Melnik et al. [Melnik, A., Shy, Oz, Stenbacka, R., 2008. Assessing market dominance. Journal of Economic Behavior and Organization 68, 63-72] have proposed a new statistic to assess market dominance. In this comment we expand their discussion of certain mathematical properties in their analysis and link their methodology to some previous approaches.Firm’s dominance; Dominant position; Measure of dominance

Panagiotis Chronis

Vasilis Droucopoulos

English

Research Papers in Economics

BANK OF GREECE
EUROSYSTEM
Working Paper
“Assessing market dominance”:
a comment and an extension
Vassilis Droucopoulos
Panagiotis Chronis
10
JANUARY 2010 WORKINKPAPERWORKINKPAPERWORKINKPAPERWORKINKPAPERWORKINKPAPER 9 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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All rights reserved. Reproduction for educational and non-commercial purposes is permitted provided that 
the source is acknowledged. 
 
ISSN 1109-669 “ASSESSING MARKET DOMINANCE”: A COMMENT AND AN 
EXTENSION 
 
 
 
 
 
Vassilis Droucopoulos 
Department of Economics, University of Athens 
Panagiotis Chronis  
Bank of Greece, Economic Research Department 
 
 
 
 
 
 
ABSTRACT 
Melnik et al. [Melnik, A., Shy, Oz, Stenbacka, R., 2008. Assessing market dominance. 
Journal of Economic Behavior and Organization 68, 63-72] have proposed a new statistic 
to assess market dominance. In this comment we expand their discussion of certain 
mathematical properties in their analysis and link their methodology to some previous 
approaches. 
 
Keywords: Firm’s dominance; Dominant position; Measure of dominance 
 
JEL classification: K21; L11; L41 
 
 
 
Acknowledgements: The authors thank Professor (Emeritus) George Krimpas and Heather 
Gibson for valuable comments. All remaining errors are solely our responsibility. The 
views expressed in this paper are those of the authors and do not necessarily reflect the 
views of the Bank of Greece. 
 
 
 
 
Correspondence: 
Vassilis Droucopoulos 
Department of Economics 
University of Athens, 
8 Pesmazoglou str. 
Athens 10559, Greece 
Tel.: +30- 210-3223148 
e-mail: vdroucop@econ.uoa.gr  
  4 “Normally, dominance destroys parity” 
Shepherd et al. (2001, p. 840). 
 
 
1. Introduction 
In a recent paper Melnik et al (2008) suggest a novel approach to the existence of 
dominance of the leading firm in an industry.
1 The formula they put forward, they claim, 
can be easily applied in antitrust cases by competition and regulatory agencies without 
the requirement of estimating demand elasticities or marginal costs, the latter being 
notoriously difficult to obtain. Their approach is to be commended since they restore the 
prominence of the market share concept that has been given (in our view, unduly) short 
shrift by some recent research.
2
By relating the paper to previous papers by LaCour and Møllgaard (2002, 2003) 
and Dobbs and Richards (2005) to which Melnik et al. (2008) do not refer, the purpose of 
the present comment is firstly, to show the interconnection of these approaches and, 
secondly, to elaborate on this paper by checking the validity of the main findings and to 
expand on them. 
The proposed single-market measure (“threshold”), which would enable 
regulators to draw inferences about dominance of the leading firm, takes the following 
form: 
)] ( 1 [
2
1
)] 1 )( ( 1 [
2
1 2
2
2
1
3
2 1 s s s s s s
N
i
i
D − − = − − − = ∑
=
γ γ                            (1) 
where  si   are firm market shares indexed in an order satisfying  s1≥s2 … ≥ ≥sN, and γ is an 
exogenous parameter interpreted “as an industry-specific assessment of the entry barriers 
relevant for the industry. Lower values of γ correspond to lower entry barriers, which 
would mean that potential (future) competition will limit the ability of firm 1 to exploit its 
market power more effectively” (ibid., p. 65). 
                                                 
1 The paper by Hellmer and Wårell (2009, p. 3239) describe the paper as “interesting and intriguing”. 
2 On this, see the earlier discussion in Shepherd et al (2000). 
  5We proceed by equating the share of the second largest firm (s2) to . 
Thus, equation (1) is transformed into: 
⎟
⎠
⎞
⎜
⎝
⎛
− −∑
=
N
i
i s s
3
1 1
 
