Bayer, R.- C. and F. A. Cowell - DARP

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and the following result is available:Proposition 5 For any d co 2 D co a linear-symmetric evasion rule satisfying(D5) implieswhere d coinXi=1nXi=1d xid xinXi=1nXi=1d coi if C 000i ( i d i ) 0 8d i 2 [0; i ] ; (17)d coi if C 000i ( i d i ) 0 8d i 2 [0; i ] ; (18)stands for the declaration of …rm i under collusion.In the case of the simple relative rule (D5), if marginal evasion costsincrease at an increasing rate then joint expected net pro…t maximisationleads to a higher aggregate declarations than under a …xed rule.How is production in‡uenced by a relative rule if collusive declarationsare possible? The declaration vector that maximises joint pro…ts at thesecond stage depends on the vector of gross pro…ts resulting from the productionstage. In the non-collusive setting (Section 5) we know that thereis an incentive to produce more than the Cournot quantity in order to increasethe evasion opportunity: the main channel for this is the fact thatincreasing your quantity reduces the pro…t and declarations of your rivals,which in turn reduces your own detection probability. The following lemmacon…rms this e¤ect under cooperative declaration behaviour:Lemma 5 Increasing the own quantity under cooperative declaration decreasesthe equilibrium declarations of the competitors at the collusive declarationstage:@d coj@q i< 0 8j 6= i:So there is still an incentive to increase production beyond the Cournotquantity in order to reduce competitors’ pro…ts, with the intention of decreasingone’s own equilibrium detection probability. Then the absence ofother e¤ects (such as cooperative saving of evasion cost in the asymmetriccase) is a su¢ cient condition for a relative rule to increase output and henceto yield e¢ ciency gains. A su¢ cient condition is symmetry.Proposition 6 If identical …rms collude at the declaration stage under asymmetric relative rule then the aggregate production quantity exceeds thequantity under a …xed rule with the same equilibrium detection probabilities.20
The incentive to sabotage the gross pro…t of competitors by choosing aquantity higher than the Cournot quantity is still present even when collusiontakes place at the declaration stage. Symmetry is su¢ cient –but notnecessary – for this e¤ect to translate into e¢ ciency gains. In a symmetricindustry where ex-ante identical …rms collude on declarations, a relativerule leads to the same declaration behaviour as a …xed rule while the secondbene…t of a relative rule, which is greater allocative e¢ ciency, still obtains.6.2 Collusion: Production stageNow focus on the question whether the relative rule (at the second stage) hasany impact on the cartel quotas chosen at the …rst stage. Does the relativerule fare better than the …xed rule with respect to economic e¢ ciency?Suppose …rms can jointly maximise expected pro…ts by choosing quantities,anticipating individually optimal choices at the declaration stage; butsuppose they cannot cooperate at the declaration stage. Then under a relativerule …rm i will declare d i as de…ned by the condition (7) dependingon the vector of gross pro…ts realised at the production stage; under a …xedrule the relevant condition is (11). If …rms can collude on production onlythen they will choose a jointly optimal vector of quantities given the anticipateddeclarations. For simplicity consider identical …rms and concentrateon a symmetric equilibrium. An equilibrium with a production vector requiringidentical quantities for all …rms implicitly assumes that the cartelof identical …rms restricts all feasible allocations to the ones where …rmsproduce identical quotas. 19First consider the case of a …xed audit rule. Denote the gross pro…tper …rm by (so that the aggregate gross pro…t is n) and the resultingaverage declaration by d. The expected joint net pro…t for equal quotas and19 In some cases …rms might be able to increase the expected joint net pro…t by using anasymmetric allocation of quotas. But there are good reasons why …rms in a cartel wouldrefuse to produce less than other …rms. If identical …rms bargaining over the collusionsurplus have equal bargaining power they should end up with the same surplus. Anallocation that implements di¤erent quotas can only be sustained if side payments arepossible. In the more likely case that side payments are not possible due to the oversightof a control agency then the best …rms can do is to choose the best cartel agreement where…rms have identical quotas. Even if side payments are possible …rms still might refuse toproduce less than other identical …rms, as they do not want to lose market share.21
Page 3 and 4: 1 IntroductionThe behaviour of …r
Page 13 and 14: allows substituting C 0 from (7) in
Page 15 and 16: is important to note that all the r
Page 17 and 18: i more scope for evasion. By increa
Page 19 and 20: two-stage nature of …rms’decisi
Page 21: Lemma 4 Any jointly optimal declara
Page 25 and 26: allocative e¢ ciency the relative
Page 31 and 32: A.2 Proposition 1Proof. If the rela
Page 33 and 34: A.5 Lemma 3Proof. From condition (1
Page 35 and 36: A.8 Proposition 4Proof. Given the u
Page 37: A.13 Proposition 8Proof. First note

Bayer, R.- C. and F. A. Cowell - DARP

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