c r e a s e zn the s t r e n g t h of t h e r e l a t i o n b e t w e e n a n y t w o e n t i t i e s y i e l d s a n i n c r e a s e of G. e) W e c o m p a r e a g a i n t w o s t a t e s of c o n c e n t r a t i o n , Z a n d Z ~, c h a r a c t e r i zed bypandR, p * a n d R ~, r e s p e c t i v e l y . L e t R = R " , but 'Pl
Pk-1 Pk" ~ Pk+ 1
p
=
( P m >-- Pk 'E > 0)
P
(8)
Pm-: pm +I Pm+:
• PN I n o t h e r w o r d s , we p r o c e e d f r o m Z to Z* by s h i f t i n g a ( s m a l l } a m o u n t f r o m T k t o T m ( P m > pk }. In [ 3 } , i t h a d b e e n s t a t e d a s a r e q u i r e m e n t w h i c h a n y c o e f f i c i e n t of ( r e l a t i v e o r a b s o l u t e ) c o n c e n t r a t i o n C m u s t f u l f i l l t h a t C* > C if s u c h a s h i f t i s c a r r i e d out•
212
As an e x a m p l e , the H-index gives
H* - H
. = ~(pi)2
_ y. pi2 = 2 E ( p m
pk+~)>
(9)
0
It c a n be shown, h o w e v e r , that this r e q u i r e m e n t is not a l w a y s met by the g e n e r a l i z e d coefficient G, and, f u r t h e r m o r e , that this is not a d e f i c i e n c y of G but r a t h e r a meaningful p r o p e r t y . F o r G, we have in this c a s e N
N
i=l
j=l
G* - G --
r
rij ¢p:p:.j- piPjl
I n s e r t i n g (8), we obtain a f t e r s o m e a l g e b r a G* - G = 2 ¢ (
~ (rim-rik) i#k, m
Pi + ( 1 - r k m ) (Pro -Pk +g )]
(10)
N
2
{Z i= 1
( r i m -rik) Pi + ( 1 - r k m ) E l
(11)
The s p e c i a l c a s e when r k m = 1 is quickly d i s c u s s e d . F r o m what has been said above u n d e r c) we expect the shift of ~ f r o m T k to T m not to a l t e r the value of G. In fact, f r o m (6) it follows that in (11) e v e r y r i m = rik (i=l, 2 . . . . . N) and we obtain i m m e d i a t e l y G* G = 0. Now let r k m < 1. I n t e r p r e t i n g (10) we can s t a t e : The i n c r e a s e in c o n c e n t r a t i o n G* will be the l a r g e r
G
1) the l a r g e r 6; 2) the l a r g e r the values of r i m (i ~ k , m ) , i . e . , the s t r o n g e r the r e l a t i o n s between T m and the entities T i (i ~ k , m ) ; 3) the s m a l l e r the values of r i k (i # k , m ) , i . e . , the w e a k e r the r e l a t i o n s between T k and the entities T i (i ~ k, m); 4) the s m a l l e r r k m ; 5) the l a r g e r P m - P k , the d i f f e r e n c e between the s h a r e s of T m and T k. It is, h o w e v e r , p o s s i b l e , that G * - G < 0. F r o m ( 1 0 ) w e obtain - b e c a u s e of £ > 0 - as a n e c e s s a r y and sufficient condition f o r G*- G < 0 ( l " r k m ) ( P m - P k + E) < ~ ( r i k - r i m ) Pi i4k, m
(12)
F r o m (4) we have r.
zm -
rik - r .lm
rik'rkm
<= r i k ( 1 - r k m )
213
Inserting in (12), we obtain an upper limit for the right hand side
(l-rkm) (Pm'Pk + ~) < 7. rik (l-rkm)P i i#k, m B e c a u s e of r k m < 1, w e f i n a l l y h a v e pm-Pk+E as a necessary versa,
<
~ rikPi i#k, m
(13)
( t h o u g h n o t s u f f i c i e n t ) c o n d i t i o n f o r G * - G < 0, o r , v i c e
Pm'Pk + 6 >
7 rikPi i#k, m
as a sufficient (though not necessary)
(14)
c o n d i t i o n f o r G * - G > 0.