2
1
33
1
1[ 2 ( 1 ) ( 1 ) ]
2
NN
D
i
ii
ss s γ
==
⎧⎫
=− − − − ⎨⎬
⎩⎭ ∑∑ i s                               (2) 
 
2. The parameter γ 
Following Dobbs and Richards (2005) we now introduce the concept of “output 
restriction test” elasticity: 
1 1
1 1
1 zs s
q
Q
q
q
Q
Q ORT =
∆
∆
=
∆
∆
= ε                                            (3) 
where Q is the industry’s output, q1 the leading firm’s output and 
1 q
Q
z
∆
∆
= . Additionally, 
according to Dobbs and Richards (ibid., p. 573): “more usually, output restriction [by the 
leading firm] will give rise to some degree of reduction in total output, and hence to a 
positive ORT elasticity”
3. These authors claim that, under reasonable assumptions, this 
elasticity will lie between zero and one. Thus, expression (2) can be written as: 
]} ) 1 ( ) 1 (
2
[ 1 {
2
1 2
3 1 3 ∑ ∑
= =
− − − − =
N
i
N
i
i
ORT
D s s
z
s
ε
γ                        (4) 
and  
2
) 1 (
) 1 (
2
3
3
∑
∑
=
=
−
+ − − =
∂
∂
N
i
i N
i
i
ORT D s
s
z
s ε
γ
                                  (5) 
For this partial derivative to become negative, as Melnik et al. (2008, pp. 66, 69) 
maintain, the following inequality should apply: 
                                                 
3 In the words of Azevedo and Walker (2002, p. 366):”Any firm can restrict its own output, but in most 
markets any unilateral output restriction would be replaced by an output expansion by other players in the 
market. This is not true of a dominant firm’s output restriction”. Also, “The greater the firm’s market share 
the less likely that other firms will be able to expand production to defeat the unilateral price increase” 
(McFalls, 1997). 
  61
2
1
3
< < − ∑
=
N
i
i
ORT
s
z
ε
, where 0<∑ <1 and as hypothesized  <  (since  =
=
N
i
i s
3
ORT ε z 1 s
z
ORT ε
 
and 0< <1) and 0< , z<1 or put differently, 1-2 < ,  1 s
ORT ε 1 s ∑
=
N
i
i s
3
1-2 <1- - , 1- <1-  which is obviously valid. In the same vein, for the partial 
derivative to be positive the inequality 1-2 > should hold. But this cannot happen 
since 1- >1-  is not valid as  >  by construction. Consequently, the finding of 
Melnik et al. (2008) that there exists a negative relationship between the parameter γ and 
the dominance threshold is validated by using the output-restriction-test elasticity (ORT) 
without actually having to estimate the latter. In sum, more significant entry barriers 
(more significant competition) are (is) associated with higher (lower) values of γ which 
decrease (increase) the dominance threshold measure ( )
1 s 1 s 2 s 1 s 2 s
1 s ∑
=
N
i
i s
3
1 s 2 s 1 s 2 s
D s
4. 
Moreover, Dobbs and Richards (op.cit., p. 574) maintain that, in the case of a 
homogenous good Cournot industry, “assuming locally linear demand and constant 
marginal costs, the ORT elasticity for any given firm, at the current Cournot equilibrium, 
is simply the market share of that firm divided by the total number of firms in the 
industry”: that is
N
si CORT = ε . In this case, expression (2) can be written as: 
]} ) 1 ( ) 1 ( 2 [ 1 {
2
1 2
3 1 3 ∑ ∑
= =
− − − − =
N
i
N
i
i
CORT D s s N s ε γ                          (6) 
and  
2
) 1 (
) 1 (
2
3
3
∑
∑
=
=
−
+ − − =
∂
∂
N
i
i N
i
i
CORT
D s
s N
s
ε
γ
                                (7) 
 
