Optimal Extortion and Political Risk Insurance

Transcription

1 Optmal Extorton and Poltcal Rsk Insurance Frédérc Koessler, Arane Lambert-Moglansky To cte ths verson: Frédérc Koessler, Arane Lambert-Moglansky. Optmal Extorton and Poltcal Rsk Insurance. PSE Workng Papers n <halshs > HAL Id: halshs Submtted on 22 Feb 2012 HAL s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not. The documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers. L archve ouverte plurdscplnare HAL, est destnée au dépôt et à la dffuson de documents scentfques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés.

2 WORKING PAPER N Optmal Extorton and Poltcal Rsk Insurance Frederc Koessler Arane Lambert-Moglansky JEL Codes: Keywords: Auctons, Corrupton, Expropraton, Extorton, Governance, Harassment, Mechansm Desgn, Poltcal Constrants, Poltcal Rsk Insurance PARIS-JOURDAN SCIENCES ECONOMIQUES 48, BD JOURDAN E.N.S PARIS TÉL. : 33(0) FAX : 33 (0) CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE ECOLE DES HAUTES ETUDES EN SCIENCES SOCIALES ÉCOLE DES PONTS PARISTECH ECOLE NORMALE SUPÉRIEURE INSTITUT NATIONAL DE LA RECHERCHE AGRONOMIQUE

3 Optmal Extorton and Poltcal Rsk Insurance Frédérc Koessler Arane Lambert-Moglansky February 21, 2012 Abstract We study the problem faced by frms that nvest n a foregn country characterzed by weak governance. Our focus s on extorton relyng on the threat of expropraton and bureaucratc harassment. The bureaucrat s power s characterzed by lookng at a general extorton mechansm adapted from Myerson s (1981) optmal aucton theory. Ths characterzaton s used to study the determnants of the qualty of governance and whether and how poltcal rsk nsurance of foregn drect nvestments mprove upon t. We fnd that t does not always mprove upon all governance ndcators. It always decreases the bureaucrat s total revenue from corrupton, but t may also ncrease the rsk of expropraton and the extorton brbes pad by some frms. Keywords: Auctons; corrupton; expropraton; extorton; governance; harassment; mechansm desgn; poltcal constrants; poltcal rsk nsurance. 1 Introducton Wth the UK BrberyAct 2010, thessueof how to resstcorrupton has becomevtal for frms. The new regme has ncreased jursdctonal scope and has gven rse to ncreased crmnal exposure for corporate enttes, partcularly under the new strct lablty corporate offence. In ths context, some corporatons complan that passve brbery, whch covers the request for a brbe by publc offcals, s very lttle prosecuted. Bascally, frms clam to be prosecuted when they actually are the vctms of extorton. Whle ths ssue has recently become a major concern for Mult-Natonal Corporatons (MNC), t s also a major development ssue snce corrupton negatvely mpact nvestment and growth (Mauro 1995). Governments n rch countres have long been aware of the larger rsk ther natonal frms were confronted wth when nvestng n weak governance countres. In most developed countres publc agences have programs provdng guarantees to protect the natonal frms nvestments through poltcal rsk nsurance as part of ther development ad polcy. Due to ther respectve natonal objectves those programs often have strct elgblty requrements. The World Bank s Mutual Investment Guarantee Agency (MIGA) ams at complementng governmentsponsored and prvate guarantee programs,..., to ncrease the flow of captal and technology to We are grateful to Jean-Edouard Collard, Phlppe Jehel, Vaslk Skreta, Danel Vllar, and especally Laurent Lamy for very useful comments and suggestons. We gratefully acknowledge fnancal support from MIGA (World Bank Group). Pars School of Economcs CNRS, koessler@pse.ens.fr. Pars School of Economcs, alambert@pse.ens.fr 1

4 developng countres. 1 The World Bank, as most nternatonal ad agences, recognzes that weak governance s a major mpedment to growth. A queston of central nterest s therefore: beyond ts mmedate mpact through rsk mtgaton, do we expect poltcal rsk nsurance to affect governance n the country hostng the nvestment? Ths artcle deals wth stuatons where the host government (we refer to t as the bureaucrat) demands brbes n exchange for refranng from expropraton, bureaucratc harassment, or for makng a legtmate decson n the frm s favor. Although extorton can be encountered n publc tenders, the ssue s partcularly acute for frms that make drect nvestments n a foregn country. The physcal presence wth own assets on the terrtory of a weak governance country makes them vulnerable to (threats of) arbtrary treatment by the local authortes. A very powerful threat where the protecton of property rghts s weak s that of expropraton, the government can smply decde to steal the frm s assets. More sophstcated can be the threat of bureaucratc harassment whch can take the form of arbtrary changes n contract condtons or creatng obstacles for the frm s actvty by varous means (e.g., blockng access to electrcty or water servce, requestng numerous permts, delayng authorzatons,...). The rsks related to extorton and bureaucratc harassment are part of what s called poltcal rsks. 2 Econometrc research about the lnks between poltcal rsk and foregn drect nvestment have produced mxed results (see World Bank Group 2009, Annex 5). When t comes to surveys of corporate offcals, the message s unambguous however. Busnessmen report to be very concerned by poltcal rsk whch they vew as a major constrant when makng ther nvestment decsons. Among those rsks two stand out: breach of contractual oblgatons by the state and expropraton actons (regulatory takngs, creepng expropraton and outrght natonalzaton) (MIGA 2011). In MIGA s poltcal rsk survey 2009, nearly 45% of the respondents menton poltcal rsk as the greatest constrant on ther busness n emergng markets. When asked about the ways by whch the frms attempt to mtgate the rsks, nearly 70% respond engagement wth the government n Russa, 65% n Inda and 55% n Chna. 3 The expresson engagement wth the government s an euphemsm for all knds of nfluence and corrupt actvtes. The frms thus report beng forced nto corrupton to mtgate some rsks, n partcular to avod expropraton and bureaucratc harassment. Although t represents a serous concern for busness and a challenge for development ad, the ssue has receved very lttle attenton n the economc lterature. Most often the knowledge and understandng about extorton remans at the level of anecdotes and case studes. Ths artcle proposes a general framework to better understand the mechansm at play when several frms are nvestng n a country where the bureaucrats have the power to exproprate some of them and can credbly threaten to do so n order 1 Preamble to MIGA s conventon. Besde natonal and multlateral provders of poltcal rsk nsurances, there exsts a smaller but developng prvate market. 2 Poltcal rsk also ncludes rsks due to war and terrorsm. 3 The proposed alternatves were jont-venture, rsk analyss, and thrd party ntermedaton. 2

