Penalizing Cartels: The Case for Basing Penalties on Price Overcharge 1

Transcription

1 Pnalizing Cartls: Th Cas for Basing Pnaltis on Pric vrcharg Yannis Katsoulacos Evgnia Motchnkova 3 David Ulph 4 Abstract In this papr w st out th wlfar conomics basd cas for imposing cartl pnaltis on th cartl ovrcharg rathr than on th mor convntional bass of rvnu or profits (illgal gains). To do this w undrtak a systmatic comparison of a pnalty basd on th cartl ovrcharg with thr othr pnalty rgims: fixd pnaltis; pnaltis basd on rvnu, and pnaltis basd on profits. ur analysis is th first to compar ths rgims in trms of thir impact on both (i) th prics chargd by thos cartls that do form; and (ii) th numbr of stabl cartls that form (dtrrnc). W show that th class of pnaltis basd on profits is idntical to th class of fixd pnaltis in all wlfar-rlvant rspcts. For th othr thr typs of pnalty w show that, for thos cartls that do form, pnaltis basd on th ovrcharg produc lowr prics than thos basd on profit) whil pnaltis basd on rvnu produc th highst prics. Furthr, in conjunction with th abov rsult, our analysis of cartl stability (and thus dtrrnc), shows that pnaltis basd on th ovrcharg out-prform thos basd on profits, which in turn outprform thos basd on rvnu in trms of thir impact on ach of th following wlfar critria: (a) avrag ovrcharg; (b) avrag consumr surplus; (c) avrag total wlfar. JEL Classification: L4 Antitrust Policy, K Antitrust Law, D43 ligopoly and thr Forms of Markt Imprfction, C73 Stochastic and Dynamic Gams; patd Gams Kywords: Antitrust Enforcmnt, Antitrust Law, Cartl, ligopoly, patd Gams. W ar gratful to Jo Harrington and Giancarlo Spagnolo for vry hlpful commnts and suggstions. W would also lik to thank th participants of a sminar at th Saudr School of Businss, Univrsity of British Columbia (Sptmbr 4) and at th sssion on th Enforcmnt of Comptition Policy in th 9 th CESSE Annual Confrnc (Corfu, 4 6 July 4) for hlpful suggstions and commnts. W ar particularly gratful for commnts and suggstions by Jim Brandr, Tom Davidoff, Tom oss and alph Wintr. All inaccuracis and omissions rmain our sol rsponsibility. Yannis Katsoulacos acknowldgs that this rsarch has bn co-financd by th Europan Union (Europan Social Fund ESF) and Grk national funds through th prational Program "Education and Liflong Larning" of th National Stratgic frnc Framwork (NSF) - sarch Funding Program: AISTEIA CoLEG. Also w acknowldg support from th Tinbrgn Institut, VU Univrsity, Shorttrm Visitor Program. This papr is basd on a rvisd vrsion of TI Discussion Papr 4-9/VII, Pnalizing Cartls: Th Cas for Basing Pnaltis on Pric vrcharg, by th sam authors. Dpartmnt of Economic Scinc, Athns Univrsity of Economics and Businss, Patission 76, Athns 4 34, Grc, katsoul@aub.gr or ysk@hol.gr 3 Dpartmnt of Economics, VU Univrsity Amstrdam, D Bollaan 5, 8 HV Amstrdam, Th Nthrlands, TILEC and Tinbrgn Institut, motchnkova@fwb.vu.nl 4 Univrsity of St Andrws, School of Economics and Financ, St. Andrws, Fif, KY6 9A, Scotland, UK; Dirctor Scottish Institut for sarch in Economics (SIE), du@st-andrws.ac.uk

2 . Introduction Cartls ar still vry activ throughout th world and prvasiv in a wid varity of markts - dspit incrasd nforcmnt in th form of much highr fins and othr sanctions and th implmntation of lnincy policis. As Lvnstin and Suslow () rport for th US, th country with probably th toughst sanctions 5 rgim in th world from 99 thr wr approximatly7 DoJ cartl convictions or ovr 36 pr yar. Indd, fifty nw criminal cartl cass wr fild in 3. 6 Empirical vidnc suggsts that whil antitrust is th most important forc lading to cartl brak-up thr ar limitations to th ffctivnss of ths policis as currntly dsignd. 7 In this papr, w argu that th widly mployd currnt dsigns of on of th most important nforcmnt tools in th fight against cartls namly montary pnaltis ar flawd and that this dos indd limit th ffctivnss of this tool. W propos an altrnativ dsign that could significantly improv th ffctivnss of montary pnaltis. Spcifically, our objctiv is to st out th wlfar conomics framwork that supports th cas for Comptition Authoritis to switch th bas on which pnaltis for cartls ar st away from th convntional bass of rvnu or illgal gains and instad to bas th pnalty on th cartl ovrcharg. ur rason for choosing to xamin this altrnativ pnalty bas is that, consistnt with th prscriptions of scond-bst wlfar conomics, this policy is targtd at th undrlying distortion gnratd by cartls th incras in pric. 8 To mak this cas w analyz th impact of various pnalty rgims that hav bn widly considrd and analyzd in th litratur on: (i) th pric chargd by any givn cartl; (ii) cartl stability and hnc th numbr of cartls that form; and finally (iii) th ovrall lvl of wlfar inducd by th diffrnt rgims. 9 W us a rpatd Brtrand oligopoly modl that allows us to compar both th pric and th dtrrnc ffcts of th four major typs of fining structurs invstigatd in th litratur. Ths four typs ar: fins basd on rvnu (s.g. Bagri t al. (3) and Katsoulacos and Ulph 5 In US sanctions tak th form of montary pnaltis as in all othr countris plus trbl damags and criminal convictions. 6 S Lvnstin and Suslow (4). 7 S in particular th rcnt sris of paprs by Lvnstin and Suslow (,, 4) that contain rviws and xtnsiv rfrncs to th rlvant litratur. 8 By contrast th altrnativ pnalty bass rvnu or illgal gains/profits whil also dpnding on th cartl ovrcharg also dpnd on th cartl output, which in turn dpnds ngativly on th cartl pric thus diluting th incntiv to lowr prics. 9 Th impact of th toughnss of th pnalty rgim on th cartl pricing bhavior has bn addrssd in Katsoulacos and Ulph (3). Howvr, thy hav rlid on a static gam, hav not analyzd th impact of th pnalty structur on cartl stability and hav not xamind th dtrrnc implication of th various pnalty structur mployd in practic, as w do hr. In his sminal articl, Harrington (5) has also shown that pric dpndnt pnaltis (that ar basd on damags) imply that th stady stat cartl pric will b blow th simpl monopoly pric and that th toughnss of th pnalty rgim (th siz of th damag multiplir) is on of th factors that rduc th quilibrium cartl pric. Howvr, h dos not provid comparisons of all th altrnativ pnalty rgims xamind hr. Also, and most importantly, th possibl dtrrnc ffcts of various pnalty structurs in conjunction with thir dirct pric ffcts hav not bn systmatically analyzd in litratur on antitrust so far.