In o t h e r w o r d s : If t h e r e l a t i o n s b e t w e e n T k a n d t h e o t h e r T i (i ~ k, m) are quite strong as compared to the relations between T m and the other T i ( i 4 k , m ) and if ~ a n d P m - P k a r e n ° t too large, the increase in conc e n t r a t i o n d u e to t h e s h i f t i n g of 6 f r o m T k to T m - r e s u l t i n g in a n i n c r e a s e of i n e q u a l i t y a m o n g s h a r e s - m a y b e m o r e t h a n o u t w e i g h e d b y t h e d e c r e a s e in c o n c e n t r a t i o n d u e to a l e s s e r e f f e c t of t h e e x i s t i n g r e l a t i o n s . E x a m p l e : N = 3; p l = . 2 5 , P2 = " 3 5 ' p3 = • 40; r 1 2 = . 6 , r 1 3 = r 2 3 = 0; k = 2, m -- 3, ~ = . 0 5 ; o r e l s e
p=
For
35 4O
p
--
30 4
R =
61
. 61
0
0
il
the H-index w e h a v e H = ~Pi2
= .345,
H* =
7.(p~)2=
.355,
H*- H = +.010
F o r t h e c o e f f i c i e n t G, w e h a v e G = ~ r i j P i P j = . 450,
G *=
. 445,
G * - G = - . 005
T h e r e l a t i o n b e t w e e n T 1 a n d T 2 ( r 1 2 = .6) i s s o s t r o n g t h a t t h e s h i f t of E = . 05 f r o m T 2 t o T 3 r e s u l t s , a l t o g e t h e r , in a s m a l l e r c o n c e n t r a t i o n than before.
V. The influence of foreign firms on (domestic) concentration Measurement of the effect on concentration exerted by foreign firms competing on the domestic market is not a methodological question but requires only a redefinition of the entities T i (or of the shares Pi, r e s -
214
p e c t i v e l y ) . I n s t e a d of u s i n g s h a r e s of t o t a l p r o c l u c t i o n o r t o t a l e m p l o y m e n t , o n e c o u l d , e . g . , d e f i n e p a s t h e v e c t o r of s h a r e s of d o m e s t i c s a l e s 5 . T h e T i , t h e n , a r e a l l f i r m s c o m p e t i n g on t h e d o m e s t i c m a r k e t , w h e t h e r d o m e s t i c or foreign. This definition allows automatically also for any possible relations b e t w e e n d o m e s t i c a n d f o r e i g n f i r m s : F o r e i g n f i r m s m a y be c o n t r o l l e d b y d o m e s t i c o n e s , o r v i c e v e r s a , a n d t h e r e m a y e x i s t a l l k i n d s of w e a k e r r e l a t i o n s b e t w e e n f o r e i g n and d o m e s t i c f i r m s j u s t as much as among domestic firms. How to d e f i n e t h e T i a n d what v a r i a b l e to b a s e p u p o n i n a p a r t i c u l a r a p p l i c a t i o n , h o w e v e r , is not a s t a t i s t i c a l p r o b l e m but r a t h e r a n e c o n o m i c one.
VI. T h e e f f e c t on G of n e g l e c t i n g s m a l l f i r m s I n p r a c t i c e , it i s o f t e n the c a s e t h a t m a r k e t s h a r e s Pi a r e k n o w n of. t h e n l a r g e s t f i r m s of a b r a n c h , w h e r e a s l i t t l e i s k n o w n of t h e d i s t r i b u t i o n of m a r k e t s h a r e s a m o n g s t the r e m a i n i n g N - n s m a l l f i r m s . It m i g h t e v e n be t h a t N i t s e l f i s u n k n o w n . If w e a r r a n g e t h e f i r m s i n d e s c e n d i n g o r d e r a n d d e n o t e the t o t a l m a r k e t s h a r e of the N - n s m a l l f i r m s a s cz, we h a v e n
N
Z Pi = i= 1
l-a:
Z Pi i= n+ 1
($5)
"-cc
I n h i s r e c e n t p a p e r , A d e l m a n [1} h a s s h o w n t h a t a n u p p e r l i m i t f o r H c a n b e o b t a i n e d if o n l y H'
=
n 2 is a v a i l a b l e : B e c a u s e Pi --< Pn ~' Pi i=l (i = n + l , n+2 . . . . . N), the n u m b e r of s m a l l f i r m s m u s t b e at l e a s t $ w h i c h i s t h e c a s e w h e n a l l s m a l l Pi = Pn (i = n + l , n+2 . . . . ). Pn In this c a s e , H a s s u m e s its m a x i m u m v a l u e as n 2 + _~ 2 = H' + CCpn (16) H m a x = i~= 1 Pi Pn Pn If M i s a n u p p e r l i m i t of the ( u n k n o w n ) t o t a l n u m b e r of f i r m s we h a v e a s a l o w e r l i m i t f o r t h e t r u e v a l u e of H n
Hmin •"
5
=
2+ i )~ = 1 Pi
M )~ (~_~_-~ c~ 2 = H' i = n+ 1
+
2 cm > M-n
H'
(17)
a
err. ; 5 }, p. 29S ff.
