We model two independents markets (A and B) with an identical demand. Goods supplied on these markets are supposed to be homogeneous. Supply is provided by 
symmetric firms in market A, and 
in B. Without loss of generality, we assume market B to be always less concentrated than market A that is
. In this industry, it exists a subset of k firms (
) which are active on both markets: we refer to them as multimarket contacts firms. This configuration means that these firms supply the good in both markets A and B.
As it is standard in the analysis of tacit collusion (see [
, the individual profit gained from a punishment strategy for all active firms in market h. We denote 
individual collusion profit. The determination of 
generally depends on the way the collusive agreement is reached as well as on various factors. Last, 
represents the individual profit gained from deviating from the collusive agreement and corresponds here to the monopolist outcome in market h, we denote
. One can thus determine a threshold for the discount factor denoted 
such as:
(1)
Whenever3
, collusion is sustainable in market
. Hereafter, we will consider that market conditions (demand, costs and so on) in markets A and B are identical, one can write
.
As a benchmark, we study collusion incentives in the case where multimarket contacts are not possible i.e.
. Because of symmetry among firms, individual collusive profits are just given by an equal sharing of the monopolistic outcome that is 
and
. Then using Equation (1), the critical discount factors in both markets are given by:
One can easily link up the incentives to collude (i.e.
) to the market concentration degree traditionally measured by the Hirschman-Herfindhal Index (HHI)4. In this benchmark case, this index for each market is:
Then we can write thresholds for the discount factor as simple linear functions of the respective HHI in each market that is:
It does appear an inverse relationship between the value of these thresholds and HHI’s. Clearly on a given market, the lower HHI is, the less sustainable collusion is.
Collusion is then sustainable in both markets if and only if the discount factor 
in the industry is such that
. We call the maximum value for these discount factors as the critical discount factor. In that case, with
, the sustainability condition in both markets boils down to 
.
Remark 1. Without multimarket contact among firms, the HHI test is not conflicting with the analysis of tacit collusion.
A competitive market structure (i.e. a weak HHI level) is linked with pro-competitive behaviors (
is high). This is the conventional wisdom in the field of competetion policy.
In the following, we study market structures with multimarket contacts and we analyze how such contacts could increase incentives for firms to collude.
) are active in both markets A and B. Here, we consider that coordination to a given collusive agreement allows firms to keep their customers with respect to the competitive (Bertrand) equilibrium. As a result in case of collusion, market shares are assumed to be “frozen”: firms “in contact” in both markets does not modify the market sharing. In short, a “business as usual” principle applies.
corresponds to active firms in the single-market A and 
corresponds to those active in the single-market B. Last 
represents the discount factor for 
active firms in both markets. Using again relation (1), we obtain easily thresholds for the discount factor for firms without multimarket contact 
and
.
or
). The total collusive profits are then given by
. Furthermoreeach firm’s deviation profit equals twice the monopoly profit that is 
Hence:
as soon as
. In that case, the critical discount factor (i.e.
) is clearly
. Therefore, collusion is sustainable in both markets A and B if6
.
and 
collusive market shares (resp. in market A and B) for a firm in contact in both markets. These shares are then defined by:
with 
This definition of market shares simply explains that a given firm which is in contact on several markets can be encouraged to reduce his market share (from 
to 
and/or from 
to
) in order to increase collusion incentives for other firms. For instance in market A, the gap between 
and 
represents the opportunity cost that an active firm in both markets has to bear in order to increase the incentives for other firms to collude.
for a single-market firm in market A (without contact), is defined by the linear equation 
Solving it in
yields
. Symmetrically, we obtain 
for a single-market firm in B.
(denoted
) as well as for active firms in both markets (
). The new definition of market shares, in Equation (3), modifies these thresholds: they do not only depend on market structures (n and m) but also on the number of active firms in both markets (k). Using relation (1), the threshold for the discount factor for active firms in the single-market A is then given by:
. Using Equation (3) one can rewrite it as a function of the market A structure (n):
and
. Indeed, we have to differentiate between situations where the market shares are “frozen” on a given single market (A or B) and situations where they are simultaneously “frozen” in both markets. More precisely, this threshold rewrites:
and defined by
. Without loss of generality, we proceed the analysis assuming
.8
where 
is defined as:
and
.
, 
and 
for the discount factor. They allow studying incentives to collude for each type of firms that depend on the values of parameters
.
) whereas collusion profits are decreasing with the number of firms.
. More precisely, it represents critical discount factors in the plane 
as defined in Lemma 1. In this figure, one can see the values of parameters for which active firms in both markets—respectively without contact in market B—have stronger incentives to deviate (area
) respectively area
.
, market B is relatively weakly competitive (
). In that case, the market B share which is supplied by active firms in both markets does not depend directly on the number of active firms in this market since
. Incentives to collude for these firms (i.e.
) are therefore independent of the intensity of competition in this market. However, incentives for firms without contact in market B does depend on m through their market share given by
.
), the active firms in both markets have strongest incentives to deviate. Within area X, collusion is then sustainable as
. However, as soon as market A is very competitive (
), this area vanishes.
), the incentives to deviate for active firms in both markets are independent from the corresponding market structure (i.e.