3 (3)); fins basd on illgal gains (s.g. Harrington (4, 5) or Houba t al. (, )); fins basd on cartl ovrcharg (s.g. Buccirossi and Spagnolo (7) and Katsoulacos and Ulph (3));and fixd fins (s.g. y (3) or Motta and Polo (3)). Whil othr paprs hav considrd th proprtis of ach of th four pnalty rgims and mad som limitd comparisons btwn thm, a major contribution of this papr is to undrtak a systmatic comparison of all four rgims in trms of both th prics st by thos cartls that form and on th dtrrnc of potntial stabl cartls (which w somtims rfr to as cartl stability in short). In stting out our argumnts w also mak two important mthodological contributions. First, w xtnd th rpatd Brtrand modl proposd in Houba t al. (, ) to captur th ffct of th cartl stability condition on cartl pricing bhavior. This allows us to bridg th standard critical discount factor approach to th analysis of collusion (s.g. Tirol (988) or Motta and Polo (3)) to profit maximizing dcisions by th cartl mmbrs (with continuum of prics, which can b chosn by th cartl). This lattr approach has bn proposd in.g. Block t al. (98) or Harrington (4, 5). Scond, w provid a framwork within which w intgrat th impact of pnalty rgims on th pric stting bhaviour of cartls that do form with thir dtrrnt ffcts and this provids an valuation of th ovrall impact of diffrnt pnalty rgims. ur first rsult is that th class of profits-basd pnaltis is idntical to th class of fixd pnaltis in trms of all th wlfar-rlvant outcoms thy produc pric, dtrrnc. Anything that can b achivd by on typ of pnalty can b achivd by th othr using an quivalnt lvl of pnalty. Consquntly w confin attntion to thr pnalty rgims thos basd on profits, thos basd on rvnu, and thos basd on th cartl ovrcharg. In trms of th pric st by thos cartls that do form, w show that proportional fins basd on ovrchargs ar mor succssful in trms of thir ffct on pric whn compard to proportional fins basd on rvnus or illgal gains. Spcifically, w show that: pnaltis basd on illgal gains lad cartls to st th monopoly pric. if th pnalty is imposd on rvnu thn th cartl pric will b abov th monopoly pric. if th pnalty is imposd on ovrcharg thn th cartl pric will b blow th monopoly pric, and, morovr, th nd to maintain cartl stability can rquir that th cartl sts a maximum pric which dcrass towards th comptitiv pric as it bcoms incrasingly difficult to maintain stability. morovr, ths conclusions do not dpnd on th toughnss of th individual pnalty rgims whr toughnss rflcts both th pnalty rat and th probability of dtction. Apart from th trivial cas whr ithr th probability of dtction or th pnalty rat is zro, in which cas thr is ffctivly no pnalty rgim and it dos not mattr on which bas th non-xistnt pnalty might hav bn basd. 3

4 Turning to th dtrrnc impact of diffrnt pnalty structurs, an important contribution of this papr is to provid for th first tim a full analysis of how this is influncd by pnalty rats and by othr nforcmnt and markt rlatd paramtrs, such as th dtction rat and th lasticity of markt dmand. To start with w show that, as xpctd, dtrrnc dpnds on th toughnss of th pnalty rgim, and that if ach rgim is mad sufficintly tough all cartls can b dtrrd. This implis that in ordr to maningfully compar th ffcts of using diffrnt pnalty bass on dtrrnc and hnc ovrall wlfar w nd to nsur that ach pnalty rgim is in som sns qually tough. W considr two concpts of qual toughnss. Th first is dtrrnc quivalnc: th sam fraction of all stabl cartls that could potntially form do in fact form. Givn th abov rsults on th prics st by cartls that do form, it is clar that undr dtrrnc quivalnc pnaltis on th ovrcharg out-prform thos on profits which in turn out-prform pnaltis basd on rvnu. Howvr comptition authoritis and courts ar not concrnd solly with dtrrnc, thy also want pnaltis that ar rasonabl and proportionat. So th scond critrion of qual toughnss that w considr is that of pnalty rvnu quivalnc: on avrag th siz of th pnalty actually paid by any cartl that forms and is subsquntly dtctd and pnalizd should b th sam. W again dmonstrat that in trms of ach of th following critria: avrag ovrcharg, avrag consumr surplus, avrag total wlfar (consumr plus producr surplus) pnaltis on th ovrcharg out-prform thos on profits which in turn out-prform pnaltis basd on rvnu. Whil, as w show, thr can b som tnsion btwn ths two diffrnt notions of qual toughnss w also show that this has no ffct on on of our cntral conclusions that, howvr on rsolvs this tnsion, pnaltis basd on th ovrcharg wlfar dominat thos basd on profits. 3 As w will show, achiving a givn lvl of toughnss undr a rvnu basd pnalty rgim rquirs th pnalty rat to vary according to th lasticity of dmand in th industry. An stimat of this can b obtaind by th us of what Farrll and Shapiro (8) hav proposd in th cas of mrgrs through th application of Critical Loss Analysis: rvald prfrnc information (to) mak infrncs about prfrncs basd dirctly on obsrvd choics. Hr, and to paraphras thir argumnt in rlation to mrgrs, on can mak infrncs about dmand snsitivity as gaugd by ral firm(s) basd on (thir collusiv) choic of pric. Th ida is capturd by th Lrnr quation. Sinc, as indicatd, th pric st by any cartl that dos form undr both a rvnu-basd pnalty rgim and an ovrcharg-basd pnalty rgim will potntially vary dpnding on th intrinsic difficulty of holding th cartl togthr, so too will th actual pnalty paid. So all w can rquir is that on avrag th pnalty paid should b th sam. 3 Mor prcisly, for any dgr of toughnss of th ovrcharg-basd rgim and for any dgr of toughnss of th profit-basd rgim which lis abov th lvl rquird to achiv quivalnt dtrrnc to th ovrcharg rgim but blow that which is rquird to achiv an quivalnt lvl of pnalty rvnu as th ovrcharg rgim, th ovrcharg-basd rgim is wlfar suprior to th profits-basd rgim in trms of avrag ovrcharg, avrag consumr surplus, avrag total wlfar. 4