                                                 
4 We doubt the correctness of equating higher strictness  of the dominance criterion with a lower value of 
 (Melnik et al. 2008, p. 65). On the contrary, we think that higher strictness is associated with a higher 
value of   and vice versa. 
D s
D s
  7For this partial derivative to become negative, as Melnik et al. (2008, pp. 66, 69) 
argue, the following inequality should hold: 1-2 <   which is equivalent to 1-
2 <1- - ; and is also valid since  > . Furthermore, as we proved previously in the 
case of the OPT elasticity and for the same reasons, the partial derivative cannot be 
positive. Therefore, the negative relationship between the parameter γ and the dominance 
threshold is established once again. 
CORT Nε ∑
=
N
i
i s
3
1 s 1 s 2 s 1 s 2 s
We now turn to the papers by LaCour and Møllgaard (2002, 2003). One of the 
factors considered in testing for dominant position is the own-price elasticity, 
1
1
1
1
q
p
p
q own
∆
∆
− = ε  which we can transform into 
Q s
p
p
q own
1
1
1
1
∆
∆
− = ε  and hence  
own
y
s
ε
= 1  
where 
Q
p
p
q
y
1
1
1
∆
∆
− = . From expression (2) we then get: 
]} ) 1 ( ) 1 (
2
[ 1 {
2
1 2
3 1 3 ∑ ∑
= =
− − − − =
N
i
N
i
i own
D s s
y
s
ε
γ                                  (8) 
and 
2
) 1 (
) 1 (
2
3
3
∑
∑
=
=
−
+ − − =
∂
∂
N
i
i N
i
i own
D s
s
y s
ε γ
                                       (9) 
 
It is easily shown, as before, that this partial derivative can only be negative, thus 
corroborating the Melnik et al. (2008) finding. 
 
3. The elasticities 
From equations (4), (6) and (8) we can get the following partial derivatives of   
with respect to the three different elasticities: 
D s
0
) 1 (
3 <
−
− =
∂
∂ ∑
=
z
s
s
N
i
i
ORT
D γ
ε
                                              (10) 
  8{where 0< <1 and N, ∑
=
N
i
i s
3
γ , >0}  z
3
(1 ) 0
D N
i CORT
i
s
Ns γ
ε −
∂
= −⋅ − <
∂ ∑                                           (11) 
 
2
3
(1 ) 0
()
D N
i own own
i
sy
s
γ
εε =
∂⋅
= −<
∂ ∑                                            (12) 
where y<0 and γ >0 
 
With reference to equations (10) and (11) we can infer that both results are 
plausible since the higher the value of the elasticity the less the remaining firms can 
counteract the change (decrease) in quantity initiated by the leading firm, reflecting a 
state of weak competition. Thus the threshold required for dominance need not be high. 
As far as the own-price elasticity is concerned, the negative value of the partial derivative 
indicates that the higher (in absolute terms - or the lower in negative terms) the elasticity, 
the lower the dominance threshold. This is, again, plausible and accords to what LaCour 
and Møllgaard (2003, p.133) point out: “A very high negative value of ε  is a sign that 
the demand curve disciplines the firm strongly: customers rush away if the price is 
increased”. 
 
4. The joint market share of all but the two largest firms 
We come to the final variable under review, namely the joint market share of all 
but the two largest firms. According to Melnik et al. (2008, p. 65) “…the dominance 
threshold is lower… the higher is the joint market share of the two largest firms” or 
alternatively, the dominance threshold is lower the lower the joint market share of the 
remaining firms  .  ) (
3 ∑
=
N
i
i s
From (4), (6) and (8) we get the three partial derivatives with respect to  :  ∑
=
N
i
i s
3
) 1 (
3
3
∑
∑
=
=
− − =
∂
∂
N
i
i
ORT
N
i
i
D
s
z s
s
γ
γε
                                           (13) 
  9where s1 =   
z
ORT ε
 
∑
=
∂
∂
N
i
i
D
s
s
3
N
CORT γε = ) 1 (
3 ∑
=
− −
N
i
i s γ                               (14) 
where s1=   N
CORT ε
and 
∑
=
∂
∂
N
i
i
D
s
s
3
own
y
ε
γ
= ) 1 (
3 ∑
=
− −
N
i
i s γ                               (15) 
where s1=
own
y
ε
 