215
T h e p r o b l e m i s a g g r a v a t e d if we c o n s i d e r G. T h e ri_. e x i s t i n g b e t w e e n a n y t w o of t h e n l a r g e r f i r m s m i g h t be a s s e s s e d , bu{ not t h e r . . b e t w e e n l a r g e a n d s m a l l f i r m s o r b e t w e e n s m a l l f i r m s . C o n s i d e r i n g (ll~), we can c o m p u t e only the c o n c e n t r a t i o n m e a s u r e f o r the n l a r g e s t f i r m s :
G'
=
n
n
l~
7. j=l
i=l
rijPiPj
H o w f a r c a n G ' d e v i a t e f r o m t h e " t r u e " v a l u e G, d e f i n e d f o r a l l N f i r m s ? A s to t h e u p p e r l i m i t , it f o l l o w s i m m e d i a t e l y f r o m t h e d e f i n i t i o n t h a t t h e v a l u e G c a n s t e a d i l y be i n c r e a s e d u n t i l a l l r e l a t i o n s b e t w e e n s m a l l f i r m s a r e e q u a l to o n e . It h a s b e e n s h o w n that, in s u c h a c a s e , s u c h f i r m s c a n b e c o n s i d e r e d a s a u n i t ; to o b t a i n G m a x w e c a n t h e r e f o r e c o n s i d e r t h e s u m of t h e s m a l l f i r m s a s o n e e n t i t y T n + 1 w i t h a v a l u e P n + l = ¢~" H e n c e , the p r o b l e m r e d u c e s to the f o l l o w i n g q u e s t i o n : What v a l u e s ri, n+l (i = 1 , 2 . . . . . n) w i l l m a x i m i z e G ? We h a v e a n o b v i o u s a d d i t i o n a l c o n d i t i o n t h a t t h e v a l u e s r i , n + l s h o u l d not a l t e r t h e g i v e n v a l u e s r i j ( i , j = 1 , 2 . . . . . n); in o t h e r w o r d s , no (unknown) v a l u e r i , n + l s h a l l be of a s i z e t h a t w o u l d a f f e c t a n y v a l u e of t h e r e l a t i o n s b e t w e e n t h e n l a r g e f i r m s . It c a n be s h o w n t h a t G a t t a i n s i t s m a x i m u m v a l u e w i t h o u t v i o l a t i n g t h i s a d d i t i o n a l c o n d i t i o n if t h e r e e x i s t s a n r . n+" = 1 b e t w e e n T n + 1 and a p a r t i c u l a r one of t h e T i (i = 1 , 2 . . . . . , n , 1 T ~ i s p a r t i c u l a r T i n e e d not be the o n e w i t h t h e l a r g e s t s h a r e . S i n c e t h e ) T i n e e d not be o r d e r e d by s i z e , w e m a y - w i t h o u t l o s s of g e n e r a l i t y a s s u m e t h a t t h i s p a r t i c u l a r T i is T n . If w e w r i t e Pi = Pi ( i = 1 , 2 , """, n - l ) , Pn Pn + ¢ t , w e h a v e n
G
max
n
= Y. )2 *~ i =1 j = l riJPiPJ
and, a f t e r s o m e a l g e b r a , n
G
max
= G"
+ 2cc~ r i n P i i= 1
+ ~2
(18)
R e m e m b e r i n g t h a t - e x c e p t in t h e c a s e of t o t a l c o n c e n t r a t i o n - at l e a s t one r . < 1, we h a v e , b e c a u s e of (15), a s a r o u g h u p p e r l i m i t in
G
max
+ 2c~(1-a}
+ o~2 = G '
+ 2 ~ - ~ 2 < G'
+ 2¢¢
(19)
T h e l o w e r l i m i t of G i s a t t a i n e d by a n a l o g y w i t h (17). If t h e r e a r e a l a r g e n u m b e r of s m a l l f i r m s (not e x c e e d i n g M - n) and if t h e r e is no r e l a t i o n r . . < 0 b e t w e e n a n y of t h e m n o r b e t w e e n s m a l l and l a r g e f i r m s , 1j
0c2 G
216
min
~= G '
+
M- n
> G' .