5 to extort brbes. We are nterested n understandng what are the determnants of the qualty of governance measured by the extent of expropraton, the magntude of the extorton brbes, and the revenue from corrupton. Wthn ths framework we nvestgate whether and how the provson of poltcal rsk nsurances can mprove upon governance n the host country. There exsts as for today a large lterature n economcs about corrupton. 4 One can dstngush two perspectves on corrupton both at the hgh and at the low level of government. The frst perspectve that has receved most attenton vews the state as beng captured by prvate nterests. At the hgher level t means that prvate nterests shape laws and regulaton n ther favor at the expense of socal economc effcency. At the lower level, t mples that regulaton s appled or abused to favor prvate nterests n conflct wth the objectves of the regulaton. Corrupton takes the form of mutually benefcal (llegal) agreements. A large lterature based on the prncpal-agent model addresses capture (see, e.g., Trole 1986, Laffont and Trole 1993). Another strand of the lterature addresses capture n aucton (see for nstance Compte et al. 2005, Burguet and Che 2005). The other much less well-developed perspectve vews the state as controlled by a rent-seeker,.e., a kleptocratc ruler who shapes laws and state nsttutons to maxmze hs own wealth by extractng rents from ctzens and frms. Because t goes aganst the nterests of most part of socety, a natural queston has been how can such a system survve (see, e.g., Acemoglu et al. 2004). Another set of questons that have been addressed relates to competton n rent-seekng between hgh and low levels government (Rose-Ackerman 1999, Shlefer 1993). Despte of the pervasveness of bureaucratc harassment of frms and ctzens to extract rents n most developng and transton economes wth very sgnfcant economc consequences (see, e.g., Klgaard, 1988), there are few theoretcal contrbutons n the mcroeconomcs of extorton. One reason may be that extorton has long been perceved as a crmnal endeavor performed by organzed group. Konrad and Skaperdas (1997) address the ssue of the credblty of the threat n racketng by crmnal gangs. A few artcles address extorton based on abuse of power. One example s Hndrcks, Keen and Muthoo (1999) who study the optmal tax collecton mechansm n a context where offcals can threaten to over-report ncome to extort brbes. Cho and Thum (2004) study extorton of one frm when the bureaucrat who ssued the lcense can come back and demand to renew the lcense n the second perod. They show that there exsts no pure strategy equlbrum of the repeated game, a feature that they nterpret as a ratonale for arbtrarness from the sde of the bureaucracy. Lambert-Moglansky, Majumdar and Radner (2007) also address extorton n lcensng but from a dfferent perspectve. They ntroduce the noton of track of bureaucrats, each of whom must gve hs approval n an ordered sequence. In each perod a dfferent frm needs a lcense to start busness and faces the track. The artcle reveals a bureaucratc hold-up problem. In a one-shot settng no lcense s ever granted. In a repeated settng the artcle characterzes two nterestng equlbra wth lcensng and dfferent levels of extorton brbes and dscusses the assocated socal economc cost 4 See for nstance Rose-Ackerman (2004, 2006, 2011) or Mshra (2005). 3

6 of extorton. The ssue of expropraton has been addressed by Thomas and Worrall (1994). They consder a blateral relatonshp between a MNC and a host country. The host country can exproprate the MNC by force of ts soveregnty. They study, n a multple perods settng, the propertes of a self-enforceable agreement where the frm gves transfer n exchange for beng left to operate. A man result s under-nvestment and delayed payment n an ntal perod. Ths pattern obtans because of the MNC s concern to ncrease the cost of expropraton to the host country,.e., the loss of future ncome. Our approach s very dfferent. We do not model nvestment decsons; nstead we consder several frms that are prvately nformed about the (exogenous) value of ther nvestments n the host country, and we characterze the extorton power of the bureaucrat when he can exproprate and harass them. Our focus s on the qualty of governance n the host country and on the mpact of poltcal rsk nsurance. The bureaucrat s power s lmted by hs mperfect of nformaton about frms profts n the host country. He may also have a poltcal constrant, meanng that he may be able to exproprate only a lmted number of frms. The bureaucrat ams at explotng hs power to extort maxmal value. Our approach reles on Myerson (1981) to characterze the optmal extorton mechansm. The dea s that the stuaton shows strong smlartes wth an aucton. The bureaucrat sells promses to leave the frm alone n exchange for a brbe. At frst sght, the settng dffers from Myerson (1981) n several respects. Frst, the bureaucrat can sell as many promses as he wants. Second, he may be forced to sell some of them by force of a poltcal constrant (he may not exproprate as many frms as he wants). Thrd, the bureaucrat s valuaton of the promses depends on the types of the frms that do not receve t,.e., that are exproprated. Last, the frm s outsde opton may be type dependent. Nevertheless, we show that Myerson s technc s applcable. Our model covers very general stuatons, n whch the value of expropraton and the cost of harassment for the bureaucrat could vary across frms, and frms may be heterogeneous wth respect to ther proft prospects and nsurance coverages. An optmal extorton mechansm s characterzed by thresholds for non-expropraton and by the magntude of the brbes the frms have to pay to avod expropraton dependng on other frms brbe proposals. Our frst result shows that the value or cost of expropraton for the bureaucrat s a determnant of the qualty of governance. The hgher the expropraton values the hgher the reserve prces below whch the bureaucrat wants to exproprate the frms and therefore the hgher the rsk of expropraton. When the poltcal constrant s not bndng all frms that are not exproprated pay a reserve prce (whch s common n case of ex-ante symmetrc frms, and ndvdualzed otherwse). Hence, the hgher the expropraton values, the hgher the extorton brbes requested to avod expropraton. Last, the revenue of the bureaucrat ncreases wth the expropraton values both drectly and ndrectly through hgher extorton brbes. Hence on all three accounts the values of expropraton negatvely affect the qualty of governance. The second determnant of the qualty of governance s the poltcal constrant. By defnton the poltcal constrant s a lmt on the number of frms that can be exproprated and therefore on the rsk 4

