Impacto medioambiental

In this section I will examine the interaction between the buyers and the antitrust authority. I will mainly investigate the incentives of the buyers to report (signal) and the antitrust authority’s reaction (to investigate). First of all, some further assumptions are necessary. It will be assumed that the buyers know with certainty if a cartel exists and that they are always worse off with the creation of a cartel (this assumption excludes cases where buyers might provide a noisy signal to the antitrust authority). If the buyers are not always worse off with the creation of a cartel, then strategic reporting might be generated, something that I am not interested in investigating in this topic. I will also assume that if the antitrust authority

investigates and a cartel exists, the cartel will be revealed with probability one. Introducing uncertainty of the antitrust authority’s successfulness will generate reporting conditional on the probability of an authority to investigate and punish (allowing for type I error), this is examined at a later stage.

I also assume that if a cartel is formed it will be able to compensate the buyers (denoted as Comp), in order to avoid being reported. More specifically, given that a cartel is formed:

Comp =Comp∗ _ifIB

R = 0 andComp= 0 ifIRB = 1, whereComp∗ is suppliers choice variable.

The actual amount of compensation will be decided by the cartel at the equilibrium. As for the antitrust authority, I will assume that its prior on the likelihood of cartel formation is equal to zero, P1 = 07. Moreover, the probability of detection equals one8 and the cost

of investigation is small enough for the antitrust authority to investigate whenever there is updating in its beliefs. The purpose of the last two assumptions is to shift the focus on the consequences of reporting and neutralize any deterrence effect that policies of the antitrust authority might have on the cartel’s behavior. As for the payoff functions, the buyer value function is defined as:

VB_(IB R;v, Q∗, Comp, I A In, Q A_{) =CS(I}B R, v;Q∗, Comp, I A In, Q A₎ −v_∗IB R+Comp∗(1−I B R) (7.1)

The value function is defined as the difference of the CS with the cost of reporting plus any compensation paid to the buyers. The CS received by the buyers depends on the actions of both the suppliers and the antitrust authority. The cost of reporting is conditional on the action of buyers9_{. The compensation, which is decided by the suppliers, is conditional on the}

7_{Where being the probability that a cartel exists and 1}

−P1 = 1 the probability that it does not. This

assumption is equivalent to the legal argument that someone is innocent until he is found guilty.

8_{In other words, if a cartel exists and the antitrust authority investigates then it will punish with probability}

equal to one.

buyers not to report10_.

The antitrust authority maximizes the social welfare function while taking into consider- ation their costs of investigation (CA). Assuming equal weight to the CS and to the suppliers

profits, I derive the following value function:

VA(IInA, Q A ;P r, CAT A, Q∗, Comp, IRB) = X j=0,1 P rj M X m=1 VmB(I B R;v, Q A , Comp, IInA, Q A ) +P rj N X i=1 Πi(Q, Comp;C, IInA, Q A_{, I}B R) +I A In∗CA (7.2)

P r refers to antitrust authority’s prior beliefs in the existence of a cartel, when P rj=0

there is no cartel andP rj=1there is a cartel. The antitrust authority’s payoff function depends

on the choices of the suppliers and the buyers and the beliefs (ex-ante and ex-post, in the case with updating) on whether a cartel exists. Equation (7.2) presents the information asymmetry between antitrust authority and the participants of the market, buyers and suppliers, on the behavior of the suppliers. Notice that the incentives of the buyers and the antitrust authority are aligned (higher CS is desirable by both), however this is not the case with the suppliers. A complete description of the value functions for each choice set is provided in the mathematical appendix. Note that the suppliers’ profit function will be defined in a later stage.

The equilibrium of the signalling game, between buyers and antitrust authority, is de- fined as a strategy that generates the max payoff of the two players given the antitrust authority’s beliefs and out of equilibrium path beliefs. The next two Lemma characterize the two equilibria that arise. However, let me first define CSc as the Consumer Surplus generated

under Cournot competition, CSm as the Consumer Surplus under Collusion and Comp as

10_{I implicitly assume that reporting is a public signal. Alternatively, the cartel members observe whether}

an antitrust authority initiates an investigation and the compensation can be conditional on investigation (At equilibrium the antitrust authority only investigates if there has been reporting. This is a result of the priors being equal to zero and the existence of cost of investigation for the antitrust authority.).

Compensation paid to the buyers.

Lemma 7.1.1 Pooling Equilibrium:

If v _≥ CSc −CSm−Comp, the buyers never report and the antitrust authority choose not

to investigate, (IB

0 ,(I0A,(I1A, QA))) with P r1 = 0 (no updating). While the antitrust authority

investigates whenever there is reporting.

