Seeking rents in the shadow of Coase

Transcription

1 Public Choice (2009) 139: DOI /s Seekig rets i the shadow of Coase Giuseppe Dari-Mattiacci Sader Oderstal Fracesco Parisi Received: 2 September 2007 / Accepted: 25 November 2008 / Published olie: 5 December 2008 The Author(s) This article is published with ope access at Sprigerlik.com Abstract I this paper, we study a two-stage ret-seekig game. I the first stage, cotestats compete à-la-tullock; i the secod stage, the wier ca resell the ret à-la-coase. We cosider a complete iformatio Tullock game i which the cotestats have differet The authors would like to thak two aoymous referees, Lloyd Cohe, Be Depoorter, Bruo Frey, David Haddock, Tom Hazlett, Joris va Hoboke, Bruce Kobayashi, Peter Meell, Ilia Raier, Has-Berd Schäfer, William F. Shughart II, Gordo Tullock, ad the participats i the 2008 aual meetig of the America Law ad Ecoomics Associatio at NYU, the 2006 Erasmus Law ad Ecoomics Workshop at Hamburg Uiversity, the 2006 aual meetig of the Public Choice Society i New Orleas, the 2006 aual meetig of the Caadia Law ad Ecoomic Associatio at the Uiversity of Toroto, ad semiars at Ghet Uiversity, Uiversity of Zurich, St. Thomas Uiversity, Istitute for Iformatio Law at the Uiversity of Amsterdam, George Maso Uiversity School of Law, Uiversity of Texas School of Law, ad Uiversity of Miesota Departmet of Applied Ecoomics for helpful commets. G. Dari-Mattiacci gratefully ackowledges the fiacial support from the NWO grat S. Oderstal gratefully ackowledges the fiacial support from the NWO grat G. Dari-Mattiacci ( ) ACLE ad Tiberge Istitute, Uiversity of Amsterdam, Roetersstraat 11, 1018WB Amsterdam, The Netherlads gdarimat@uva.l url: S. Oderstal Amsterdam School of Ecoomics ad Tiberge Istitute, Uiversity of Amsterdam, Roetersstraat 11, 1018WB Amsterdam, The Netherlads a.m.oderstal@uva.l url: F. Parisi School of Law, Uiversity of Miesota, 229 South 19th Ave., Mieapolis, MN 55455, USA parisi@um.edu url: F. Parisi Departmet of Ecoomics, Uiversity of Bologa, Via Zamboi, 33, Bologa, Italy

2 172 Public Choice (2009) 139: valuatios for the ret. The aalysis focuses o the ex ate effects of a secodary market o efforts, payoffs, ret-dissipatio ad ret-misallocatio. We show that the secodary market, while correctig possible misallocatios, may exacerbate ret dissipatio. I some situatios, the icrease i ret dissipatio more tha offsets the allocative advatage, so that a secodary market might reduce welfare. We further show how the effect of ex post tradeability o welfare depeds o the parties bargaiig power ad valuatios of the ret, also cosiderig the case of edogeous bargaiig power. Keywords Ret-seekig Asymmetric ret valuatios Ret-dissipatio Ret-misallocatio JEL Classificatio C72 D72 1 Itroductio Oce upo a time, two kights were joustig for the had (ad the love) of a pricess. The kights were of equal stregth, valor, ad courage, but oe of them oly aimed at the kigdom, while the other also secretly loved the pricess. The literature o ret-seekig has aalyzed these types of cotests ad has poited to the fact that the participats will dissipate socially valuable resources i the attempt to wi the prize. However, this literature has focused maily o ret-dissipatio ad has give little credit to the idea that there is a additioal social cost associated with such touramets: the pricess may marry the wrog kight. Whe parties have differet valuatios of the ret, allocative iefficiecies may come about i additio to ret-dissipatio. Ret-seekig cotests do ot guaratee that rets will be appropriated by parties who value them the most (Lott 1987; Hurley 1998; Nti 1999). Glaeser ad Luttmer (2003) show, for istace, that misallocatio of housig occurs i cotrolled markets, where the allocatio of apartmets does ot fully rely o the price system. Ret-misallocatios amout to a social cost but, ulike ret-dissipatio, these forms of allocative iefficiecy ca be corrected after the fact. While i our example the kights caot subsequetly reallocate the pricess, Coase (1960) poited out that other sought-after rights ca be traded ex post. 1 Real life examples of tradeable rets rage from homestead property to emissio permits, trasferable public procuremet cotracts, ad the ew Europea Iteret domai ames dot-eu. 2 I the case of tradeable rights, if a low-valuig party 1 The possibility of ex post bargaiig differs from the possibility of a ex ate collusio betwee the retseekers. If ret seekers could collude ex ate, the etire ret-seekig problem would disappear. Rets would be allocated through bargaiig, without dissipatio. I several real-world settigs, however, participats i a ret-seekig game do ot have a opportuity to bargai or collude with oe aother prior to the ret-seekig cotest. Impedimets to bargai may be the result of legal prohibitios (e.g., prohibitio of collusio betwee bidders i a public procuremet cotest) or factual impedimets (e.g., large trasactio costs i cotests with a large umber of competitors). I several such situatios, these impedimets to bargaiig are reduced or elimiated after the cotest. Thus, for example, a wiig bidder is allowed to resell or subcotract to a losig competitor. Similarly, impedimets to bargaiig or collusio amog a large umber of cotestats due to high trasactio costs are likely to be reduced after the ret-seekig cotest. 2 See the iformatio available at