7 of expropraton. But we also show that t reduces the level of the extorton brbes requested to avod beng exproprated. When the poltcal constrant s bndng the bureaucrat s forced to sell promses not to exproprate at lower prces than the reserve prces. Fnally, snce ths s a constrant on the bureaucrat optmzaton problem, the tghter t s (the less frms he can exproprate) the lower the bureaucrat s revenue. In the second part of the artcle we ntroduce Poltcal Rsk Insurance (PRI). More precsely, we ntroduce a guarantee of compensaton for frms ncurrng losses due to abuse of power,.e., expropraton and bureaucratc harassment. In lne wth common practce, the compensaton s calculated as a share of the nvestment. 5 We frst show how to ntroduce nsurance nto our model wthout alterng the basc analytcal tools. The optmal mechansm s defned for sutably modfed varables, and t nvolves agan threshold values or reserve prces whch may be frm specfc, and a rule to determne the magntude of the brbe each frm has to pay to avod expropraton. The central queston s how does the level of nsurance affects the outcome of the optmal extorton mechansm and the resultng qualty of governance. We frst llustrate n a leadng symmetrc example wth postve value for expropraton that ntroducng nsurance has two effects. On the one hand t rases the overall rsk for expropraton; on the other hand, t reduces the extorton brbes. The ntuton s that because the bureaucrat cannot extract as much rents by means of extorton brbes, the mechansm s readjusted so he opts for more expropraton. Ths s consstent wth the standard result n the nsurance lterature that establshes that because of moral hazard, nsurance ncreases the rsk because the agent reduces ts effort to reduce the rsk. In our context, the frm s effort to reduce rsk corresponds to payng brbes. The overall effect of nsurance on the revenue from corrupton s unambguously negatve. We next consder general stuatons n whch frms may choose varous extent of nsurance coverage. We establsh that the mpact on the overall rsk of expropraton of an ncrease of a frm s nsurance coverage depends crtcally on the sgn of the frm specfc value to the bureaucrat of abusng power. When the value to the bureaucrat s postve, more nsurance ncreases the rsk. On the contrary, when the value to the bureaucrat s negatve, the rsk decreases wth an ncrease of nsurance coverage. Snce our model covers stuaton wth asymmetrc frms, t also allows to establsh some nterestng results regardng ndvdual and cross effects of nsurance. An ncrease of nsurance coverage for a gven frm leads to a reallocaton of the rsk between dfferent frms. When for a frm the value to the bureaucrat of abusng power s negatve, an ncrease of ths frm s nsurance coverage always decreases ts rsk of expropraton; and, f the poltcal constrant bnds, an ncrease of ths frm s nsurance coverage ncreases the rsk faced by the other frms. On the contrary, when the value to the bureaucrat of abusng power s postve, the mpact of an ncrease of nsurance coverage vares across the nterval of possble realzed profts (types) of the frm. It decreases for low types and t ncreases for types above some threshold value. Ths also mples 5 For smplcty we assume that compensaton s an exogenous varable and frms pay no premum. The ssues related to the market for PRI s left for future research. 5

8 that when the frm ncreases ts nsurance coverage, the rsk of expropraton of other frms may ncrease or decrease dependng on the realzed proft of the frm that ncreases ts own nsurance coverage. We next address the queston as to how nsurance affects the magntude of the extorton brbes. In our smple leadng example the effect of a symmetrc ncrease of nsurance coverage s unambguously vrtuous,.e., t decreases the extorton brbes. In other examples, we show however that ncreased nsurance may have the reverse effect: t may be optmal for the bureaucrat to compensate for the nsurance by askng for hgher brbes from very hgh valued frms. We formulate condtons that secure that the extorton brbes always decrease wth nsurance. Fnally, we establsh that an ncrease of nsurance coverage always reduces the bureaucrat s expected ncome from corrupton,.e., from expropraton and brbery optmally combned. The remanng of ths artcle s structured as follows. In the next secton we present the framework for extorton mechansms, we derve optmal mechansms, and we provde comparatve statc results for the qualty of governance wth respect to the expropraton value and the poltcal constrant. In secton 3 we ntroduce poltcal rsk nsurances and characterze the modfed optmal extorton mechansm. In Secton 4 we nvestgate the mpact of an ncrease of nsurance coverage on the qualty of governance. Secton 5 concludes. 2 Optmal Extorton Mechansm 2.1 Basc Model Consder a set N = {1,...,n} of (rsk neutral) frms exertng some actvty n a weak governance country, and one (rsk neutral) bureaucrat (government offcal) who can ask some brbes to the frms n exchange of not expropratng them. Let N 0 = N {0} be the set of all players (the n frms and the bureaucrat). We study the rsk of expropraton, the extorton power and the revenue from corrupton of the bureaucrat wth a mechansm desgn approach, assumng that the bureaucrat can fully commt to an extorton mechansm, so that frms barganng power s mnmzed. But we also assume that the bureaucrat s unsure about the values that frms attach to non-expropraton or the maxmum amount of brbes that each frm s able to pay. If the bureaucrat perfectly knew each frm s value, he would be able to fully extort them by threatenng any non-obedent frm of expropraton. Snce we assume that each frm prvately knows ts own value of non-expropraton (.e., of operatng), the bureaucrat has a role smlar to a desgner of an aucton mechansm wth prvate values. The optmal extorton mechansm that we characterze s ndeed obtaned by slghtly adaptng the desgn of an optmal aucton n Myerson (1981). The dfferences between our settng and hs aucton model are the followng: 6