(20)
Summarizing, limits G'
we can state that the "true" value G will lie between the
+ ~
cc
2
n
+ 2ccmax (k)
rikP k + Cc2 < G ' +2 c~
(21)
i=l
I n p r a c t i c e , h o w e v e r , it it w i l l a l m o s t a l w a y s b e p o s s i b l e t o a r r i v e a t a n a r r o w e r r a n g e f r o m w h a t l i t t l e i n f o r m a t i o n on t h e s m a l l T i i s a v a i l a b l e . A s a r o u g h ( u p p e r ) e s t i m a t e of G one m i g h t a s s u m e t h a t e a c h of t h e l a r g e f i r m s d o m i n a t e s ( r i j = 1) a n u m b e r of s m a l l f i r m s , t h e s h a r e of w h i c h i s p r o p o r t i o n a l to t h e s h a r e of t h e r e s p e c t i v e l a r g e f i r m (p~ -- Pi ' 1-c~' i = 1 , 2 . . . . . n) : n
G " 7. i=l
n
~
*~ "1 riJPiPJ
1 = ( 1 . ~ ) 2 G"
•
G"
(1+2cc)
(22)
Sometimes it may be possible to estimate, at least approximately, the average relation existing a) between big and small firms and b) between small firms. L e t u s d e n o t e t h i s a v e r a g e v a l u e a s p . In o u r n o t a t i o n , t h i s m e a n s t h a t allrij = p f o r i * j e x c e p t f o r t h e c a s e in w h i c h b o t h T l a n d Tj b e l o n g to t h e s u b s e t of t h e n l a r g e e n t i t i e s . Proceeding
a s in (7), we o b t a i n in t h i s c a s e
G = G'
+ p{(l-(1-c~)2} + ( l - p )
N Y. p,2 .L i =n+ 1
(23)
B y a n a l o g y w i t h (16), N 2
_=
~" Pi < OCPn < n i= n+ 1
Therefore, the third term on the right hand side is small as compared to the second term and can be neglected. Hence, if p can be assessed, a fair estimate of G is given by
G-- G' + p {~-(I-~) 2]
(24)
VII. Application In p r a c t i c a l a p p l i c a t i o n , o b v i o u s l y , t h e e l e m e n t s of R a r e not d i r e c t l y a v a i l a b l e . W h e r e a s t h e p; c a n v e r y o f t e n be o b t a i n e d " f r o m o f f i c i a l s t a t i s t i c s , t h e s t r e n g t h o~ t h e r e l a t i o n b e t w e e n t w o f i r m s c a n n o t b e i m m e d i a t e l y m e a s u r e d , e x c e p t in t h e c a s e r . = 1 ( o n e f i r m t o t a l l y d e p e n d e n t on a n o t h e r ) . 13
217
T h e a u t o n o m o u s e l e m e n t s of R m u s t , t h e r e f o r e , b e a s s e s s e d t o t h e b e s t of t h e k n o w l e d g e of t h e i n v e s t i g a t o r , f r o m w h a t e v e r i n f o r m a t i o n on relations between firms is available. E c o n o m i s t s w i l l f i n d it e a s y t o p o i n t out a n u m b e r of r e a s o n s w h y a n a c c u r a t e a s s e s s m e n t of t h e r i j i s n e v e r p o s s i b l e . It i s f a l l a c i o u s , h o w e v e r , to r e j e c t t h e u s e of G f o r t h e s e g r o u n d s ; u s i n g H i n s t e a d , m e a n s s e t t i n g r i j = 0 f o r a n y i = j. T h e e r r o r c o m m i t t e d b y s e t t i n g r i j = 0 is certainly greater than the error committed by a somewhat faulty a s s e s s m e n t of t h e v a l u e of r i j , w h e n e v e r s t r o n g r e l a t i o n s b e t w e e n f i r m s actually exist. F o r t h e s e r e a s o n s , t h e i n d e x G, b a s e d on e v e n t h e r o u g h e s t e s t i m a t e s of t h e a u t o n o m o u s e l e m e n t s of R, w i l l r e f l e c t t h e c o n c e n t r a t i o n a m o n g t h e T i f a r b e t t e r t h a n H, a n i n d e x t h a t d i s r e g a r d s e n t i r e l y t h e i n f l u e n c e on c o n c e n t r a t i o n of r e l a t i o n s b e t w e e n f i r m s . I h o p e e c o n o m i s t s w i l l not c o n s i d e r it t h e t y p i c a l h u b r i s of a s t a t i s t i c i a n if I c a l l it a c h a l l e n g e t o e c o n o m i s t s to t r y a n d f i n d a g r e e d r u l e s f o r a s c r i b i n g n u m e r i c a l v a l u e s to c e r t a i n k i n d s of r e l a t i o n s e x i s t i n g in e c o nomic practice.