). Second, as market B becomes more competitive (
), the share of market B supplied by firms with contacts in both markets corresponds to the situation where 
i.e. a freeze of market shares. In that case, their incentive to collude is now linked to the structure of market B. Within this area Y, collusion is sustainable as soon as
.
is moving upward. In this case, the share of firms that are supplying in both markets is very high and the number of 
active firms in the single market B is very low; in that case, their benefit from a deviation is diminishing since their collusion profit is growing. One can also underline that the critical discount factor decreases with the number of active firms in both markets. Parameter k can be considered therefore as a structural factor facilitating collusion in markets9. This critical threshold decreases as a function of k. As the number of firms with contacts in both markets is relatively high, it becomes constant with respect to k since it is equal to
.
) and the benchmark case without multimarket contact (
defined in Section 2).
, 
and 
according to the values of parameters. Within locus 
and
, multimarket contacts do not increase the incentives for firms to collude. In both regions, the critical discount factor threshold does not decrease with the number of active firms in both markets:
. In the area
, presence of firms with contacts in both markets entails more incentives for firms to collude in market A and also in market B. One can precise these results when market B becomes more competitive (m increases with n being fixed).
)we have 
(in area
): no collusion transfer occurs. Indeed initially, without multimarket contact, the critical discount factor is already very low. Active firms in market B have thus strong incentives to collude since collusion profits are shared between few firms. In that case, transferring collusion is too costly for active firms in both markets since they should give up a too large market share in market A but also in market B. In this situation, active firms in both markets make collusion less sustainable since
.
), our results show that firms with contacts in both markets are able to transfer collusion on this market: 
(area
). We distinguish between two situations according to values of m: 1) whenever 
and 2) whenever
, market B becomes less concentrated. Individual collusion profits for active firms in this market are relatively low. In that case, the critical discount factor is defined using the incentive constraint for firms with multimarket contacts. These firms are then able to transfer collusion at a lower cost giving up a relatively small part of their market share. On the other side, if 2)
, the critical discount factor in the multimarket contacts framework is defined using the incentive constraint for active firms in the single-market B (
).
, market B is very competitive (
); active firms in this market have then initially a low market share that is
. It is then impossible to transfer collusion since de facto firms with contacts cannot concede an additional part of their market share.
).
for which a collusion transfer is made. More precisely, for n and 
given, the figure shows the market B structure for which firms with multimarket contacts are able to transfer collusion.
), more concentration in market B can reduce incentives to collude for firms in market A.
high), collusion profits for active firms in single market B must be weak; this is the case when these firms are numerous (
high). The idea is obvious: ability for firms to transfer collusion depends on their ability to give up market shares that depends on the value of 
or on the number of firms (
) without contact in market B.
, the difference between both critical factors denoted 
is not a monotonic function of m. When the market B structure is not very competitive
), the gap 
is increasing in m.
), incentives to transfers collusion reverse: they decrease with the number of active firms in market
. Last, one can see that this value 
is increasing in the number of firms with contacts in both markets (i.e. k). 
varies with m and k.
), incentives to transfer collusion are higher for more competitive market B: 
is increasing with k. Moreover, one can note that maximum incentives (corresponding to a market structure
) are increasing in k.
) and market 
is weakly concentrated (
), concentration of market A entails more collusive behaviors for firms in market B.
, 
and 
depicted in 
are decreasing functions11 of each
. On the other hand, using relation (7) in Lemma 1, we can see that 
is an increasing function of 
whereas 
is a decreasing function of
,
. Hence, it follows that 
is a
but 
is an increasing function of HHI in markets A and B.
), we just have proved the following proposition.
, a low HHI level in market A or B corresponds to a market structure which facilitates collusion.
, the critical discount factor (which would be the correct but unobservable collusion test) increases whereas HHI decreases. This reflects the situation where more competitive markets A or B could strengthen incentives to collude for some firms in these markets.
Let us define three difference
and
and study them according to
.
and 
then we turn back in the situations of Sections 2 and 3 where
for 
then 
and
. This implies 
and
hence 
so we have
. Second, suppose that 
then from Equations
which is bounded below by
. Thus 
is of no relevance here.
, moreover 
when 
where 
This value 
is an increasing concave function of n which takes values
if 
and 
if
. Last as
, 
is monotonic and decreasing in m, so 
when
and when 
then 
To sum up 
if 
and
if
. Third if
then again using Equations (4)-(6):
is again bounded below by
, so 
and studying 
is useless but 2)
. Hence
.
and 
then 1)
is decreasing in n and is zero for
. Moreover 
since
Consequently 
that is
; 2) 
is decreasing in m and is zero for
, but 
since
. Hence 
if 
and 
if
; 3)
is monotone increasing in m and is zero for
where 
is increasing in n and take the values 
if 
and
if
. Hence 
and thus 
if 
and 
if
. In summary 1) if 
then
; 2) if
then 
and 3) if 
then
.
if 
where
if
that is
is an increasing function of k. We see that 
and 
in their respective definition domains (see above).
.
and 
and 
then 
is decreasing in m and is equal to zero for 
Moreover, we have 
since 
In that case, 
if
. (b) If 
and 
then 
is decreasing in m and is equal to zero when 
We have therefore 
since 
Moreover, if 
then 
and if 
then
. In that case, 
if
.
as a subset 
such that
.
and
. (a) If
and 
then
in that case
. Idem if
and
. We can define area 
as a subset of 
such that
then 
and 
, subset of 
so that
in locus
. We denote in the text
. If 
and if 
then, we shown above that:
, 
is increasing in m.
, 
is decreasing in m since
. Moreover 
and we show that
.
then 
is an increasing function of 
since
. On the other side, 
is a decreasing function of 
and 
since 
for
.