9 1. The seller (bureaucrat) sells several (homogeneous) goods (a good n our model beng a guarantee of not beng exproprated) and each buyer (frm) needs one good at most (each frm demands at most one guarantee not to be exproprated); 2. The seller has some poltcal constrant on the mnmal number of goods he should sell (.e., the bureaucrat may not be able to exproprate all frms); 3. The seller s valuaton for a good depends on the types of the buyers who do not receve the good (.e., the bureaucrat s payoff from expropraton may depend on the values of the frms that are exproprated); 4. Wth a poltcal rsk nsurance, a buyer s outsde opton may be type dependent (.e., a frm s payoff when t s exproprated may depend on ts type when compensaton accordng to the nsurance contract s type dependent). Despte those dfferences we wll show that the formal analyss s very close to the one of an optmal aucton mechansm. Wthout extorton we assume that frm s proft from operatng n the country s exogenously gven by t T [a,b ], where 0 a < b < +. We assume that only frm knows the true realzaton of ts proft or type t. For smplcty we assume as n Myerson (1981) that frms types are ndependently dstrbuted. Let f : T R + be the contnuous densty functon for s type, and F the correspondng cumulatve dstrbuton functon. When frm s exproprated ts payoff s ndependent of ts type and s normalzed to zero (ths assumpton s relaxed latter, n Secton 3, when we ntroduce poltcal rsk nsurance. 6 ) In contrast, the expropraton value of frm for the bureaucrat may depend on frm s type and s denoted by e (t ) R (we mght allow for negatve value of expropraton, for example f the bureaucrat ncurs a reputaton or harassment cost). A (drect revelaton) mechansm s gven by outcome functons p : T [0,1] n and x : T R n +. Gven a profle of announced types t = (t 1,...t n ), p (t) s the probablty of not expropratng frm and x (t) s the expected amount of money, or brbes, pad by frm. As n Myerson (1981), the optmal mechansm wll turn out to be determnstc. Remark 1 Infullgeneralty weshouldspecfypandxwhensomefrmsdecdenottopartcpate to the mechansm. We do not model ths stuaton explctly because t s not an ssue n our model. It s easy to check ex-post that the partcpaton constrant wll be satsfed n equlbrum by consderng a mechansm that exproprate wth probablty one any frm that unlaterally devates by non partcpatng, so that the partcpaton constrant s mpled by the ndvdual ratonalty condton (3) below. It s also easy to see that partcpaton by all frms can be mplemented n any 6 The compensaton for the case the frm s exproprated s defned as a share of proft whch makes the model wth nsurance equvalent to a model wth partal expropraton. 7

10 Nash equlbrum of the extorton mechansm by consderng mechansms that exproprate frms n an asymmetrcway ncase of nopartcpaton: for example, f morethanonefrmdonot partcpate then the frm wth the smallest ndex s exproprated wth probablty one. Gven a mechansm (p,x) the (nterm) expected utlty of frm when ts type s t T s gven by U (p,x;t ) = (t p (t) x (t))f (t )dt, (1) T and the (ex ante) expected utlty of the bureaucrat s ( ) U 0 (p,x) = (1 p (t))e (t )+x (t) f(t)dt. (2) T N A mechansm s feasble f t satsfes the ndvdual ratonalty (IR) constrant U (p,x;t ) 0, for all N, (3) and the ncentve-compatblty (IC) constrant U (p,x;t ) (t p (s,t ) x (s,t ))f (t )dt, for all N, s,t T. T (4) Condton (3) means that frms cannot be forced to partcpate to the mechansm: they should get an expected payoff whch s at least the expected payoff they obtan when they are exproprated wth probablty one (n whch case ther expected payoff would be zero). Condton (4) means that frms have no ncentve to msreport ther types to the bureaucrat when they expect that all other frms truthfully report ther types. In addton to these standard constrants, there may also be poltcal constrants for the desgner: the bureaucrat s requred to exproprate at most K {1,...,n} frms, so that p (t) n K, for all t T. (5) N The partcular case n whch there s no poltcal constrant s smply K = n. 2.2 Feasble and Optmal Mechansms The objectve of the bureaucrat s to choose the mechansm (p, x) that maxmzes hs expected payoff U 0 (p,x) under the above IR constrant (3), IC constrant (4) and poltcal constrant (5). 7 Let Q (p,t ) = p (t)f (t )dt, T 7 As n Myerson (1981) the revelaton prncple apples: there s no loss of generalty for optmalty by focusng on drect and truthful mechansms. 8

11 be the nterm probablty that frm of type t s not exproprated gven the mechansm (p,x). Then, the IC constrant can be equvalently rewrtten as U (p,x;t ) U (p,x;s )+(t s )Q (p,s ). As n Myerson (1981) t can be shown that the followng lemma apples. Lemma 1 (p,x) satsfes the IR constrant (3) and the IC constrant (4) ff for all N and s,t T the followng condtons hold: Q (p,s ) Q (p,t ) f s t ; (6) t U (p,x;t ) = U (p,x;a )+ Q (p,s )ds ; (7) a U (p,x;a ) 0. (8) Proof. The proof exactly follows the proof of Lemma 2 n Myerson (1981). Hence, the problem of the bureaucrat s to maxmze U 0 (p,x) gven by (2) under the constrants (5), (6), (7) and (8). Usng the prevous lemma, the bureaucrat s objectve can be rewrtten: expresson (2) of the bureaucrat s expected utlty can be rewrtten as U 0 (p,x) = e (t )f(t)dt+ (t)(t e (t T T Np ))f(t)dt N N Usng (7) and then ntegratng by parts we get: (t p (t) x (t))f(t)dt = U (p,x;t )f(t )dt T T t = U (p,x;a )f(t )dt + f(t ) Q (p,s )ds dt T T a = U (p,x;a )+ (1 F (t ))Q (p,t )dt T 1 F (t ) = U (p,x;a )+ p (t)f(t)dt. f (t ) Hence: U 0 (p,x) = T N T ( t e (t ) 1 F ) (t ) p (t)f(t)dt+ f (t ) T T (t p (t) x (t))f(t)dt. e (t )f(t)dt NU (p,x;a ). N we get Choosng t x (t) = p (t)t p (s,t )ds, (9) a t U (p,x;t ) Q (p,s )ds = U (p,x;a ) = 0, a 9

12 so (7) and (8) are satsfed. Snce T N e (t )f(t)dt s a constant, ndependent of the mechansm (p,x), the optmal mechansm s gven by (9) and p : T [0,1] n that maxmzes ( t e (t ) 1 F ) (t ) p (t)f(t)dt, T f (t ) N}{{} c (t ) subject to the constrants (5) and (6). For the rest of the artcle we make the followng standard regularty assumpton, whch guarantees that the state by state maxmzaton of the program above mples that p (t,t ) s ncreasng n t, and thus that the monotoncty condton (6) s satsfed. Assumpton 1 (Regularty) For every N the vrtual type of frm, s strctly ncreasng n t. c (t ) = t e (t ) 1 F (t ), (10) f (t ) From the prevous analyzes we mmedately get the followng characterzaton of the optmal mechansm: Proposton 1 (Optmal Extorton wthout Insurance) Under regularty, the optmal extorton mechansm (p,x) s such that p : T [0,1] n maxmzes c (t )p (t) subject to n K Np (t) n for all t T, N where the vrtual type c (t ) of frm s gven by (10). That s, p (t) = 0 for the frms wth the (up to) K lowest vrtual types below 0, and p (t) = 1 for the others. The payment of frm to the bureaucrat s gven by: t x (t) = p (t)t p (s,t )ds, for all N. a For any fnte set {x 1,x 2,...} of real numbers, denote by mn K x the Kth smallest element of ths set. That s, f x 1 < x 2 < < x K <, then mn K x = x K. Let y (t ) = mn{s T : c (s ) 0 or c (s ) mn K c j (t j )}, (11) j be the smallest type of frm such that frm s not exproprated when other frms types are gven by t. The optmal mechansm can therefore be reformulated as follows: { { 1 f t > y (t ), y (t ) f t > y (t ), p (t) = and x (t) = 0 f t < y (t ), 0 f t < y (t ). For each frm the optmal mechansm nvolves a (possbly frm specfc) threshold value c 1 (0) for non-expropraton whch s determned so that the vrtual type of frm s equal to zero. The (12) 10

