Incomplete Cartels Iwan Bos Norwich, 2008. Introduction Incomplete cartels are cartels with less than one hundred percent market share. In other words,

Incomplete Cartels

Iwan Bos

Norwich, 2008

Introduction

Incomplete cartels are cartels with less than one hundred percent market share. In other words, there is at least one non-participating firm.

Most theories of collusion assume the perfect or full cartel to be all-inclusive. Yet, “Many of the cartels that have been uncovered in the past are really “incomplete

cartels”, i.e., they do not include all the firms in the industry but just some fraction of them.” (Pakes, 2006).

Cartels were not all-inclusive in a significant number of cases in Europe since 1964 (average market share around 75%).

Hay and Kelly (JLE, 1974) analyzes 65 horizontal cartel cases. Market share data are available in 45 cases, 32 of which deal with cartels that are not all-inclusive (average market share of around 88%). In 14 cases the market share exceeded 90%.

Average CR4 is around 75%, while average number of firms in these cases is 12, suggesting that market shares were unevenly spread.

‘…it is not necessary for a conspiracy to include all the firms in the market to exist, or indeed, to be successful. In general, they [non-conspirators] seem to be the smallest competitors…’

Introduction

DeRoos (IJIO, 2007) takes a structural dynamic approach to analyze the well-known Vitamin C cartel. His analysis suggest that a cartel will persist only if fringe competitors remain small.

Vitamin conspiracy collapsed due to growing fringe production from China. A similar situation occurred in the Citric Acid industry.

In short, evidence, although relatively scarce, suggests that: Many cartels are not all-inclusive Incomplete cartels are often dominant (significant market share)

Firms not included in the cartel are typically the small players in the market.

The Theoretical Problem

Most literature on collusion assumes cartels to be all-inclusive. Theoretical reasons for why firms fail to form the full (perfect) cartel: incentive to

cheat ex ante (participation) and ex post (stability). Most of the literature assumes symmetric firms, which is problematic, because it

is difficult to see why similar firms would take non-similar decisions (some should participate in equilibrium, some should not)

This problem is particularly severe because in equilibrium outsiders are typically better off than insiders. Theory predicts the incentive to free-ride on a cartel formed by others is strong.

To provide a rational basis for incomplete cartels one has to model a setting in which some firms have an incentive to collude, while others simply best respond. At a minimum, this requires firms to differ in at least one respect. Yet, many oligopoly models are not very convenient to study collusion with asymmetric firms.

Towards a Theory of Incomplete Cartels

We aim to provide a rationale for the existence of cartels that are not all-inclusive. In particular, we address the question “who is in and who is out?”

We take the following approach Dynamic game with firm heterogeneity in terms of production capacity (proxy

for firm size). Take into account both the participation problem and the incentive problem. Overt as opposed to tacit collusion (we assume colluding is costly) Aim to relate the theoretical outcomes with partial cartels in practice.

Main Results

Small cartels are not viable. Optimal cartel size is all-inclusive when cartelizing is costless. Optimal cartel size is incomplete when cartelizing is costly and the smallest

firm(s) are sufficiently small. Sufficiently small firms have no incentive to take part in any cartel Sufficiently large firms have no incentive to free-ride on a cartel formed by rivals. The most profitable cartel is formed by the largest firms in the industry. Under certain conditions, firms have an incentive to form the cartel for which

total profits are highest.

A Model of Collusion with Asymmetric Capacity Constraints

Let N be a set of n profit-maximizing sellers which simultaneously set price. Homogeneous products produced at constant marginal cost c > 0. Market demand function with and . Production capacity is denoted by and is assumed fixed for all .

Furthermore, without loss of generality, capacities are indexed such that:

Define two sets: and No particular rationing scheme, but we impose the following mild condition

0)(' pD)( pD 0)('' pDNi

ik

nkkk ...21

ppNjp j :)( ppNjp j :)(

NikpDppDipj jiiii ,0,)(max),()(

Assumptions

Assumption 1 (some symmetry across firms):

If , then

If , then

Assumption 2 (no very large firms in absolute terms):

Assumption 3 (limited power for largest firm(s)):

Note that Assumption 2 in conjunction with Assumption 3 implies , but this has no bite in our analysis because our focus is on incomplete cartels.

)()()(0

pj jpj j kkpD )(,),(0 pikppD iiii

)()()(

pj jpj j kpDk )(),,( pippDk iiii

NikpD im ,)(

iNj j NicDk\

),(

3n

Static Nash Equilibrium

Firms choose a price from the set:

Individual profit function:

Proposition 1: In competition, there exists a symmetric equilibrium in which all firms price at marginal cost. Moreover, as , this equilibrium is unique.

,...,,,...,,0 ccc

),()( iiiii ppDcp

0

Collusion Let denote a (sub)set of firms that engage in a price-fixing cartel

and set a cartel price Lemma 2: If a profitable cartel sets a price , then the best

response of an individual profit-maximizing outsider is to sell units at a price , for sufficiently small.