The proof of the above lemma and the one that follows will be given in the mathematical appendix. The first lemma states that there will be no reporting if the cost of reporting is higher than the difference of CS’s, under Cournot and collusion, and the compensation. In other words, if the cartel manages to make the buyers just indifferent from the Cournot equilibrium (i.e. by choosing accordingly the amount of compensation or choosing sufficiently low prices) then there will be no reporting and the cartel will remain hidden from the antitrust authority. Alternatively, it might be the case that the costs of reporting may be forbiddingly high for the buyers to report.

Lemma 7.1.2 Separating Equilibrium:

If v < CSc −CSm −Comp, the buyers report under collusion and the antitrust authority

investigate, ((IB

1 , I0B),((I1A, QA), I0A)) with P r1 equal to 1 if there is reporting P r1 and equal

to zero if not.

A direct observation that arises from the above lemmas is that there are only truth telling strategies used by the buyers. This is driven by the assumption that buyers are always worse off under cartel formation, as well as from the fact that there is no noisy signal and that the buyers have no uncertainty of the competitiveness of the market. As for the last lemma, it states that collusion is not possible if the cost of reporting is low enough, consequently there is only one equilibrium with reporting and it is conditional on the cost of reporting and the compensation the buyers will receive. Therefore, reporting will not be observed if

the cartel members are able to compensate the buyers. An interesting result that emerges is that the absence of reporting, in industries where the buyers are able to report (have insight information), does not mean that a cartel does not exist. It might be the case that the cartel is successfully controlling for reporting.

The condition used to define the two lemmas can be used to define the Incentives Compatibility Constraint (ICC from hereafter) of the buyers. A cartel will need to choose its compensation in order for the buyers not to report, otherwise the suppliers will not collude (this is derived with basic backward induction arguments of the cartel members’ participation behaviour11_{). The ICC is found by rearranging the condition of lemma 7.1.1:}

Comp        ≥CSc−CSm−v if v < CSc−CSmbe, = 0 if v > CSc−CSmbe.

The above condition states that the compensation decreases as the cost of reporting increases12_{. Furthermore, notice that there is a max value for v such that no compensation}

needs to be paid, this can be seen by the second part of the ICC. The max point is found by comparing the loss of the CS under the benchmark case13_{, superscript (be) is for benchmark,}

where no reporting is observed and no compensation needs to be paid. This is equivalent to setting the ICC equal to zero and finding the cost of reporting that is greater than the difference of the CS under Cournot and collusion.

The above analysis has implicitly assumed that the antitrust authority requires all buyers

11_{If the buyers are allowed to report then the cartel members will know with certainty that the antitrust}

authority will dissolve the cartel. Hence, if there is a specific date that the cartel will dissolve then the suppliers (cartel members) will deviate in the period before the break-down. The last argument suggests that the incentives of deviation will be such that the cartel becomes unstable from period zero and as a result will not be formed at all.

12_{It is not optimal, as it will be argued later on, for the cartel to overcompensate. Hence, the ICC will hold}

with equality.

13_{Under linear demand (P =a−bQ) the benchmark collusive quantity isQ}

m=a2−bc. Notice that the under

linear demand can be written asCS= bQ₂2 hence the CS under collusion (benchmark case) isCSm=(a−c)

2

to report, thus excluding the possibility of a free-rider’s problem between buyers. However, this problem is present due to the existence of the cost of reporting. The cost of reporting needs to be paid by at least one buyer but the benefits (CS under Cournot) will be shared by all. A buyer might not be willing to report if someone else is going to report, thus free-riding on other buyers. An example of this dilemma is presented in the game that follows.

Buyer i / Buyer j Report Do not report Report CSi,c−v, CSj,c−v CSi,c−v, CSj,c

Do not report CSi,c, CSj,c−v CSi,m,CSj,m

Table 7.1: Reporting Dilemma

The row player corresponds to buyer i while the column are buyers j wherejǫ_{1, .., i₋

1, i+ 1, ..., n_} and j ₆= i. A similar issue is examined by Palfrey and Rosenthal (1984). In their paper they show that there are M (number of buyers) equilibria in pure strategies with M-1 buyers not reporting and one buyer reporting. They also show that if not all but some players use pure strategies then there is at most one equilibrium in mixed strategies and if all players choose mixed strategies there are at most two equilibria in mixed strategies with reporting.

As for the antitrust authority, it is worthwhile to mention that requiring all buyers to report is not optimal. Since reporting is a wasteful activity. At the same time Palfrey and Rosenthal (1984) have shown that if there is a rule that requires only one player to pay the cost (report) for a project that benefits all (competition in the next stage) then there will be at least one buyer that will report. Thus, the optimal rule should be set to one buyer to report.