5 ur clar policy rcommndation is thrfor that Comptition Authoritis should switch to a pnalty structur that uss th pric ovrcharg as th bas on which th pnalty is imposd. 4 In ssnc th rason is that ovrcharg basd fins ar prfrabl, sinc thy targt th pric, which is causing th damag to consumrs. Profit basd fins ar a wakr instrumnt sinc thy do not targt th pric dirctly, but targt firms arnings, whil rvnu basd fins hav strongly countrproductiv ffcts as originally also shown in Bagri t al. (3) and Katsoulacos and Ulph (3). vrcharg-basd fins ar suprior to currntly mployd pnalty rgims at not just a thortical wlfar-conomics lvl. It is likly that implmntation of ovrcharg-basd fins in practic is no mor difficult than th nxt bst altrnativ (in trms of wlfar inducd) a profits-basd pnalty. 5 Although stablishing th countrfactual can b tricky, comptition authoritis hav to obtain stimats anyway during th invstigation in ordr to stablish whthr a group of firms rally has drivn up th pric. And crtainly such information is ndd in ordr to obtain stimats of th ovrcharg, during damag claim procdurs - dvlopmnts in conomics and conomtrics mak it possibl to stimat cartl ovrchargs with rasonabl prcision or confidnc. 6 W furthr discuss implmntation issus blow. Th rst of this papr is organizd as follows. Sction discusss th currnt sntncing guidlins. Sction 3 outlins th modl. In Sction 4 w driv all th main formula for pricing, dtrrnc and various wlfar indicators undr ach of th four pnalty rgims. In sction 5 w undrtak a systmatic comparison of th various rgims in trms of prics, dtrrnc and various masurs of ovrall wlfar. Sction 6 concluds.. Brif viw of th Currnt Sntncing Guidlins 7 This sction dmonstrats through a brif rviw that rvnu-basd pnaltis is th norm in all major jurisdictions with caps that ar basd on ithr rvnu (EU) or on illgal gains (US). To start with, in th EU, a violation of th cartl prohibition constituts an administrativ offnc. In ordr to nsur transparncy of this nforcmnt procdur, th EC publishd nw pnalty guidlins in 6 rfining th mthodology that has bn applid so far (sinc 998). Undr ths guidlins, fins ar calculatd in th following way: First, th Commission dtrmins a basic amount which may b adjustd aftrwards du to aggravating and mitigating lmnts. Th basic amount is calculatd by taking into account th undrtaking s rlvant turnovr (of th last yar of th cartl), th gravity and th duration of th infringmnt, as wll 4 It is important to not that in this papr w ar concrnd solly with th qustion of which of th various altrnativ pnalty bass is suprior in trms of its wlfar implications and not with th diffrnt issu of whthr currnt cartl pnalty rats ar or ar not too high. Thr is an xtnsiv thortical and mpirical litratur on this lattr qustion which is rviwd, for xampl, in Katsoulacos and Ulph (3). Thir rsults support th rcnt vidnc by Allain t.al () and Boyr t.al () that currnt rats ar not too low and indicat that highr rats (on a rvnu bas) will not ncssarily lad to lowr cartl prics. 5 Bagri t. al. (3) provid additional argumnts to thos prsntd blow for prfrring a profit-basd pnalty rgim to a rvnu-basd rgim (thy do not considr an ovrcharg-basd rgim). 6 For dtails s Brandr and oss (6). 7 S also Bagri t.al. (3). 5

6 as an additional amount of about 5% - 5% of th valu of sals in ordr to achiv dtrrnc. For cartls, th proportion of th rlvant turnovr is st at th highr nd of th scal 8 which is 3%. Additional uplifts or rductions ar thn mad whn crtain aggravating or attnuating circumstancs xist. Howvr, th maximum amount of th fin imposd shall not xcd th cap of % of annual worldwid turnovr of th undrtaking in th prcding businss yar. In th US, cartls ar proscutd as criminal offncs, and sntncs ar imposd by a nonspcializd court. Th courts us th US Sntncing Guidlins (USSG) as a consulting tool rgarding th appropriat form and svrity of punishmnt for offndrs. According to ths guidlins, both pcuniary and non-pcuniary pnaltis may b imposd: fins on firms and individuals, as wll as imprisonmnt of individuals involvd in th cartl. With rgards to fins on firms, th procss of thir assssmnt bgins with th calculation of a bas fin. To dtrmin th bas fin, a prcntag of th volum of affctd commrc, that is, of total sals from th rlvant markt, is takn into account. Th USSG suggsts that % of th volum of affctd commrc can b usd as a good proxy. This volum of affctd commrc covrs th ntir duration of th infringmnt. nc th amount of th bas fin has bn calculatd, aggravating and mitigating lmnts ar takn into considration. Howvr, th final fin for undrtakings must not xcd a maximum statutory limit which is th gratst of million USD or twic th gross pcuniary gains th violators drivd from th cartl or twic th gross pcuniary loss causd to th victims. Most othr ECD countris follow th lad of th US and EU on on or both dimnsions. For xampl, in th UK th starting point for calculating antitrust fins is a fraction of th rlvant turnovr, i.. affctd commrc; th cap on fins is st at % of th undrtaking s global turnovr, xactly as is th cas in th EU. 3. Th Modl W considr an infinitly-rpatd Brtrand oligopoly modl in th prsnc of antitrust nforcmnt. 9 Antitrust nforcmnt consists of th probability to dtct a cartl and a fin schdul. If th firms collud, thy will b dtctd probabilistically and find according to th fin structur. Givn th dtction probability and th fin schdul, th firms will collud at a pric that maximizs thir futur profit, supportd by a simpl triggr stratgy profil. In ach of infinitly many priods, n firms compt in prics in a homognous oligopoly modl with linar dmand function of th form Q p,, Whr p dnots pric and Q is th quantity supplid to th markt. Symmtric marginal costs ar dnotd by c and, consistnt with th structur of th dmand function, ar normalizd to. 8 6 EU Guidlins. 9 Svral lmnts of this modl ar borrowd from th analysis in Houba, Motchnkova and Wn (,, ). Similar dmand structur has bn analyzd in Katsoulacos and Ulph (3). 6