13 threshold value plays a role smlar to the reserve prce n optmal aucton mechansms, and s chosen by the bureaucrat n order to maxmze hs expected revenue. If the bureaucrat has no poltcal constrant (K = n) he never sells a promse not to exproprate a frm for a brbe below that threshold value. He sells to each frm a promse not to exproprate at prce c 1 (0), and f frm does not pay then t s exproprated. When the bureaucrat cannot exproprate as many frms as he wshes,.e., when he s forced to sell a promse of non-expropraton to at least n K frms, he cannot obtan the threshold values from all non-exproprated frms. Instead, he must decrease the prce for a non-exproprated frm to y (t ) gven by Equaton (11),.e., the hghest prce acceptable to the lowest non-exproprated frm s type. As a result the role of the threshold value n the extorton mechansm s somehow more lmted the more tght the poltcal constrant (the smaller K) and the larger the total number of frms. When frms are ex-ante symmetrc,.e., e ( ) = e j ( ) and f ( ) = f j ( ) for every,j N, we denote by t 0 = c 1 (0) the optmal and common threshold value for non-expropraton. In that case, the optmal mechansm s much smpler. Any frm whose type t s above t 0 s never exproprated (p (t) = 1). When the poltcal constrant K s not bndng (.e., { N : t < t 0 } < K), every frm whose type t s below t 0 s exproprated (p (t) = 0) and pays nothng, and the others are not exproprated and pay the threshold value t 0. When the poltcal constrant K s bndng (.e., { N : t < t 0 } K), then only the K frms whose types are the K lowest types below t 0 are exproprated and pay nothng, and the others are not exproprated and pay the same prce: mn K j N t j, the Kth lowest type n {t 1,...,t n }. Notce that contrary to standard auctons, when the poltcal constrant s bndng the effectve brbe (mn K j N t j) may be strctly lower than the bureaucrat s reserve prce (t 0 ). When frms are ex-ante symmetrc, the optmal mechansm can also be mplemented wth the followng smple aucton-brbng game: each frm N smultaneously and voluntarly submts a bd b (t ) 0 as a functon of ts type t T ; then, up to K frms wth the lowest bd below t 0 are exproprated, and the others are not exproprated and pay mn{t 0,mn K j N b j(t j )}. Observe that, lke n second-prce auctons, t s a weakly domnant strategy for each frm to bd ts value: b (t ) = t for every t T. Lke n aucton mechansms, f frms are not ex-ante symmetrc, then the optmal mechansm should also take nto account each frm s dentty. The optmal extorton mechansm s llustrated n the followng symmetrc example wth unformly dstrbuted types and wth lnear and symmetrc expropraton values for the bureaucrat. The effect of asymmetres between frms s analyzed after the example. Example 1 As an example, assume that for every N the proft t of frm s unformly dstrbuted on [0, 100], and the expropraton value for the bureaucrat s lnear n the proft of frm : e (t ) = γt, wth γ < 2. Then, the vrtual type of frm s gven by c (t ) = t γt 1 t /100 1/100 = (2 γ)t 100, 11

14 whch s ncreasng n t for γ < 2, so regularty s satsfed. The (common) threshold value for non-expropraton s c 1 (0) = t 0 = γ. Notce that when γ > 1 the vrtual type c (t ) of every frm s strctly negatve for every t, and therefore the K frms wth the smallest types are always exproprated. On the contrary, when γ < 1 the poltcal constrant may not always be bndng: t s only bndng when the types of the K frms wth the K lowest types are below t 0. For nstance, f K = 1 (.e., f the bureaucrat can exproprate at most one frm), then for every frm the optmal probablty of expropraton for the bureaucrat characterzed n Proposton 1 s gven by: 8 p (t,t ) = { 1 f t > mn j N0 t j, 0 f t = mn j N0 t j. In other words, frm s exproprated (.e., p (t) = 0) f and only f ts type s the smallest among all frms types (.e., t = mn j N t j ) and s below the threshold value for non-expropraton (.e., t < t 0 = γ ), n whch case frm pays no brbe (x (t) = 0). Otherwse, f t > mn j N0 t j, then frm s not exproprated (.e., p (t) = 1) but pays the followng extorton brbe: t t x (t) = p (t)t 0 p (s,t )ds = t 1 ds = mn t j. mn j N0 t j j N 0 So f t > t 0 for every, then no frm s exproprated and they all pay t 0 = γ. Otherwse, f t < t 0 for some frm, then frm argmn j N t j s exproprated and all the other frms pay mn j N t j. More generally, when the bureaucrat can exproprate at most K frms, up to K frms wth the K lowest types below t 0 = γ are exproprated, and the others pay mn{t 0,mn K j N t j}. Not surprsngly, f the poltcal constrant s weaker (.e., K s larger), then the rsk of expropraton and the brbes extracted from non-exproprated frms ncrease. The revenue of the bureaucrat also ncreases wth K snce K only enters as a constrant n hs optmzaton program. Fnally, observe that the threshold value t 0 does not depend on the number of frms (n) and on the poltcal constrant (K), but t s ncreasng n γ: the hgher the expropraton value for the bureaucrat, the hgher the threshold value for non-expropraton. When frms are not ex-ante symmetrc as n the example above, the optmal mechansm dscrmnates among dfferent frms dependng on ther proft dstrbutons and on the bureaucrat s values form expropraton. The characterzaton of the optmal mechansm n Proposton 1 shows that frms heterogenety smply ntroduces heterogenety n frms vrtual type functons. Ths heterogenety of vrtual type functons affects frms relatve probabltes of expropraton and the extorton brbes they have to pay. To see ths, consder two dfferent frms and j wth the same proft (type) y and notce that c j (y) c (y) e j (y)+ 1 F j(y) f j (y) e (y)+ 1 F (y). f (y) 8 Recall that N 0 = N {0} s the set of all players (the n frms and the bureaucrat). 12