VIII. A n e c o n o m i c a p p l i c a t i o n W e a n a l y s e d a c e r t a i n b r a n c h of t h e A u s t r i a n i n d u s t r y c o n s i s t i n g of ( a t t h e t i m e of t h e i n v e s t i g a t i o n ) 56 s m a l l a n d m e d i u m s i z e d f i r m s . F o r p, w e u s e d t h e n u m b e r of e m p l o y e e s a s a s h a r e of t o t a l e m p l o y m e n t . The following relations existed: a) 46 of t h e 56 f i r m s w e r e m e m b e r s of a ( r a t h e r l o o s e ) a s s o c i a t i o n to p r o m o t e c o m m o n i n t e r e s t s ; we v a l u e d t h i s r e l a t i o n a t . 1 . b) 32 of t h e 56 f i r m s w e r e m e m b e r s tion (value . 2).
of a s l i g h t l y s t r o n g e r a s s o c i a -
c) 3 f i r m s (of t h e t o p f i v e f i r m s ) c o o p e r a t e d to p r e v e n t m a r k e t d i s ruptions, leaving the individual firms' in any other respect (value . 3).
decision power untouched
d) T h r e e g r o u p s of f i r m s , c o n s i s t i n g of 4, 3 a n d 2 f i r m s , r e s p e c t i v e l y , h a d e x c h a n g e d s u b s t a n t i a l p o r t i o n s of t h e i r s h a r e s ( b e t w e e n 30To a n d 45% a n d h a d m u t u a l l y a s s u m e d c e r t a i n m a n a g e m e n t f u n c t i o n s ( v a l u e . 5) . e) A n e v e n s t r o n g e r
a l l i a n c e b e t w e e n 3 o t h e r f i r m s w a s v a l u e d at . 7 .
f) One f i r m h o l d s t h e m a j o r i t y of t h e s t o c k of t w o o t h e r f i r m s , we valued as . 9 .
218
which
W h e n e v e r s e v e r a l r e l a t i o n s e x i s t e d b e t w e e n two f i r m s , l u e of t h e s t r o n g e s t r e l a t i o n .
w e took t h e v a -
A f t e r h a v i n g r e v i s e d the m a t r i x a s o u f l i n e d i n s e c t i o n III, w e found G = .2540 whereas H -- . 1 1 7 4 F u r t h e r m o r e , w e u s e d t h i s e x a m p l e to c h e c k t h e e f f e c t on G of n e g l e c t i n g s m a l l f i r m s ( s e c t i o n VI). W e a s s u m e d t h a t t h e p. a n d t h e r,.. w e r e L ~J k n o w n f o r t h e 12 l a r g e s t f i r m s o n l y ; of t h e 44 r e m a i m n g f i r m s , t h e o n l y t h i n g known is t h e i r total s h a r e 56 Pi = cc = . 1 6 7 3 i=13 We then have
12
H'
G'
2
= Z Pi i= I
-_ • 1164
and
12 12 = Y ~ ri~pip~jj = . 2 2 2 0 i--I j = l
It i s not s u r p r i s i n g t h a t H' d e v i a t e s l e s s f r o m H t h a n G ' d e v i ~ e s f r o m G; the d i f f e r e n c e b e t w e e n H a n d H ' c o n s i s t s of t h e 44 v a l u e s Pi (i = 13, 14 . . . . . 56), w h e r e a s t h e d i f f e r e n c e b e t w e e n G a n d G ' c o n s i s t s of t h e
562 - 122= 2992 values ri.Pi p . J J not contained in G'
i
F o r (16) a n d (17) w e h a v e H
H
m~ min
=
.1191
= .1171
(18) and (I 9) yield G
max
= .4147 < •5286
B o t h u p p e r l i m i t s g r a v e l y o v e r e s t i m a t e G, a s a s s u m p t i o n of t o t a l d e p e n d e n c e a) b e t w e e n a l l a l l s m a l l f i r m s a n d one of the l a r g e f i r m s ; i n a r e valued as . 1 or .2 . F o r the a n a l o g o u s • 3202, o v e r e s t i m a t e s G.
t h e y a r e b a s e d on t h e s m a l l f i r m s a n d b) b e t w e e n reality, most relations r e a s o n , a l s o (22), y i e l d i n g
F o r (20) we h a v e
G ~ .2227
219
(23) and (24), depending on the additional knowledge of an "average" relation between the small and big firms, come much closer: For p = . 1, we obtainG --- .2535 and G ~ .2527; for p = .2 we obtainG--- .2848 and G ~ .2841. Hence, the true value G = .2540 is quite well hit inspire of the rough and "faulty" assessment of p = .1 or p = .2 . In practical applications, therefore, (24) will serve as a useful estimate.