7 In th absnc of a cartl, th comptitiv quilibrium would b th uniqu Brtrand- N N Nash quilibrium in which all firms st pric at marginal cost, so p c. p is thrfor what is somtims rfrrd to as th but-for pric th countrfactual pric that would hav arisn had thr not bn a cartl. As is wll known, th assumptions of homognous products and Brtrand comptition imply that: In th but-for quilibrium all firms mak zro profits. In ordr to b abl to charg a highr pric all n firms in th industry nd to b in th cartl. Any firm that dviats will obtain for priod a profit qual to th industry profits at th cartl pric. This implis that th actual profits arnd by th cartl ar also th illgal gains profits in xcss of thos that would hav bn arnd had th cartl not bn in plac. Accordingly, throughout th papr w us th trms illgal gains and profits intrchangably.notic that, givn our assumptions, if a cartl formd that did not includ vry firm in th industry, thn Brtrand comptition btwn th cartl and th fring would driv pric down to marginal cost, and rproduc th comptitiv quilibrium. So if any cartl is to form and driv up pric it must includ all firms in th industry. Th socially worst outcom is whn all firms collud at th M monopoly pric p. W now assum that in vry priod th n firms dcid whthr to collud and if so, at what pric. If th firms collud at pric p c, total cartl illgal profits will b ( p) ( p)( p), whil total cartl rvnu is ( p) p( p). Both functions ar continuous and concav in p. Th pric-ovrcharg is th xtnt to which pric is raisd abov its "but-for" lvl (or Nash N lvl). This can b xprssd ithr as th absolut incras in pric p p p or as N p p N th prcntag incras. Givn our normalization assumption that p, ths N p two intrprtations tak th sam numrical valu. Givn th spcification of th dmand function, in this comptitiv quilibrium output will b. Givn th normalization of pric, ε also masurs th industry rvnu arnd in th comptitiv quilibrium, that is p Q Q p N N N N p Q. In addition N Q N is th invrs pric lasticity of dmand valuatd at th comptitiv quilibrium. So, in this modl, ε is a paramtr that rflcts th undrlying comptitivnss of any givn industry in which a cartl might form. It has thr diffrnt, though rlatd, intrprtations.. Th particular intrprtation will b rflctd in th dimnsions of th units in which th particular intrprtation is bing mad. N Thus as a masur of comptitiv output, Q, th units that apply will b of th quantitis in which th output of N that industry ar producd; as a masur of industry rvnu in th comptitiv quilibrium,, th units will b units of currncy that apply in that particular industry; as a masur of th invrs lasticity it will b a pur dimnsionlss numbr. 7

8 W assum that: du to limitd rsourcs of th CA, cartls will not b dtctd for sur and w dnot th probability of dtcting a cartl by, ; β is constant across all cartls in particular it is indpndnt of th pric chosn by a cartl and of whthr or not a givn cartl in a givn industry has prviously bn dtctd; if a cartl is dtctd and invstigatd, th CA will dcid for sur that th cartl has actd in anti-comptitiv fashion and impos a pnalty; all this is common knowldg. Givn ths assumptions in particular our assumption that th probability of dtction is constant and indpndnt of prvious dtction - profit maximization implis that, having bn dtctd, a cartl should just r-form. W assum that this is indd what happns. A similar assumption that th cartl rstablishs aftr ach conviction has bn adoptd in Motta and Polo (3). Howvr othr assumptions could b mad. For xampl Harrington (4, 5) assums that a cartl cass to xist aftr dtction. Mor gnrally Houba, Motchnkova and Wn (, 5) assum that, aftr dtction, thr is a constant probability call it, - that th cartl will continu in xistnc aftr dtction. Howvr, nothing of substanc is affctd by this mor gnral tratmnt. Lt F dnot th pnalty imposd on a cartl that has bn dtctd and succssfully proscutd. In th subsqunt sctions w will considr a numbr of altrnativ fin schduls/ structurs, in all of which th fin actually paid by a cartl potntially dpnds on th pric st by th cartl. So w considr 3 fins on illgal gains, F ( p) ( p) p p fins on rvnu, F p p p p ; fixd fins, F( p) ; ( ) ( ) F. Hr and ar th pnalty rats that apply, rspctivly, in th profit and rvnu-basd rgims. 4 Notic that for ach of ths thr pnalty rgims th pnalty bas and th pnalty itslf ar dnominatd in units of currncy. W will contrast all of ths pnalty bass with fins basd on th ovrcharg. Howvr, as notd abov, th ovrcharg, θ, is not dnominatd in units of currncy, and, givn our normalization of th comptitiv pric, can b givn two intrprtations. If w think of th In th mor gnral framwork with, w rplac th trm n( ) that appars in our analysis blow with th trm n ( ). 3 Not that in practic pnaltis also dpnd on th duration of th cartl. In our analysis w ar focusing on th stady-stat pnaltis. Howvr, incorporating duration introducs a numbr of complications that mak th analysis lss tractabl without fundamntally affcting th rsults. For xampl w los stationarity s Harrington (4), a papr that focuss on th implications of taking into account cartl duration. Katsoulacos and Ulph (4) also rcognis that comptition authoritis may dtct a cartl ithr whil it is still in xistnc or aftr it has falln apart and show how this affcts th optimal pnalty rat. 4 Ths ar pur numbrs that convrt a bas dnominatd in currncy into a pnalty that is also dnominatd in currncy 8

9 ovrcharg as masuring th absolut incras in th pric, thn, in ordr to hav an ovrcharg pnalty bas that is dnominatd in units of currncy w hav to multiply th ovrcharg by N som masur of output, and th natural on to us is th comptitiv output, Q. So th ovrcharg pnalty bas will b p p N. Q N p. Q N. 5 If w lt dnot th pnalty rat in an ovrcharg-basd pnalty rgim, thn th fin undr an ovrcharg-basd rgim is F ( p ) p. In th analysis that follows w will tak th pnalty rats,, to b constants. So whn w talk about diffrnt pnalty rgims w ar talking about th us of diffrnt pnalty bass. 6 If th firms collud at pric p in any priod, th xpctd pr priod profit to vry firm is ( p) F( p). This procss thn rpats vry priod, and ach firm taks account of th n discountd continuation payoff of rmaining in th cartl with discount factor. 7 W focus on th class of quilibria that ar supportd by simpl grim-triggr stratgis: firms collud at pric p in vry priod- rcognizing that thir collusion will b dtctd by th CA in any givn priod with probability, 8 and that, if dtctd, th cartl will crtainly b proscutd and hav th pnalty F( p) imposd. Howvr, if any firm dviats in any priod by undrcutting prics, th firms will rvrt to th static Nash/comptitiv quilibrium in all futur priods. With such stratgis, th prsnt valu of a firm's xpctd profit from bing in th cartl is givn by ( p) F( p) Vp, F(.). n( ) In ordr to support such an quilibrium, no firm should hav incntiv to dviat, which is th cas if and only if 5 Altrnativly, if w think of th ovrcharg as th prcntag incras in pric and so a pur numbr in ordr to hav a pnalty bas dnominatd in currncy w nd to multiply this by somthing which is also masurd in N N N units of currncy, and th natural on to us is rvnu arnd in th comptitiv quilibrium,. In p p N p this cas th pnalty bas for an ovrcharg-basd pnalty rgims will b. N N p Q. So, in ithr intrprtation th pnalty bas for th ovrcharg-basd rgim is, and is dnominatd in units of currncy. 6 n could also think of pnalty rgims in which th pnalty rats dpnd on th ovrcharg vn though th pnalty bas is somthing othr than th ovrcharg. Houba t al. (), Jansn and Sorgard () and Katsoulacos and Ulph (3) hav shown that th pric-rducing ffct of th profit basd fins can b improvd, whn th pnalty rat dpnds on th ovrcharg. In this papr w rul out this possibility and analyz pur ffcts of th diffrnt pnalty bass. Howvr it is asy to show in our modl as wll that if th pnalty rat is proportional to th ovrcharg, th rsulting cartl prics will b lowr undr ach pnalty structur that w considr. In addition, pnalty rats which vary with pric ovrcharg ar not in lin with th currnt sntncing guidlins as discussd in sction. 7 As notd abov w adopt Motta and Polo (3) assumption that cartl rstablishs aftr ach conviction. 8 W rul out th possibility of β = sinc thn thr would b no nforcmnt rgim and so th qustion of what bas to us for pnaltis would b irrlvant. 9