15 Hence, frm j, wth a smaller value of expropraton e j ( ) for the bureaucrat or a hgher hazard rate f j (y) 1 F j (y), wll be exproprated less often and wll pay less brbes than frm wth the same proft y as frm. We wll see n Secton 4 that the effect of nsurance on the vrtual types, and therefore on the probablty of expropraton and on extorton brbes, s more complex and may be ambguous. 2.3 The Qualty of Governance In ths subsecton we study beyond the specfc example above the mpact of the poltcal constrant and of the expropraton value for the bureaucrat on the qualty of governance. In the present context t seems approprate to capture the qualty of governance n the host country by three ndcators. The frst s the overall rsk of expropraton mpled by the threshold values for non-expropraton c 1 (0). Recall from Proposton 1 that frm s never exproprated f c (t ) > 0,.e., t > c 1 (0); otherwse, f t < c 1 (0), then frm s exproprated whenever the bureaucrat s poltcal constrant s not bndng. Hence, the threshold values determne the ex-ante probablty of expropraton and thus contrbute to determne the extent of the rsk of expropraton. The hgher the thresholds the more lkely the frms types are lower than the thresholds and therefore the more lkely they could be exproprated. The second ndcator of qualty of governance s the magntude of the brbes that frms pay to avod beng exproprated. The larger the extorton brbes,.e., the morethe frmsmust pay to avod bengthevctms of abuseof power by the bureaucrat, the lower the qualty of governance. Fnally, a thrd ndcator s the expected revenue of the bureaucrat, whch comes both from extorton brbes and expropraton. The larger the revenues from abusng or threatenng to abuse power, the lower the qualty of governance. Intutvely, one expects lke n Example 1 that under the optmal extorton mechansm the values of expropraton (e (t ), = 1,...,n) and the slackness of poltcal constrant (the maxmal number of feasble expropratons, K) affect negatvely the qualty of governance. In the two propostons below we show that ths s generally true The Expropraton Value A determnant of the vrtual type of frm, and hence of the threshold value c 1 (0), s the functon e ( ) : T R that determnes the value to the bureaucrat for expropratng frm as a functon of frm s type t T. In Example 1 we assumed e (t ) = γt and the threshold value t 0 = γ was common to all frms and ncreasng n γ. When t 0 < 100,.e., γ < 1, ths mples that the ex-ante probablty that any frm s exproprated s ncreasng wth the value of the frm to the bureaucrat. It also results n hgher brbes (equal to t 0 ) pad by frms that are not exproprated when the poltcal constrant s not bndng (brbes are constant, equal the hghest frm s type among the set of exproprated frms, when the poltcal constrant s bndng). The next proposton shows that ths s a general comparatve statcs property of the optmal mechansm, for arbtrary dstrbutons of types and for values of expropraton that are not necessarly symmetrc and lnear n frms 13

16 types. Proposton 2 For each frm, the probablty of expropraton of ths frm, the extorton brbe pad by ths frm when t s not exproprated, and the revenue of the bureaucrat are ncreasng wth the expropraton value of ths frm for the bureaucrat. Proof. Consder an expropraton value ẽ ( ) of some frm N such that ẽ (t ) > e (t ) for every t T and such that regularty s stll satsfed. Then, the vrtual type of frm s gven by c (t ) < c (t ) for every t T, whch mples that the threshold value for non-expropraton s c 1 (0) > c 1 (0). Hence, the probablty of expropraton of frm and the extorton brbe pad by frm when t s not exproprated are hgher wth ẽ ( ) than wth e ( ). To show that the revenue of the bureaucrat s also hgher wth ẽ ( ) than wth e ( ) t suffces to notce that the optmal extorton mechansm wth e ( ) s also feasble wth ẽ ( ) because the expropraton values do not enter nto frms utltes, and yelds the same brbe revenue but hgher expropraton values. Therefore, the optmal extorton mechansm wth ẽ ( ) necessarly yelds a hgher total expected revenue for the bureaucrat. Notce that the overall rsk of expropraton and the revenue of the bureaucrat are always smaller for smaller values of expropraton of any frm j. But decreasng the value of expropraton of some frm j may be detrmental for another frm when the poltcal constrant s bndng and s type s below ts threshold for non-expropraton (t < c 1 (0)). Indeed smaller values of e j ( ) mply hgher values of j s vrtual types c j ( ), and hence the vrtual types of any other frm becomes smaller relatve to j s vrtual type. The resultng brbe and rsk of expropraton of frm could therefore ncrease as y (t ) of Equaton (11) ncreases wth c j ( ) The Poltcal Constrant The poltcal constrant lmts the bureaucrat s rght to exproprate. When he can exproprate as many frms as he wants, the optmal mechansm calls for expropratng all the frms wth c (t ) < 0 andeach remanngfrmthat snot exproprated pays afxedbrbec 1 (0), whch s ndependent of other frms types. When the bureaucrat can exproprate at most K frms, the K frms wth the K lowest vrtual types below 0 are exproprated and the others pay the value of ther smallest possble types allowng them not to be exproprated gven others types. Hence, the weaker the poltcal constrant (the larger K) the larger the probablty of expropraton and the brbes extracted from each frm. Snce K only appears as a constrant n the bureaucrat s optmzaton program (through Equaton (5)), hs revenue s also ncreasng n K. Thus, we have: Proposton 3 The rsk of expropraton, the extorton brbes pad by the frms when they are not exproprated, and the revenue of the bureaucrat are ncreasng wth the number K of frms the bureaucrat has the power to exproprate. 14