These relations can be summarized in one expression: 
3
 
D
N
i
i
s
s
=
∂
∂∑
 
>
<   0  when s1 
>
<      ∑
=
−
N
i
i s
3
1
But the positive partial derivative is not validated since s1>s1+s2 cannot hold, thus 
the higher the joint share of the remaining firms   the lower the dominance 
threshold. This is, however, the opposite of the Melnik et al. result. Thus, a word of 
caution and further elaboration are needed. 
) (
3 ∑
=
N
i
i s
The partial derivatives presented above are valid only when s1 is kept constant. 
This means that if the ∑ decreases (increases), as s
=
N
i
i s
3
1 remains constant, the share of the 
second largest firm (s2) goes up (goes down) resulting in the decrease (increase) of (s1-s2) 
which in turn tends to raise (lower) the threshold (sD). At the same time though, the 
increase (decrease) of   exerts a negative (positive) influence on the threshold 
(s
) 1 (
3 ∑
=
−
N
i
i s
D) but not as strong as the positive (negative) contribution of the diminution 
(enlargement) of the market share difference of the two largest firms, hence the overall 
  10increase of the threshold (sD). Consequently, Melnik et al. err in stating “…the 
dominance threshold is lower…the higher is the joint market share of the two largest 
firms…” (p.65) as the comparison of market E with either market C or market D of their 
Table 1 (p. 66) indicates. The diminution of the market share difference predominates, 
causing the increase of the dominance threshold (sD) despite the increase of (s1+s2). 
If however, the market share difference remains constant, which is not an option 
in our equations (4), (6) and (8), when  varies, then the threshold increases 
(decreases) as  goes up (goes down). Obviously, if all three shares, i.e. s
∑
=
N
i
i s
3
∑
=
N
i
i s
3
1, s2 and 
, vary concurrently, then it is not possible to attribute the new value of the 
dominance threshold to the change of a single variable. 
∑
=
N
i
i s
3
The following table provides a self-explanatory numerical example of our 
previous arguments: 
 
 
Table1 
Market shares and firm dominance thresholds 
Market s1 s2 s1- s2 ∑
=
N
i
i s
3
  s
D
A  0.40 0.15 0.25 0.45 0.43 
B  0.40 0.25 0.15 0.35 0.45 
C  0.45 0.20 0.25 0.35 0.42 
D  0.45 0.30 0.15 0.25 0.44 
E  0.35 0.20 0.15 0.45 0.46 
 
 
5. Conclusion 
It has been argued, echoing the Chicago-school view, that “the presumption 
associating market power with high market shares is rebuttable” (Hay, 1992, p. 822). 
Along the same lines, “…the case law puts a great deal of stock in the size of market 
shares in inferences of market power - a general inference that modern economics tells us 
is not supported by either theory or empirical evidence” (Scheffman, 1992, p. 919). 
  11However, opponents of this view have maintained that “… market share is the leading 
fact, the most basic determinant of the degree of competition or monopoly” (Shepherd et 
al., 2001, p. 841). Neither view elevates “market share” into the absolute criterion or 
denigrates it to such an extent as to render it completely inoperable and useless.
5 As 
Melnik et al. suggest and as the present comment has insisted other factors also come into 
play. Generally speaking, the fast-track formula seems to rest on solid grounds. But the 
use of supplementary variables, whose estimation is admittedly more demanding, would 
impact on the study of dominance in a more useful manner. 
 
 
                                                 
5 “Though market shares are recognized to be important, no serious scholar has claimed that they control 
market outcomes precisely. They are simply the best single indicator, as business experience has also fully 
recognized. Their evidence establishes a presumption about market power’s possible effect in raising price, 
which secondary conditions may modify” (Shepherd et al., 2000, p.852, fn. 26) and “The relationship 
between market share and market power is not exact or tight….But within any market, larger market shares 
give significantly more market power; and in almost every market, market shares above 30% provide 
substantial market power. That is one reason why firms struggle so relentlessly to gain more market share” 
(ibid., p. 860, fn. 42). 
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  15

“Assessing market dominance”: a comment and an extension

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