10 ( p)( p) F( p) D Vp, F(.) ( p) n( ) () whr th right hand sid of th condition is th profit a firm would rciv for just on priod from dviating by undrcutting th cartl pric. 9 To b mor prcis, w dnot D ( p) MAX p p b th profits mad by a firm that dviats from a cartl that has pp c st a pric p c, and p D ( p) argmax p p dnots th associatd pric chargd pp c by th dviator. Thn th dviation tchnology is givn by 3 M D p, p p D ( p)( p), p p p ( p) ; ( p) M M M M p, p p, p p M. () W ar not awar of any contributions analyzing this bfor for th cas whr pric is abov th monopoly lvl (as will b th cas undr th rvnu basd rgim). Lt n( ) masur what w call th intrinsic difficulty of kping a cartl togthr. This is incrasing in n sinc th mor firms thr ar in th cartl th smallr th shar of cartl profits accruing to any on firm, whras by dviating all th cartl profits accru to just on firm. It is also a dcrasing function of δ sinc th mor wight firms put on th futur th gratr th valu of staying in th cartl and not just grabbing th on-priod profits from dviating. Notic that as long as F thn th cartl stability condition () can only hold if, so in all that follows w will confin attntion to valus of,. W dfin th maximum critical difficulty,, as th valu of Δ at which th cartl stability condition just holds. 3 Notic that if thr is no antitrust nforcmnt i.. - thn th cartl stability condition just rducs to, and so w hav th standard rsult that. It is obvious that whn thr is nforcmnt (i.. ) - th assumption mad throughout this papr 9 As in Motta and Polo (3), w assum that pric-dviating firms will not b proscutd. Altrnativ assumptions ar xamind in Spagnolo (4), Buccirossi and Spagnolo (7), Chn and y (), Jansn and Sorgard (4). Proscution of pric dviating firms will not chang th main conclusion of this papr about supriority of ovrcharg basd fins. Pric ffcts rsults rmain similar to th currnt st up, whil dtrrnc ffcts of all thr pnalty structurs bcom idntical. 3 Not that this dviation tchnology is idntical to standard dviation tchnology, whr a firm undrcuts th M collusiv pric by a small amount and gts total cartl profits for on priod whn p p (s.g. Tirol, 988), whil it xtnds th notion of standard dviation tchnology for cartl prics abov monopoly lvl, i.. whn M p p. 3 Not that any maximum critical difficulty translats into minimum critical discount factor, which in n th absnc of antitrust nforcmnt can b rwrittn as, which is th standard critical discount factor in n infinitly rpatd Brtrand stting with n symmtric firms (s.g. Tirol 988).

11 - it will b mor difficult to kp a givn cartl togthr and th maximum critical lvl of difficulty will fall, so w will hav. In th nxt sction w will xplor prcisly how varis both within and across pnalty rgims. For any givn stabl cartl that forms w dfin th cartl pric inducd by a givn pnalty C C rgim as th pric, p, p that yilds th highst xpctd profit from bing in th cartl subjct to th sustainability condition (). Formally: C ( p) F( p) ( p) F( p) D p F(.) arg max subjct to ( p) p Th consumr surplus and total wlfar inducd by a cartl in a particular industry charging a pric p ar givn by ( p) CS( p), TW ( p) ( p ).3 (3) ur objctiv is to undrstand how th particular structur of fins i.. th choic of pnalty bas - affcts: C C (i) th pric, p, p chargd by any givn stabl cartl that dos form; (ii) th cartl stability condition as rflctd in th maximum critical difficulty,, for which stabl cartls will form, which dtrmins th numbr of cartls that form and so th dtrrnc ffct of a givn pnalty rgim; (iii) th ovrall pric that mrgs undr a givn pnalty rgim, which is dfind as th C pric, p, chargd by th cartl ovr thos valus of, for which stabl cartls xist, and th comptitiv pric p for thos valus of, for which no stabl cartl xists; (iv) th avrag 33 lvl of wlfar inducd by that pnalty structur, as rflctd in (a) th avrag pric/ovrcharg; (b) th avrag Consumr Surplus; (c) th avrag total wlfar. 4. Cartl Pricing, Cartl Stability and Wlfar In this sction w driv th impact of ach of th four altrnativ typs of fin structurs idntifid abov on cartl pricing and on cartl stability (by which w man th maximum 3 Notic that as long as th pric st by th cartl lis strictly btwn th comptitiv pric and th chok pric, that is as along as p, ths ar both strictly dcrasing functions of p. 33 Th avrag hr, and throughout th papr is ovr thos valus of, for which stabl cartls xist, and thos valus of, for which no stabl cartl xists and so th pric is just th comptitiv pric. In th absnc of any thortical or mpirical vidnc on th distribution of Δ, in calculating ths avrags w assum it is uniformly distributd on [,].