17 Proof. Drectly from Proposton 1 and the observatons above. We conclude that the qualty of governance s unambguously decreasng wth the value that the bureaucrat can obtan from expropraton and wth the number of frms that he has the rght to exproprate. The next secton studes how the optmal extorton mechansm and the qualty of governance are modfed by the ntroducton of poltcal rsk nsurances for the frms. 3 Optmal Extorton Mechansm wth Insurance In ths secton we assume that poltcal rsk nsurance s avalable for governmental expropraton of frms nventores, equpment, or other assets located n the country. More precsely, we assume that when frm of type t s exproprated t s compensated by an amount A (t ) 0 from an nsurance. We make the followng monotoncty assumpton so that v (t ) t A (t ) can be nverted and the optmal mechansm wth nsurance can be characterzed wth the same method as n the prevous secton wth a change of varable. 9 Assumpton 2 For every, v (t ) t A (t ) s strctly ncreasng n t. A typcal example n whch ths assumpton s satsfed s when the amount of assets covered by the nsurance n case of expropraton s lnear n the frm s loss of ncome: A (t ) = λ t wth λ [0,1), so that v (t ) = (1 λ )t. Wth nsurance, the nterm expected utlty of frm s gven by T (t p (t)+a (t )(1 p (t)) x (t))f (t )dt. (13) The ndvdual ratonalty constrant requres that ths expected utlty s hgher than A (t ) or, equvalently, T (v (t )p (t) x (t))f (t )dt 0. Notce that, normalzng frms utltes to 10 U (p,x;t ) = (v (t )p (t) x (t))f (t )dt, (14) T we get the same IR and IC constrants as wthout nsurance, except that the value for nonexpropraton for frm s v (t ) = t A (t ) nstead of t. Consder ndeed the followng change 9 Notce that ths trck s not possble n other extensons of Myerson (1981) n whch the type-dependent outsde optons only enter the agents partcpaton constrants (see, e.g., Fgueroa and Skreta, 2009). In our model, the outsde opton A (t ) s perceved by the frm both when t does not partcpate to the mechansm at all and when t partcpates but s exproprated. 10 A (t ) s a constant, ndependent of the mechansm, so t can be subtracted from frm s nterm expected utlty gven by (13) wthout modfyng ncentve-compatblty constrants. 15

18 of varable: t = v (t ). Let F ( t ) = F (v 1 ( t )) and f ( t ) = f (v 1 dstrbuton and densty of t over T = [ã, b ] [v (a ),v (b )]. ( t )) v (v 1 ( t )) be the correspondng Wth ths changeof varables, andfor amechansm ( p, x), where p : T [0,1] n and x : T R +, players expected utltes (1) and (2) can be rewrtten as: Ũ ( p, x; t ) = ( t p ( t) x ( t) ) f ( t )d t, (15) T and Ũ 0 ( p, x) = T ( ) (1 p ( t))ẽ ( t )+ x ( t) f( t)d t, (16) N where ẽ ( t ) = e (v 1 ( t )). The optmal mechansm can therefore be characterzed exactly as n the prevous subsecton, whch yelds the followng vrtual type for each frm : c (t ) = t ẽ ( t ) 1 F ( t ) f ( t ) = v (t ) e (t ) v (t ) 1 F (t ). f (t ) Agan, we make the followng regularty assumpton: Assumpton 3 (Regularty wth nsurance) For every N the vrtual type of frm wth nsurance, s strctly ncreasng n t. c (t ) = v (t ) e (t ) v (t ) 1 F (t ), (17) f (t ) The optmal mechansm (p,x) defned on the orgnal types, p : T [0,1] n and x : T R +, s such that p (t) = p (v 1 (t 1 ),...,v n (t n )) and x (t) = x (v 1 (t 1 ),...,v n (t n )). We therefore get the followng characterzaton of the optmal extorton mechansm wth nsurance: Proposton 4 (Optmal Extorton wth Insurance) Under regularty the optmal extorton mechansm (p,x) wth nsurance s such that p : T [0,1] n maxmzes c (t )p (t) subject to n K Np (t) n for all t T, N where the vrtual type c (t ) of frm s gven by (17). That s, p (t) = 0 for the frms wth the (up to) K lowest vrtual types below 0, and p (t) = 1 for the others. The payment of frm to the bureaucrat s gven by: x (t) = p (t)v (t ) v (t ) v (a ) p (s,t )ds. (18) The optmal mechansm wth nsurance can be reformulated n terms of the ntal types: f y (t ) = mn{s T : c (s ) 0 or c (s ) mn K c j (t j )}, j 16

19 s the smallest type of frm that would result n frm not beng exproprated when other frms types are gven by t, the optmal mechansm can be rewrtten as: { { 1 f t > y (t ), v (y (t )) f t > y (t ), p (t) = and x (t) = 0 f t < y (t ). 0 f t < y (t ). Example 1 contnued Consder agan Example 1 and let v (t ) = t A (t ) = (1 λ)t, where λ [0,1), so all frms have the same nsurance coverage. Then, the vrtual type of frm as a functon of ts type t s gven by: c (t ) = (1 λ)t γt (1 λ)(100 t ) = [2(1 λ) γ]t 100(1 λ). Assume that λ and γ are not too hgh: 2(1 λ) γ > 0,.e. λ < 1 γ/2, so that regularty s satsfed. Then, the (common) threshold for non-expropraton s gven by: t 0 = c 1 (0) = 100(1 λ) 2(1 λ) γ = γ, 1 λ and up to K frms wth the K lowest types below t 0 are exproprated. The non-exproprated frms pay mn{(1 λ)t 0,mn K j N (1 λ)t j} ; that s, they pay (1 λ)t 0 when the poltcal constrant s not bndng (.e., strctly less than K frms are below t 0 ), and they pay mn K j N (1 λ)t j when the poltcal constrant s bndng. The threshold t 0, and therefore the rsk of expropraton, s ncreasng wth λ when γ > 0, but decreasng wth λ when γ < 0. The extorton brbes pad by the frms when the poltcal constrant s bndng, mn K j N (1 λ)t j s obvously always strctly decreasng wth λ. When the poltcal constrant s not bndng, the payment (1 λ)t 0 apples and t s also strctly decreasng n λ when λ (1 λ)t 0 = 100(1 λ) λ 2 γ < 0, that s λ < 1 γ. 1 λ When the nequalty above s not satsfed (.e., when λ 1 γ) we have t 0 100, so the rsk of expropraton s constant and the poltcal constrant s always bndng. Hence, the payment mn K j N (1 λ)t j always apples for the n K non-exproprated frms, and therefore the brbes and the expected revenue of the bureaucrat decrease wth λ. We conclude that, n ths example, extorton brbes pad by the frms that are not exproprated always decrease wth the extent λ of the nsurance coverage. These comparatve statcs results wth respect to the coverage of the poltcal rsk nsurances are generalzed n the next secton. (19) 4 The Qualty of Governance wth Insurance In ths secton we study the qualty of governance as a functon of frms nsurance coverage. More precsely, we study the mpact of the extent of nsurance on the three ndcators of the qualty of governance defned n Subsecton 2.3: () the rsk of expropraton, () the magntude of the extorton brbes, and () the bureaucrat s expected revenue from corrupton. 17