IX. A n example from political science Rae 14 } has applied the H-index to measure concentration of a legislative body.
Let us c o n s i d e r as an example the F r e n c h p a r l i a m e n t of 1968. Of the 482 s e a t s , the C o m m u n i s t P a r t y (T1) held 34, the F e d e r a t i o n of the Left (T2) 57, the C e n t e r (T3) 29, the Gaullists (T4) 296, the Independent Republicans (T5) 54 and the C o n s e r v a t i v e s (T6) 12. Converting into s h a r e s , we have 071] 118| 060~
P =
614| 112] 025J
and H -- .41. If, however, the Gaullists and the Independent Republicans were considered one party, we have
i0711 p
W
=
t18] 060| 726] 025J
and H -- .55 . The G-index p e r m i t s a m o r e p r e c i s e a n a l y s i s , c o n s i d e r i n g any r e l a t i o n s between the six p a r t i e s . a) The C o m m u n i s t P a r t y and the F e d e r a t i o n of the Left s o m e t i m e s coo p e r a t e in the National A s s e m b l y . Usually, they join t o g e t h e r for the second ballot in elections to the National A s s e m b l y (r12 = .3) b) Especially in the field of foreign policy, the Federation of the Left and the Center are connected closely in the National Assembly. In many cases the two parties form coalitions for the second ballot, mainly against the Gaullists. Often, the behavior of the two parties in the Natio-
220
nal A s s e m b l y looks like the b e h a v i o r of a combined m o d e r a t e left opposition a g a i n s t a m o d e r a t e right g o v e r n m e n t (r23 = .5) c) T o c o m b a t the e l e c t o r a l c o a l i t i o n s f o r m e d by the C o m m u n i s t s and the F e d e r a t i o n of the Left, the C e n t e r and the Gaullists f r e q u e n t l y f o r m a c o a l i t i o n for the second ballot. T h e r e a r e a l s o v e r y c l o s e p e r s o n a l l i n k s between f o r m e r M R P - m e m b e r s , who a r e now G a u l l i s t s , and the m a j o r i t y of f o r m e r M R P - m e m b e r s now in the C e n t e r (r34 = .3) d) Between the C e n t e r and the Independent R e p u b l i c a n s , the G i s c a r d d' E s t a i n g - w i n g of the g o v e r n m e n t , t h e r e is an a d d i t i o n a l connection. T h e only i m p o r t a n t d i f f e r e n c e between the Independent R e p u b l i c a n s and the Gaullists is the question of E u r o p e a n policy; the p o l i c y of the C e n t e r d i f f e r s f r o m the Gaullist policy and is a l m o s t the s a m e as the policy of the Independent Republicans (r 35 = .5) e) A r a t h e r s m a l l c o m m o n base b e t w e e n the C e n t e r and the C o n s e r v a tives is the position taken t o w a r d s c e r t a i n a s p e c t s of the Gaullist policy, f o r example the p r o - A r a b policy of the g o v e r n m e n t (r36 = .1) f) The G a u l l i s t s and the Independent R e p u b l i c a n s a r e united in the UDR ( D e m o c r a t i c Union for the Republic) and in the g o v e r n m e n t . On m o s t i s s u e s the r e l a t i o n between t h e s e two g r o u p s is p r a c t i c a l l y 1 (r45 = .9) g) The G a u l l i s t s and the C o n s e r v a t i v e s (mainly the r e s t of the old Independents Antoine P i n a y ' s) a r e a l l i e d in t h e i r o p p o s i t i o n a g a i n s t all left i s t p a r t i e s , e s p e c i a l l y the C o m m u n i s t s . The C o n s e r v a t i v e s , for e x a m p le, f a v o r e d the g o v e r n m e n t ' s p o l i c y a f t e r the " r e v o l u t i o n " of May 1968 (r46 = . 3) h) The Independent Republicans and the C o n s e r v a t i v e s have a c o m m c n origin, the old Independent P a r t y . Many p e r s o n a l r e l a t i o n s still exist between the P r o - G a u l l i s t and the A n t i - G a u l l i s t wing of the Independents (r56 = .5) . C o n s t r u c t i n g the ( p r e l i m i n a r y ) m a t r i x of a u t o n o m o u s r e l a t i o n s f r o m t h e s e v a l u e s , we obtain the m a t r i x that was used as a s e c o n d e x a m p l e in s e c tion III. F r o m this m a t r i x , we obtain the final m a t r i x as c o m p u t e d at the end of s e c t i o n III and G = .65 Comparing G and H, it can be seen thai a substantial part of the .concentration existing in the French parliament of 1968 is to be attributed to the relations between parties.