12 critical difficulty of holding a cartl togthr). 34 W combin ths into a numbr of masurs of ovrall/avrag wlfar. 4.. Pnalty on Profits A pnalty on illgal profits is givn by F (p) (p )( p), whr is th constant pnalty rat. Th stability condition for ach individual cartl mmbr is givn by ( )( p)( p) D ( p). (4) In what follows w lt dnot th toughnss of th profits-basd pnalty rgim. This masur rflcts both th probability of dtction and th pnalty rat. For thos stabl cartls that do form, th cartl pric is givn by: C M p arg max Vp, F( ) p. (5) p M Thus fins basd on illgal profits induc th cartl to st th monopoly pric p indpndntly of th toughnss of th pnalty rgim. 35 Hnc, according to dviation tchnology spcifid in (), th stability condition can b rwrittn as ( )( p)( p) ( p )( p). (6) Notic that in ordr for thr to b any p, p, for which th cartl stability condition (6) holds it is ncssary that. It also clarly follows from (6) that th maximum critical difficulty undr a profits-basd pnalty rgim that is implmntd with toughnss, is. (7) Furthrmor, a toughr rgim lowrs th maximum critical difficulty and so maks it lss likly that stabl cartls will form. In ordr to combin ths two lmnts and to provid an ovrall pictur of how a particular pnalty rgim affcts both th pric st by thos stabl cartls that do form and th xtnt to which crtain cartls that might possibly hav formd ar dtrrd from doing so w procd as follows. For all valus of [,] w dfin th ovrall pric that,, to b th pric that would b st by th would mrg undr a profits-basd rgim, p cartl for thos rangs of, pric for thos valus of, for which stabl cartls xist, and th comptitiv for which no stabl cartl forms. Formally: 34 Th xisting litratur has not xamind th implications of antitrust pnaltis for cartl stability in a systmatic mannr. 35 Houba t al. () show that if th probability of dtction and th pnalty rat ar constant thn th sam rsult holds undr homognous products for mor gnral dmands.

13 , p,. (8), This is shown in Figur blow. Figur : vrall Pric Undr Profits-Basd Pnalty Figur illustrats that for thos stabl cartls that do form, i.. for thos for whom, th cartl pric is qual to monopoly pric. This is bcaus th pnalty just producs an qui-proportionat rduction in nt xpctd profits. So maximizing nt profits is quivalnt to maximizing gross profits and th pric that is st is th sam as that which would hav arisn had no pnalty rgim bn in plac th monopoly pric. Notic that if, for som rason Cournot comptition, product diffrntiation tc. -, th but-for pric wr abov marginal cost thn illgal gains would b smallr than actual profits. This will imply that undr a pnalty basd on illgal gains th collusiv pric will b blow that producd by a pnalty on profits bcaus th rduction in pnalty is proportional to cartl output, so giving th cartl an incntiv to xpand output and lowr pric. 36 Howvr, bcaus th ovrcharg-basd pnalty focuss dirctly on th distortion causd by cartl pric-stting bhaviour th intuition givn abov suggsts that it will still outprform a pnalty basd on illgal gains. 37 Nxt, basd on th assumption that Δ is uniformly distributd on [,], w can dfin th avrag ovrcharg, consumr surplus and total wlfar undr a profits-basd rgim that is implmntd with toughnss, as: p, d (9) CS CS p, d 3 () W ar gratful to Jo Harrington for hlpful discussions on this issu. 37 A thorough analysis of this mor gnral stting rmains an intrsting rsarch qustion. 3

14 3 TW TW p, d 3 () 8 8 Th formula that appar on th HS of (9), () and () follow from straightforward intgration using (8) and (3). Finally, notic that th profits mad by a stabl cartl that has formd and has st th monopoly pric ar. Consquntly, undr a profits-basd pnalty rgim that is 4 implmntd with toughnss,, th avrag fin that is paid by any cartl that is dtctd and pnalizd is F () 4 4 and so is dirctly proportional to th toughnss of th profits-basd pnalty rgim. 4.. Fixd Pnalty A fixd pnalty is givn by F( p) F, whr F is th constant absolut pnalty. Undr this rgim th cartl stability condition () bcoms ( p)( p) F D ( p) For thos stabl cartls that do form, th cartl pric is: C M pf arg max Vp, F( ) p. (3) p M This implis that fixd fins induc th cartl to st th monopoly pric p, indpndntly of th toughnss of th pnalty rgim. 38 Th rason is clar: a fixd pnalty acts just lik an xpctd fixd cost, and, as w know, fixd costs hav no ffct on pricing dcisions. Howvr, a toughr rgim lowrs th maximum critical difficulty and so maks it lss likly that stabl cartls will form. Now, by () stability condition abov can b rwrittn as ( p )( p) F ( p )( p) (4) F ( )( p )( p) 38 Harrington (5) shows that if th probability of dtction and th pnalty rat ar constant thn th sam rsult holds for mor gnral dmands. 4

15 In ordr for thr to b any pric that a cartl can st and still satisfy th stability condition it must b th cas that ( ) 4F F ( ). MAX( p)( p) p (5) 4 This implis that th maximum critical difficulty of a fixd pnalty rgim is: 4F F. (6) Consquntly th ovrall pric that would mrg undr a fixd pnalty rgim is calling that p F 4F,, F. (7) 4F,, thn from (8) and (7) w hav th following: Proposition Th class of fixd pnalty rgims is idntical to th class of profit-basd pnalty rgims in th sns that: (i) for vry valu of arising in som profits-basd pnalty rgim, thr is a fixd pnalty F such that th associatd fixd pnalty rgim will induc 4 xactly th sam wlfar-rlvant outcoms: cartl pric, dtrrnt ffcts (maximum critical difficulty), ovrall pric tc. (ii) Convrsly for vry fixd pnalty F associatd with som fixd pnalty rgim, 4F thr is a pnalty rat on profits such that th associatd profits-basd pnalty rgim will induc xactly th sam wlfar-rlvant outcoms: cartl pric, dtrrnt ffcts (maximum critical difficulty), ovrall pric. In what follows w will thrfor ignor fixd pnalty rgims and confin our attntion solly to a comparison of profits-basd pnalty rgims, rvnu-basd rgims and ovrchargbasd rgims Pnalty on vnu A pnalty on rvnu is givn by F ( p) p( p), whr φ is th constant pnalty rat. 39 Th cartl stability condition () now bcoms 39 By constant w man that, within a givn industry, it dosn t vary dpnding on th amount of rvnu a cartl maks. As discussd blow it may howvr diffr across industris dpnding on th invrs lasticity of dmand, ε. 5

16 ( p)( p) p( p) D ( p). (8) If th cartl stability condition (8) is not binding, th unconstraind profit maximizing cartl pric is C M p arg max Vp, F( ) p. (9) p Notic that this unconstraind cartl pric is a strictly incrasing in and is abov simpl monopoly pric for all. This confirms that fins basd on rvnus not only do not rduc th cartl pric blow th monopoly pric, but actually push th pric abov th monopoly pric. This distortionary ffct of antitrust fins basd on rvnus was first idntifid in Bagri t al. (3). 4 Morovr, it follows from (9) that th toughr th pnalty rgim, th highr will b th cartl pric and th gratr th distortion. This was also prviously shown in Katsoulacos and Ulph (3). Th intuition is straightforward a pnalty on rvnu lowrs xpctd avrag and marginal rvnu but not marginal costs, so lading cartls to rduc output and driv up pric. Th toughr th pnalty th biggr th rduction in marginal rvnu and so output. Now, lt b th toughnss of th rvnu-basd pnalty rgim. Sinc cartl sts th pric abov th monopoly pric p M, by () th cartl stability condition bcoms ( p)( p) p( p) M () Not that th right hand sid of () is indpndnt of th pric st by th cartl. Hnc, hr stabl cartls oprat as if thy had costs. So for thr to b any quilibrium in which th cartl producs positiv output it must b th cas that this cost is blow th chok pric, which rquirs:. Providd this holds, stabl cartls that form st th pric that maximizs th LHS of (), i.. C C p, p. Not also that stabl cartls xist on th intrval, whr M M. () 4 Howvr this was don in th contxt of a static modl that did not prmit th xamination of stability conditions and how ths ar affctd by pnaltis. In particular thy did not stablish th xistnc of a minimum pric undr a rvnu-basd pnalty rgim. 6

