SOME NOTES ON APPLYING THE HERFINDAHL-HIRSCHMANINDEXAKIO MATSUMOTO, UGO MERLONE, AND FERENC SZIDAROVSZKY
Abstract. The Herfindahl-Hirschman Index is one of the most commonlyused indicators to detect anticompetitive behavior in industries. In fact, anincrease in the value of the index is usually interpreted as an indicator of actions
which may lessen competition or even create a monopoly. In this paper weshow that this is not always the case. If all firms cooperate, then the index
cannot be used since it does not depend on the cooperation level. We also
show an example when competition even has a decreasing effect on the valueof the index.

1. Introduction
The theory of oligopoly has a very large and diverse literature. Many different
variants and extensions of the classical Cournot (1838) model were introduced andanalyzed. A comprehensive summary of the most important results can be foundin Okuguchi (1976), Okuguchi and Szidarovszky (1999) and Bischi et al. (2009).
Almost all earlier studies examined the oligopolies with competing firms and very
few works were devoted to the effect of cooperating groups of firms. Different kindsof cooperation were examined in Merlone (2007), his general model included different profit formulations in cross shareholding. Recently Matsumoto et al. (2009a,b)
extended this model for partial cooperation and developed a nondifferentiable dynamics in which the firms stop cooperating when they suspect that their activity
may draw the attention of the antitrust authorities. One of the commonly usedmeasure is the Herfindahl-Hirschman Index, HHI, especially when examining horizontal mergers (Whinston, 2006), nevertheless this concentration index has beenused by other authors to study cooperative behavior in other contexts (Porter andZona, 1999). Theoretical and empirical evidence shows that higher HHI value indicates higher price-cost margin (Viscusi et al., 2005). Anticompetitive behavior canbe much more diverse than horizontal mergers, see for instance Whinston (2006).
Furthermore, Flath (1992) proved the anticompetitive effects of horizontal shareholding whereas it is shown in Matsumoto et al. (2009a) that shareholding interlocks
are mathematically equivalent to partial cooperation. Therefore shareholding interlocks can also be interpreted as partial cooperation among the firms, so the HHI
can be assumed to be a practical indicator of conspiracy. The literature has examined several drawbacks of the HHI index, for example Kwoka (1977) argues that
this index embodies both size inequality and firm numbers with weights which are
Key words and phrases. Oligopolies, partial cooperation, Herfindahl-Hirschman Index, shareholding interlocks.
The paper was written while the first two authors were visiting the Systems and IndustrialEngineering Department of the University of Arizona, Tucson. They appreciated its hospitality
over their stay. The usual disclaimer applies.
12

MATSUMOTO, MERLONE, AND SZIDAROVSZKY

assumed a priori instead of being derived. Most of the other critiques are empirically grounded: for instance Borenstein et al. (1999) have shown that at least inthe case of electricity markets the HHI is a poor measure of competitiveness. Furthermore, other authors (Foncel et al., 2008), (Liaukonyte, 2007) have questioned
the use of this index even for analyzing mergers. In this paper we approach this
topic from a theoretical perspective. We will examine some fundamental propertiesof the HHI index and show that it cannot be used in certain situations as it may
lead to incorrect conclusions.
2. The Mathematical ModelConsider an N -firm single-product oligopoly with homogeneous product andA
hyperbolic price function f (Q) = Q, where A > 0 is a constant and Q is thetotal output of the industry. The cost function of firm k is assumed to be linear,ck (xk ) = ck xk + dk , where xk denotes the output of firm k. The profit of firm k is
the difference of its revenue and cost,Axk(2.1)
ϕk (x1 , ...xN ) =− (ck xk + dk )xk + Qk
PN
where Qk = l6=k xl is the output of the rest of the industry.
It is next assumed that firms k = 1, 2, . . . , m partially cooperate with a common
cooperation level δ and firms k = m + 1, . . . , N have Cournot behavior. The payoff
functions of these firms are given by (2.1), and the payoffs of the cooperating firms
k = 1, 2, . . . , m are given as(2.2)
mPψk (x1 , . . . , xN ) = ϕk (x1 , . . . , xN ) + δϕl (x1 , . . . , xN )l=1l 6= k

Axk +δA=mP

l=1l 6= k

xk +Qkxl− (ck xk + dk ) − δmP(cl xl + dl ) .

l=1l 6= k

It is easy to see that the best responses of the competing firms arerAQk(2.3)
xk =− Qkck
and those of the cooperating firms are given assA (Qk − δqk )(2.4)
xk =− Qkckby assuming interior optimum, where qk =mPxl . Equations (2.3) for k =l=1
l 6= km + 1, . . . , N and equations (2.4) for k = 1, . . . , m provide N equations for theN unknown xk output levels. A simple but lengthy calculation shows that theindustry output at the equilibrium is(2.5)

Q=A [(1 + δ (m − 1)) (N − m − 1) + m]
,NmP
P
[1 + δ (m − 1)]ck +ckk=m+1k=1

SOME NOTES ON APPLYING THE HERFINDAHL-HIRSCHMAN INDEXand the total output of the cooperating firms is(2.6)"NP

A [(1 + δ (m − 1)) (N − m − 1) + m] mq=

ck − (N − m − 1)

k=m+1"(1 + δ (m − 1))

NPmP

ck +

k=m+1

3

#m
P

ck

k=1

#2.