To m e a s u r e c o n c e n t r a t i o n in l e g i s l a t i v e bodies by G r a t h e r than by H has s e v e r a l a d v a n t a g e s : 1) When two l e g i s l a t i v e bodies a r e c o m p a r e d , one m a y be m o r e d i s p e r s e d , y i e l d i n g a l o w e r H-index; s o m e of its p a r t i e s , h o w e v e r ,
221
m a y h a v e m u c h s t r o n g e r r e l a t i o n s (like t h o s e b e t w e e n Independent R e p u b l i c a n s and G a u l l i s t s in F r a n c e } . T h e G - i n d e x , m e a s u r i n g t o t a l c o n c e n t r a t i o n , r e s e m b l e s both f e a t u r e s .
2) T h e s a m e is t r u e when c o m p a r i n g one l e g i s l a t i v e body at two different periods.
3} T o s p l i t a (large) p a r t y into s u b - d i v i s i o n s (factions) g r e a t l y e f f e c t s H . On the o t h e r hand, such a split has no influence on the value of G if the r e l a t i o n s b e t w e e n the f a c t i o n s a r e 1; the f a r t h e r below 1 the value of the r e l a t i o n s , the s t r o n g e r its influence on the value of G.
References 1 2 3 4 5
A d e l m a n , M . A . , " C o m m e n t on the " H " C o n c e n t r a V i o n M e a s u r e as a N u m b e r s - E q u i v a l e n t " , Rev. Ec. S t a t . , LI (Feb. 1969), 99-101. A r n d t , H. (editor), " D i e K o n z e n t r a t i o n in d e r W i r t s c h a f t " , ( B e r l i n : D u n c k e r + Humblot, 1961). B r u c k m a n n , G . , " E i n i g e B e m e r k u n g e n zur s t a t i s t i s c h e n M e s s u n g d e r K o n z e n t r a t i o n " , M e t r i k a XIV (2-3, 1969), 183-213. R a e , D . W . , " T h e P o l i t i c a l C o n s e q u e n c e s of E l e c t o r a l L a w s " , (New Haven and London: Yale U n i v e r s i t y P r e s s , 1967). Theil, H . , " E c o n o m i c s and I n f o r m a t i o n T h e o r y " ( A m s t e r d a m : N o r t h Holland P u b l i s h i n g C o . , 1967).
Zusammenfassung G. B r u c k m a n n : E i n a l l g e m e i n e s K o n z e p t d e r K o n z e n t r a t i o n und s e i n e M e s s u n g G e g e b e n die V e r t e i l u n g eines M e r k m a l s b e t r a g e s auf M e r k m a l s t r ~ i g e r , wird u n t e r " K o n z e n t r a t i o n " ~ b l i c h e r w e i s e die Ungleichheit d i e s e r V e r teilung auf die (grSf]ere o d e r k l e i n e r e ) Z a h l d e r M e r k m a l s t r ~ g e r v e r standen. H 6 h e r e o d e r g e r i n g e r e " K o n z e n t r a t i o n " kann jedoch auch d u t c h die T a t s a c h e b e w i r k t werden, da~ z w i s c h e n den M e r k m a l s t r ~ i g e r n Bindungen b e s t e h e n . Von zwei I n d u s t r i e z w e i g e n , yon denen d e r e r s t e aus 5 echt k o n k u r r i e r e n d e n U n t e r n e h m u n g e n b e s t e h t , d e r zweite aus 10 U n t e r n e h m u n g e n , yon denen 8 in e i n e m Kar~ell v e r e i n i g t sind, w~irde m a n wohl den zweiten I n d u s t r i e z w e i g als s t a r k e r " k o n z e n t r i e r t " b e z e i c h n e n wollen. Alle fiblichen M a ~ z a h l e n d e r K o n z e n t r a t i o n ( G i n i - K o e f f i z i e n t , H e r f i n d a h l - l n d e x , EntropiernaI~ etc. ) m e s s e n l e d i g l i c h den e r s t g e n a n n t e n A s p e k t . I m folgenden wird eine Ma~zahl d e r K o n z e n t r a t i o n eingef~ihrt, die b e i d e A s p e k t e in e i n e m mi~t. Die E i g e n s c h a f t e n d i e s e r Ma~zahl w e r d e n d i s k u t i e r t und an zwei k o n k r e t e n B e i s p i e l e n d a r g e s t e l l t .