17 Notic that and as. W can now dfin th ovrall pric that would mrg undr a rvnu-basd rgim for all, as th pric that would mrg taking account of both th pric st by thos cartls that do form and th (comptitiv) pric that would prvail wr no cartl to form. It follows from () that th ovrall pric undr th rvnu-basd pnaltis taks th form:, ( ) p (, ) (), ( ) It is illustratd in Figur. p p ( ) Figur : vrall Pric Undr vnu-basd Pnalty Basd on th assumption that Δ is uniformly distributd on [,],w can dfin th avrag ovrcharg, avrag consumr surplus and avrag total wlfar undr a rvnu-basd rgim that is implmntd with toughnss as: p, d (3), CS CS p d (4), TW TW p d. (5) Finally undr a rvnu-basd pnalty rgim that is implmntd with toughnss, w can dfin th avrag fin that is collctd from stabl cartls that form and ar subsquntly dtctd and pnalizd as F p, d (6) 7

18 4.4. Pnalty on vrcharg A pnalty on th cartl ovrcharg is givn by F ( p) ( p ), whr is th constant pnalty rat. Th cartl stability condition now bcoms ( p)( p) ( p) D ( p). (7) Providd that th cartl stability constraint is not binding, th unconstraind profit maximizing cartl pric is C M p arg max Vp, F( ) p. (8) p Notic that this unconstraind cartl pric is strictly dcrasing in and is blow monopoly pric for all. Hnc, according to dviation tchnology in (), th cartl stability condition can b rwrittn as ( p)( p) ( p) ( p )( p). (9) Now, providd p and so th cartl sts a pric abov th but-for pric, w can divid both sids of (9) by ( p ) to obtain: Q p p So, dpnding on th magnitud of th trm, in ordr to maintain stability, th cartl may b forcd to st a minimum lvl of output (quivalntly a maximum pric). Thrfor w hav th following rsult: Proposition : Whn pnaltis ar imposd on th ovrcharg thn cartl stability rquirs that th cartl should not st th pric abov th maximum pric MAX p. (3) Corollary: Th maximum pric is a strictly dcrasing function of: (i) th toughnss of th pnalty rgim, as rflctd in th paramtrs, ; (ii) th difficulty of holding cartl togthr,. Proof: Straightforward implication of partially diffrntiating th xprssion in (3) with rspct to,,. 8

19 In ordr for th cartl to hav any chanc of making profit th maximum pric must b gratr than or qual to th marginal cost (= ). Hnc, from (3), th maximum critical difficulty is:. If w now lt rgim, w hav dnot th masur of toughnss of th ovrcharg-basd pnalty, (3) and sinc, as notd abov, for th cartl stability condition to hold w must hav it follows that th toughnss paramtr must satisfy. Notic that th masur of toughnss capturs th probability of succssful dtction and proscution, β, and th pnalty rat, η. Using this notation th maximum pric as dfind in (3) bcoms MAX p, (3) Hnc, taking into account th maximum sustainabl cartl pric, w s that th pric st by thos stabl cartls that do xist is: C M p, MIN, p. (33) Notic that th critical valu of Δ at which th maximum pric constraint bits, i.. th trms in (33) ar qual, is. (34) As bfor for all, w dfin th ovrall pric that would mrg undr an ovrcharg-basd rgim to b th pric that would mrg taking account of both th pric st by thos cartls that do form and th pric that would prvail wr no cartl to form. From (33) and (34) this is givn by:, p(, ), (35), Notic that, for, th ovrall pric is a strictly dcrasing function of Δ, C with p as. It is illustratd in Figur 3. 9

20 Figur 3: vrall Pric Undr vrcharg-basd Pnalty Furthr, basd on th assumption that Δ is uniformly distributd on [,], w can dfin th avrag ovrcharg, avrag consumr surplus and avrag total wlfar undr an ovrcharg-basd pnalty rgim that is implmntd with toughnss, as: p, ln ln( ) d (36) CS CS p, 6 3 d 8 (37) TW TW, 3 8 ln ln( ) p d 8 (38) Finally undr an ovrcharg-basd pnalty rgim that is implmntd with toughnss, w can dfin th avrag fin that is collctd from stabl cartls that form and ar subsquntly dtctd and pnalizd as Notic that F d, lnln( ) F F 4 (39) (4) Th first rsult ariss bcaus, although th pnalty bas is positiv, th pnalty rat is zro, whil th scond rsult ariss bcaus, although th pnalty rat is positiv, th pnalty bas is 4 This follows from th formula in (39) by using l Hopital s rul.

21 zro sinc, as th dgr of toughnss tnds to, th cartl pric is drivn down to th comptitiv pric and thr is no ovrcharg. Th drivation of th formula that appar on th HS of (36)-(39) is givn in Appndix. 5. Comparisons within and across Pnalty gims In this sction w draw on th rsults of th prvious sction to undrtak an analysis of how th various wlfar indicators in which w ar intrstd pric, dtrrnc, avrag surplus tc. ar affctd by both within-rgim factors such as toughnss and th natur of th industry (as capturd by th invrs lasticity of dmand, ε) but also by across rgim changs that aris from switching to diffrnt pnalty bass. W bgin with a numbr of background rmarks. call that from Proposition a profits-basd pnalty rgim is quivalnt to a fixd pnalty rgim whn implmntd with th sam toughnss. So in this sction w focus on a comparison btwn: profit-basd rgims; rvnu-basd rgims and ovrchargbasd rgims. If ffctivly thr ar no pnaltis i.. if - thn, undr all thr rgims, cartls xist for all,, and just charg th monopoly pric M p 4 - so all rgims ar idntical. In othr words, if no pnaltis ar imposd it dosn t mattr on which bas you don t impos thm. Similarly if all rgims ar implmntd with maximum toughnss - i.. if - thn, undr all thr rgims th maximum critical difficulty is zro and th ovrall pric is just th comptitiv pric, 43 and so, onc again, all thr rgims ar idntical. In othr words, if no stabl cartls vr form it dosn t mattr on which bas you would hav pnalizd a non-xistnt cartl So in what follows w undrtak comparisons on th assumption that all rgims ar implmntd with a dgr of toughnss lying btwn and Comparison of prics st by stabl cartls that form W start with a comparison across th thr rgims of th prics st by thos cartls that do form. 45 This largly summariss rsults stablishd in Sction 4. W start with: 4 This can b asily sn by stting 43 This can b asily sn by stting 44 Formally ; ;. 45 Strictly spaking, for this comparison to b valid w hav to assum MIN,, into quations (7), (8), (), (), (3), and (35) into quations (7), (8), (), (), (3), and (35)