20 4.1 Rsk of Expropraton In our leadng symmetrc example, the (common) threshold value for non-expropraton, t 0, s ncreasng n λ when the value of expropraton for the bureaucrat s postve (γ > 0). Snce a larger threshold mples a hgher probablty that any frm s type s below ths threshold, a larger nsurance coverage mples an overall hgher probablty of expropraton. Recall that the optmal mechansm calls for the K-lowest types below that threshold to be exproprated. Therefore, whenever the poltcal constrant bnds at a gven level of nsurance coverage, t bnds at any hgher level. In such a case an ncrease n nsurance coverage has no mpact on the rsk of expropraton. We show below that the result s true beyond the specfc example whenever the value for expropraton s postve for the bureaucrat (.e., when e (t ) s always postve), at least wth lnear nsurance coverage. On the contrary, the threshold for expropraton decreases wth nsurance coverage when the value of expropraton for the bureaucrat s negatve, and nsurance has no mpact on the rsk of expropraton when expropraton has no value for the bureaucrat. At the end of the subsecton we study precsely the mpact of asymmetrc nsurances coverage on frms relatve rsk of expropraton when the poltcal constrant of the bureaucrat s bndng. When nsurance s lnear n frms types (.e., A (t ) = λ t, λ [0,1)), the ndvdual threshold value for non-expropraton of frm, c 1 (0), s the soluton t 0 of c (t 0 ) = 0,.e., t 0 e (t 0 ) 1 λ 1 F (t 0 ) f (t 0 ) = 0. The LHS of ths equaton s decreasng n λ when e (t 0 ) > 0, so when the regularty condton s satsfed (.e., c ( ) s ncreasng) a larger nsurance coverage λ requres a hgher threshold value for the LHS of the equaton to be equal to 0. Smlarly, when e (t 0 ) < 0, the threshold value decreases n λ, and when e (t 0 ) = 0 the threshold value does not depend on the percentage of nsurance coverage. Ths gves us the followng proposton. 11 Proposton 5 Assume that nsurance s lnear: A (t ) = λ t, λ [0,1). The mpact of the percentage λ of nsurance coverage of frm on the threshold value c 1 (0) of non-expropraton of frm depends on the sgn of the expropraton value functon e ( ) for the bureaucrat: The threshold for non-expropraton ncreases (decreases) wth λ when e ( ) > 0 (e ( ) < 0), and does not depend on λ when e ( ) = 0. Thus, we learn that the mpact of nsurance on the rsk of expropraton crtcally depends on the sgn of the expropraton value for the bureaucrat. The nterpretaton s that f the bureaucrat s extorton powerbuldsonanablty tosezetheassetsandbeneftfromthem, thentherskncreases wth nsurance. In contrast f hs power bulds on a capacty to create costly nusance for the frms, the rsk s unchanged f harassment s costless and t decreases f harassment s costly. 11 The result can also be obtaned as a drect applcaton of the mplct functon theorem. 18

21 The threshold values have an mpact on the rsk of expropraton snce any type above ths threshold s certan not to be exproprated. When the poltcal constrant s not bndng the rsk of expropraton of a frm j s not affected by the nsurance coverage of frm because the vrtual type, and therefore the threshold for non-expropraton of frm j, c 1 j (0), does not depend on λ. But when the poltcal constrant s bndng there exsts nterestng cross effects among dfferent frms. The rsk of expropraton of a gven frm whose type s below ts threshold depends on the relatve poston of ts vrtual type wth respect to other frms vrtual types below zero. To see the effect of nsurance on the vrtual type c (t ) of frm, and hence on the relatve probabltes of expropraton, assume agan lnear nsurances. Then, we have: Notce that by regularty, t 1 F (t ) f (t ) s strctly ncreasng n t. c (t ) < 0 t > 1 F (t ). (20) λ f (t ) Denote by t the soluton of t 1 F (t ) f (t ) = 0. Snce c 1 (0) solves (0) 1 F (c 1 (0)) f (c 1 (0)) c 1 e (c 1 (0)) 1 λ = 0, we have c 1 (0) > t when e ( ) > 0, n whch case c 1 (0) ncreases wth λ n accordance wth Proposton 5. We also have c 1 (0) < t when e ( ) < 0, n whch case c 1 (0) decreases wth λ, so the rsk of expropraton never ncreases n that case (t strctly decrease when t < c 1 (0) and s null and constant when t > c 1 (0). Ths s llustrated n Fgure 1 wth two levels of nsurance coverage λ and λ, wth λ > λ and the correspondng vrtual type functons c ( ) and c ( ) and threshold values t 0 = c 1 (0) and t 0 = c 1 (0). In the fgure on the LHS we have e ( ) > 0, and n the fgure on the RHS we have e ( ) < 0. Notce that even when e ( ) = 0, the rsk of expropraton of frm decreases wth λ for t < t = c 1 (0) when the poltcal constrant s bndng (because c (t ) ncreases for t < t ), although the threshold value c 1 (0) does not change wth λ (so that the rsk of expropraton s null and ndependent of λ for t > t = c 1 (0)). AsanexampleconsderthecaseoftwofrmswthK = 1, andassumethatbothfrmsarentally symmetrc, e 1 ( ) = e 2 ( ) > 0, t 1 = t 2 = t, wth a low nsurance coverage λ and c 1 ( ) = c 2 ( ) = c( ). Consder frst the case n whch frm 1 s type s t 1 and frm 2 s type s t 2, wth t 1 < t 2 < t < c 1 (0) as n Fgure 2. Under the common low nsurance coverage, frm 1 s exproprated, but not frm 2, because c(t 1 ) < c(t 2 ) < 0. But f frm 1 ncreases ts nsurance coverage to λ (yeldng the vrtual type functon c( ) represented n Fgure 2), then the role are swtched: frm 2 s exproprated because c(t 2 ) < c(t 1 ) < 0. The contrary may obtan for hgher types. Indeed, consder now the case n whch frm 1 s type s s 1 and frm 2 s type s s 2, wth s 1 > c 1 (0) > s 2. Wth a common low nsurance coverage, frm 2 s exproprated because c(s 2 ) < 0 < c(s 1 ). If frm 1 ncreases ts nsurance to λ as before, the role are reversed: frm 1 s exproprated whle frm 2 s not snce c(s 2 ) > c(s 1 ). The followng proposton summarzes some of the observatons above. 19