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Summary G. B r u c k m a n n : A G e n e r a l i z e d C o n c e p t of C o n c e n t r a t i o n and i t s M e a s u r e m e n t F o r a c e r t a i n g r o u p of e n t i t i e s ( f i r m s ) one a s p e c t of " c o n c e n t r a t i o n " is given by t h e i r s i z e d i s t r i b u t i o n and n u m b e r . A n o t h e r a s p e c t of " c o n c e n t r a t i o n " , h o w e v e r , is g i v e n by the fact t h a t t h e s e e n t i t i e s ( f i r m s ) m a y be i n t e r l i n k e d in s o m e way or o t h e r . Of two i n d u s t r i a l b r a n c h e s , one c o n s i s t i n g of 5 h i g h l y c o m p e t i t i v e f i r m s , the o t h e r of I0 f i r m s , 8 of which a r e c o n t r o l l e d by one h o l d i n g c o m p a n y , we would tend to c o n s i d e r the s e c o n d b r a n c h as m o r e " c o n c e n t r a t e d " . A l l t r a d i t i o n a l m e a s u r e s of c o n c e n t r a t i o n ( G i n i - c o e f f i c i e n t , H e r f i n d a h l - i n d e x , e n t r o p y m e a s u r e , e t c . ) t a k e into a c c o u n t the f i r s t a s p e c t only. In t h i s p a p e r , a c o e f f i c i e n t is i n t r o d u c e d that m e a s u r e s t h e f i r s t and the s e c o n d a s p e c t s i m u l t a n e o u s l y . P r o p e r t i e s of t h i s index a r e d i s c u s s e d and e x e m p l i f i e d in two c o n c r e t e a p p l i c a t i o n s . Rtsum~
G. B r u c k m a n n : Un c o n c e p t g 6 n 6 r a l de la c o n c e n t r a t i o n et son m e s u r a g e G 6 n 6 r a l e m e n t , p o u r un g r o u p e d' e n t i t 6 s ( e n t r e p r i s e s ) , un a s p e c t de la " c o n c e n t r a t i o n " e s t l e u r d i s t r i b u t i o n p a r g r a n d e u r et n o m b r e . Un a u t r e a s p e c t de l a " c o n c e n t r a t i o n " e s t c e p e n d a n t 1' i n t e r d ~ p e n d a n c e qui e x i s t e e n t r e l e s e n t i t 6 s ( e n t r e p r i s e s ) . En c o n s i d ~ r a n t deux b r a n c h e s i n d u s t r i e l l e s dont l ' u n e c o m p t e 5 e n t r e p r i s e s v r a i m e n t c o n c u r r e n t e s et l ' a u t r e 10 e n t r e p r i s e s dont 8 s o n t c o n t r b l 6 e s p a r un holding on e s t i n c l i n 6 ~ d i r e que la d e u x i ~ m e s o i t plus c o n c e n t r 6 e . T o u t e s l e s m e s u r e s de c o n c e n t r a t i o n t r a d i t i o n e l l e s (le c o e f f i c i e n t de Gini, 1' i n d i c e de H e r f i n d a h l , 1' e n t r o p i e , e t c . ) ne c o n s i d 6 r e n t que le p r e m i e r a s p e c t . Le p r 6 s e n t a r t i c l e i n t r o d u i t un c o e f f i c i e n t qui m e s u r e l e s deux a s p e c t s e n m ~ m e t e m p s . L e s p r o p r i & t 6 s de ce c o e f f i c i e n t sont d i s c u t 6 e s et i l l u s t r 6 e s p a r deux e x e m p l e s c o n c r e t s . PesmMe
G. Bruckmann 0606H~eHHaH K o H ~ e n t ~ n a KOH~eHTpaUHH H e e HsMepeHHe
~JIYI 143BeCTHO~I r p y n n M e~HHHI~ (npe~npHsTH~) OAHH acneKT ((KOHI~eHTpaLU4H), saKJI~oHaeTc~I B HX pacnpe~eneHHH no pa3Mepy H HHCny. ~pyro~ acneKT (