22 Proposition 3: Pnaltis basd on ithr profits (illgal gains) or on rvnu ar inffctiv in rducing th cartl pric blow th monopoly pric. In particular: (i) With pnaltis on illgal gains (profits) th cartl pric rmains qual to th monopoly C M pric, p p and, morovr, is indpndnt of th toughnss of th profits-basd rgim,. (ii) Pnaltis on rvnu ar distortionary and produc a cartl pric that M C a. lis btwn th monopoly pric and th chok pric i.. p p ; b. is strictly incrasing in th toughnss of th rvnu-basd pnalty rgim,,. c. tnds towards th chok pric as. Ths rsults can all b radily stablishd formally by an inspction of quations (5) and (9). Thir intuition is clar. Whatvr th dgr of toughnss, fins basd on illgal gains just produc a proportional rduction in xpctd profits. Consquntly maximizing nt profits is quivalnt to maximising gross profits and so lads th cartl to st th sam pric that would hav prvaild had thr bn no nforcmnt. 46 Howvr, pnaltis basd on rvnu lowr xpctd marginal rvnu but do not affct marginal cost, thus inducing cartls to cut output and driv up pric. Morovr, th toughr th pnalty rgim th gratr th rduction in xpctd marginal rvnu and consquntly output and so th highr th pric. W nxt hav: Proposition 4: Pnaltis on ovrchargs ar ffctiv in producing a cartl pric that C M (i) lis btwn th monopoly pric and th comptitiv i.. p p ; (ii) is a strictly dcrasing function of th toughnss of th ovrcharg rgim, ; (iii) tnds towards th comptitiv pric as. Ths rsults can radily b stablishd formally by an inspction of quation (33). Again, th intuition is clar. Whn pnaltis ar basd on th ovrcharg th quantity bas is fixd at th but-for lvl and so th pnalty is linar in pric. With pnaltis on illgal gains th quantity bas is th collusiv quantity. Thus as pric riss th quantity falls giving th cartl an incntiv to st a highr pric than undr th ovrcharg-basd pnalty. Th toughr th rgim, th gratr is th incntiv to kp pric down. Th conclusions of Propositions 3 and 4 can b combind in th following C M C C Proposition 5: For all,,, and, p p p p. 46 As notd abov this rsult dpnds on homognous goods assumption. If products ar diffrntiatd, th illgal gains pnalty schm will chang th collusiv valu as wll as tightn th ICC. This will rduc th ovrall cartl pric blow th monopoly pric.

23 This shows that, in rlation to th prics 47 st by stabl cartls that form, w can gt a clar ranking across th thr pnalty rgims that is indpndnt of th prcis dgr of toughnss of ach rgim. This rmarkably strong conclusion will b important in th analysis in Sction Dtrrnc Effcts: Comparativ Static Proprtis Th dsirability of altrnativ pnalty rgims dpnds not just on thir ffct on th pric chargd by any givn cartl, but also on thir ffcts on th numbr of cartls that ar formd and rmain stabl - thir ffct on dtrrnc. In this sub-sction w bgin to xamin ths lattr ffcts by undrtaking a comparativ static analysis of how th maximum critical difficulty of holding a cartl togthr is affctd by th pnalty rgim and markt conditions. W hav: Proposition 6: (i) For all thr pnalty rgims, th maximum critical difficulty of holding a cartl togthr is dcrasing in th toughnss of th pnalty rgim, and gos to zro as th dgr of toughnss gos to ; (ii) For rvnu-basd pnalty rgim th maximum critical difficulty is also incrasing in th invrs lasticity paramtr. Proof: Follows immdiatly from quations (7), () and (3) dfining th maximum critical difficulty undr all thr rgims Wlfar Effcts of Using Diffrnt Pnalty Bass Prcisly bcaus, as w hav just sn, th dtrrnt ffcts of any givn pnalty rgim is so snsitiv to th toughnss with which it is implmntd, it follows that th ovrall pric and hnc th various masurs of avrag consumr surplus tc. ar also going to b vry snsitiv to th toughnss with which any givn rgim is implmntd. So using, say, rvnu rathr than th ovrcharg as a pnalty bas may produc bttr wlfar outcoms if th rvnu-basd pnalty is implmntd vry toughly whil th ovrcharg-basd pnalty is implmntd vry wakly. Consquntly if w want to undrtak a maningful analysis of th consquncs for various indicators of wlfar pric, dtrrnc tc. - of using diffrnt bass on which to impos pnaltis, w hav to do so holding th dgr of toughnss constant in som sns. Thr ar a numbr of possibl intrprtations of what it might man for rgims to b qually tough. In what follows w considr two: dtrrnc quivalnc and pnalty rvnu quivalnc Dtrrnc Equivalnc n fairly natural intrprtation of what it might man for ach rgim to b qually tough is that th fraction of cartls dtrrd is xactly th sam across all thr rgims i.. th 47 And hnc th associatd lvls of consumr surplus and total wlfar 3

24 maximum critical difficulty is th sam across all thr rgims. Formally w rquir that th * * toughnss paramtrs,, ar such that, for som, * * *. (4) If w dnot th toughnss paramtrs for which this is tru by *, *, *, thn clarly * * * and. But, from Proposition 5 this immdiatly implis: and, : * * C * * C * * C *,,,,,, p p p p p p This givs us: * * * * (4), : p, p, p, (43) Proposition 7: If w impos dtrrnt quivalnc for any (i) * * * (ii) * * * CS CS CS (iii) * * * TW TW TW. * *, thn Proof: (i) follows by using (4) and (43) and intgrating ovr all [,]. (ii) and (iii) follow by noting from (3) that consumr surplus and total wlfar ar strictly dcrasing functions of pric. Figur 4 shows th supriority of th ovrcharg basd rgim for ovrall prics undr dtrrnc quivalnc. p * * p p p * Figur 4: vrall Prics Undr Dtrrnc Equivalnc 4