ck

k=1

For the sake of simplicity we will consider the semi-symmetric case when ck = cx
for k = 1, . . . , m and ck = cy for k = m + 1, . . . , N , that is, the members of bothcooperating and competing groups have identical marginal costs. By introducing
the notation c = cx /cy as the cost ratio it is easy to verify that at the equilibrium
point the common output of the cooperating firms is(2.7)

xC =A [N − m − (N − m − 1) c] [N − 1 + δ (m − 1) (N − m − 1)]
q=2m[(1 + δ (m − 1)) (N − m) + cm] cy

and the common output of the competing firms has the form
(2.8) xN =Q−q
A [cm − (1 − δ) (m − 1)] [N − 1 + δ (m − 1) (N − m − 1)]=.
2N −m[(1 + δ (m − 1)) (N − m) + cm] cy

In order to have a meaningful model we have to assume that N > 2, m ≥ 2 andN ≥ m + 1; to guarantee the positivity of the equilibrium outputs in both thepartially cooperative and the noncooperative cases we also have to assume thatm−1
N −m<c<m
N −m−1(2.9)

if N 6= m + 1, otherwise the upper bound can be ignored.
The Herfindahl-Hirschman Index is the sum of squared market shares of thefirms in the industry1; in our case we have¶2N µX¡ ¢2 ixk
1 h ¡ ¢2HHIC == 2 m xC + (N − m) xN
QQk=1(2.10)
2=2m [N − m − (N − m − 1) c] + (N − m) [cm − (1 − δ) (m − 1)][(1 + δ (m − 1)) (N − m) + cm]

2.

If all firms have Cournot behavior, then δ = 0, so2(2.11)HHIN =2m [N − m − (N − m − 1) c] + (N − m) [cm − (m − 1)][(N − m) + cm]

2.

It is well known that in any nonsymmetric N -firm industry HHI has the unit upperbound which occurs in the case of a single monopolist, and the lower bound of HHI
is 1/N which occurs, for example, in the noncooperative symmetric case (HHINwith c = 1).
1We recall that in the empirical applications the convention is to multiply the resulting sumof squared market shares by 10,000.

4MATSUMOTO, MERLONE, AND SZIDAROVSZKY

3. Three cases of the HHI index
In this section three special cases will be shown in order to illustrate the applicability of this index.
(1) Consider first the case of c = 1, when the firms have identical marginalcosts. Then2(3.1)

HHIC =m + (N − m) [1 + δ (m − 1)][(1 + δ (m − 1)) (N − m) + m]

2andHHIN =1
N

and simple calculation shows that HHIN < HHIC . Therefore in this casethe increase of the index indicates cooperation.
(2) Consider again the previous case and assume that all the firms cooperate,that is, m = N . In this case HHIC = 1/N regardless of the value of δ.
That is, the cooperation of all firms cannot be detected.
(3) Now we return to the semi-symmetric case and examine the difference between HHIC and HHIN . Simple but lengthy calculation shows that with
δ > 0 the difference HHIC − HHIN has the same sign as(3.2)
2 (c − 1) (N − 1) [N + (c − 1) m] + (m − 1) δ [(N − m) (N − 2) (c − 1) + cN ] .
If c ≥ 1, then this expression is positive, so HHIC > HHIN and partial
cooperation is detected by the authorities.
Notice that c ≥ 1 if and only if cx ≥ cy , that is, the cooperating firms
are less efficient than the competing firms. Assume next that c < 1, andintroduce the notation γ = 1 − c. From condition (2.10) we know thatγ ∈ (0, 1/m). Expression (3.2) will be denoted by g (δ) and can be simplified
as(3.3) −2γ (N − 1) (N − γm) + δ (m − 1) [− (N − m) (N − 2) γ + N − N γ] .

(3.4)

Notice that this function is linear in δ, and g (δ) converges to a negative limit
as δ −→ 0. So if the cooperative firms select a sufficiently small cooperationlevel δ, then their cooperation is undetectable. If g (1) ≤ 0, then no partial
cooperation can be detected, since g (δ) ≤ 0 for all 0 ≤ δ ≤ 1. If g (1) > 0,then there is a threshold
2γ (N − 1) (N − γm)δ∗ =(m − 1) [− (N − m) (N − 2) γ + N − N γ]
such that partial cooperation with δ < δ ∗ is undetectable and for δ > δ ∗it can be detected by the authorities. Notice that condition g (1) > 0 isequivalent to relation(3.5)

2γ 2 m (N − 1) − γ [2N (N − 1) + (m − 1) (N − m) (N − 2) + (m − 1) N ] ++N (m − 1) > 0.
The left hand side converges to N (m − 1) as γ −→ 0, so with sufficiently
small value of γ this relation necessarily holds. Small value of γ means thatthe cost ratio is sufficiently close to unit.
4. Conclusion

One of the most frequently used indicator of anticompetitive behavior amongfirms in an industry is the Herfindahl-Hirschman Index. There are however cases
when it may be inappropriate. Two such cases were shown in the case of N -firm

SOME NOTES ON APPLYING THE HERFINDAHL-HIRSCHMAN INDEX

5oligopolies with partially cooperative firms. The cooperation of all firms cannot be
detected since the value of the HHI index does not depend on the cooperation levelin the case of symmetric oligopolies, that is, the index always shows no cooperation.
In the semi-symmetric case there is the possibility that cooperation has a decreasing
effect on the index.
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Akio Matusumoto, Department of Economics, Chuo University, 742-1, Higashi-Nakano,Hachioji, Tokyo, 192-0393, JapanUgo Merlone, Department of Statistics and Applied Mathematics, University ofTorino, Corso Unione Sovietica 218/bis, Torino I-10134, Italy, phone +39 011 6705753,
fax +39 011 6705783
Ferenc Szidarovszky, Systems and Industrial Engineering Department, University ofArizona, Tucson, AZ 85721-0020, U.S.A. AZ 85721-0020, U.S.A.