Individual cartel profits then amount to:

Infinitely repeated version of this game with ‘grim-trigger strategies’. The incentive compatibility constraint of is then given by:

Lemma 3: As ,

N cpc

cpc

jk c

j pp 0

i

icd

i kcp )(

di

t

ci

tcci pV

1

1),(

0

.,

ikpDcp ij

jcc

i

Collusion

Note that as long as which due to Assumption 3 always holds.

Small cartels are not viable! The cartel problem is:

Subject to:

Lemma 4: The optimal cartel price is strictly increasing in total cartel capacity.

ikkpDk

pDk

i jNi ic

i

i

cNi i

ci

di

,0)1()(

)(

ci

di )( c

Ni i pDk

i jNi icccc kkpDcppV )()(

1

1),(max

ikkpD

k i jNi ic

i

i ,0)1()(

The Model Visualized Let total industry capacity be given by K. The optimal cartel prize is determined

by total cartel capacity.

0 K

Cartel Capacity Fringe Capacity

Optimal Cartel Size when Cartelizing is Costless

Proposition 5: A cartel can always allocate its profits in such a way that it Pareto dominates every smaller cartel.

This result is non-trivial. For example, in a standard linear Cournot model Pareto improving cartel expansion requires.

Which does not always hold (e.g., n=10, x=8, t=1) The intuition underlying Proposition 5 is that outsiders benefit from the cartel,

but not ‘too much’. Corollary 6: The optimal cartel size is all-inclusive.

22 )2(

1

)2(

1

xn

t

txn

Collusion among the Few

The assumption of cartelizing being costless is arguably restrictive: Establish the ‘right division’ of profits typically will require negotiations Taking part in a cartel is often illegal and therefore costly (chance of being caught,

fines, etc.) Monitor each other to ensure compliance

Although not much is known about magnitude of these costs we may safely assume these to increase in the number of participants. Negotiations and reaching an agreement more difficult in larger groups Threat of ‘race to the courthouse’ (leniency, it matters to be (one of) the first) Monitoring efforts Risk of detection, because larger cartels are presumably ‘more visible’.

Optimal Cartel Size when Cartelizing is Costly

Take account of these costs by introducing a general cost function , with x being the number of cartel participants. In light of the previous discussion we naturally assume that .

Introducing costs tightens the incentive compatibility constraint for all cartel members.

Proposition 7: If the smallest firm in the industry is sufficiently small, then the optimal cartel size is less than all-inclusive.

0' xT

xT

Incentives to Collude

So far, we did not consider the incentives of firms to take part in a cartel or to remain an independent outsider instead. Given that the cartel is not all-inclusive, the question is ‘who is in and who is out?’.

The participation problem has already been formulated by Stigler (AER, 1950):

“…the promoter of an incomplete cartel is likely to receive much encouragement from each firm – almost every encouragement, in fact, except participation…”

The incentive to free-ride is due to the fact that cartel members do not fully utilize their capacity, while outsiders produce up to capacity and sell their products at approximately the same price.

In the current setting, firms are characterized by their capacity stock and this might cause a diversity in free-riding incentives.

Incentives to Collude

Extend the game with a participation stage at t = 0. We assume that firms play an open membership game as proposed by d’Aspremont et al. (1983).

Outsiders have no incentive to join a cartel if: Insiders have no incentive ‘to leave’ if: These two conditions together form the ‘participation constraint’. The participation constraint is a refinement criterion in the sense that it narrows

the set of viable cartels. We further assume that firms receive a proportional share of total cartel profits

(at least three reasons for this: natural way to divide ‘the cake’, empirical evidence (Harrington, 2006) and it facilitates collusion).

\, Niic

io

i ii

oi

ci ,

Incentives to Collude

Lemma 8: As , a sufficiently small firm has no incentive to join any cartel. Lemma 9: A sufficiently large firm has an incentive to join any cartel. Proposition 10: A cartel agreement between the largest firms in the industry is a

solution of the game. Note, however, that this result is typically not unique. There may well exist

equilibria in which the largest outsider is larger than the smallest insider. A cartel that is of special interest is the cartel for which total cartel profits are

highest, i.e., the most profitable cartel. We denote this cartel by . Proposition 11: For any given cartel size, the most profitable cartel comprises the

largest firms in the industry. Proposition 12: If the smallest member of is sufficiently large, then is a

solution of the game.

0

* *

*

Concluding Remarks

This paper provides a rationale for the existence of incomplete cartels. Main conclusions:

If cartelizing is costless the optimal cartel size is all-inclusive. If cartelizing is costly the optimal cartel size is less than all-inclusive when the

smallest firms are sufficiently small. Sufficiently small firms have no incentive to take part in any cartel. Sufficiently large firms have no incentive to free-ride on any cartel. Under certain conditions, firms have an incentive to form the most profitable

cartel, which may or may not be all-inclusive.

It may be interesting to allow for large firms in absolute terms, but this significantly complicates the analysis. Note, however, that this is unlikely to

change any of the main results.