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

More documents

Recommendations

Info

similar to that in the non-cooperative case:@d coj= @dco j@q i @ j (q)@ j (q)= 1 Ci 00 [f + t] @ j (d co ) @j (q);@q i @d j @q iwhere = @ 2 E j =@d 2 j . Since < 0; C00 i () > 0, @ j () =@d j@ i (q) =@q i < 0 the result is established.< 0; andA.11 Proposition 6Proof. The FOC at the production stage is given by (8). Using (14) givesnXj=1@ i (d ) @d j[@d j @q ii i ]+ @ i(q) @i (q)[1 t]+@q i @q i@d nX@ij (d) j j = 0:@q i @dj=1 i(44)Contrast this with (10) in the non-collusive case. Invoking symmetry –i.e. i i = 8i –using (D3) and rearranging reduces (44) to@ i (q)@q i=@d j (q) @ i (d (q))[ ]@q i @d j1 t + P : (45)n @ i (d)j=1[ ]@d jP nj6=iContrast this with (34) in the non-collusive case. For the same reasons as inthe proof of Lemma 3 we must have @ i (q) =@q i < 0 8i 2 N in (45), whichimplies q coi > q C i 8i, where q C i stands for …rm i’s Cournot quantity.A.12 Proposition 7Proof. Maximising (19) with respect to Q gives the FOC:n 0 [f + t]t @d@Q +n 1 0 [f + t] @@Qn @@Q@dCi 0 @Q d = 0:(46)Substituting Ci 0 from the FOC (11) for optimal declaration into (46) yields[1 t] @@Q = 0;which, given that t < 1, requires @=@Q = 0. This implies that under thestated assumptions made a …xed rule leads to the monopoly quantity.34
A.13 Proposition 8Proof. First note that symmetry ensures that all …rms increase their declarationby the same amount if they all earn an additional dollar of grosspro…t. Therefore increasing the pro…t for all by the same amount does notin‡uence the individual ex post detection probability, and an individual dollarof gross pro…t will never lead to an additional declaration of more thanone dollar if the detection probability does not change. Therefore@d@Q = @d @@ @Q 1: (47)Di¤erentiating (20) with respect to Q gives the FOCn [f + t]t @d@Q + n 1 [f + t] @@Qn @@Q@dCi 0 @Q d = 0:(48)Substituting the FOC for an optimal declaration from (5) into (48) andsimplifying yields1 [f + t] @@Q + @@Qwhich further simpli…es to1 t @d @i (d )[f + t] d [f + t] + t = 0;@Q @d i(49)@ i (d )@d i[f + t] d 1@d @= 0: (50)@ @QGiven that t < 1, conditions (D1) and (47) imply that (50) holds if and onlyif @=@Q = 0.35
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 and 22: Lemma 4 Any jointly optimal declara
Page 23 and 24: The incentive to sabotage the gross
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: A.8 Proposition 4Proof. Given the u

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?