3 Public Choice (2009) 139: wis the cotest he is likely to sell the prize to a high-valuig party, thereby correctig ret-misallocatio. 3 While correctig misallocatios, ex post resale may icrease ret-dissipatio. The prospect of ex post resale makes the prize more appealig to low-valuig parties. Eve parties with very low valuatios may fid the cotest profitable if they ca sell the prize to high-valuig parties at a price that may be sigificatly higher tha their ow valuatio. I some situatios, the possibility for ex post resale might be the oly reaso why parties egage i ret-seekig activities. Examples o poit are commo. We routiely observe idividuals with o iterest i sports or music egagig i scalpig of tickets before a decisive game or a importat cocert. Likewise, i the dot-com era, domai ames have ofte bee purchased by idividuals with o itrisic valuatio, solely attracted by resale opportuities. A example is foud i Avery Deiso Corp. v. Sumpto, 4 where a compay had registered over 12,000 Iteret domai ames for o other apparet purpose tha that of resellig them. Coutless other firms or idividuals have made fortues by simply resellig Iteret domai ames. 5 By iducig larger ivestmets by low-valuig parties, the possibility of ex post resale may exacerbate the ret-dissipatio problem. Coversely, forbiddig resale, as it is for the Australia dot-com-dot-au domai ames, 6 might geerate large misallocatio costs. I this paper, we explore the effects of secodary markets i a ret-seekig model. We observe three possible outcomes: (i) ex post resale reduces ret-dissipatio; (ii) ex post resale icreases ret-dissipatio while still reducig total social costs; (iii) ex post resale icreases ret-dissipatio to the poit that also total social costs icrease, sice the icrease i dissipatio exceeds the reductio i misallocatio. The outcome is show to deped o the relative bargaiig power of the parties. 7 If the bargaiig power of the low-valuig party is limited, a secodary market reduces social costs. Our model shows that the possibility of ex post resale has ambiguous effects o icetives, equilibrium payoffs ad ret-dissipatio. Without a secodary market, ot all parties ecessarily participate i the ret-seekig cotest: parties with the lowest valuatios might stay out of the game. Amog the participatig parties, those with higher valuatios ivest more ad achieve higher wiig probabilities. Istead, with a secodary market, all parties participate ad exert the same level of effort, splittig the ret equally amog them. Payoffs follow a similar, but ot idetical patter, the payoff of the party with the highest value beig always larger tha that of the other parties. 3 Eve if the prize itself is otrasferable, such as some liceces, permits, or import quotas, de facto trasferability may result from the sale of the whole firm, the sale of the products to a itermediary, or subcotractig. I the case of import quotas, for example, a authorized importer could either distribute the product to cosumers directly or sell the etire stock ad correspodig moopoly ret to a more effective distributor F.Supp C.D.Cal., See postversio= See the 2005 Australia Domai Name Admiistrator s Policy o Clarificatio of Domai Name Licece Prohibitio o Sale of Domai Names, at: 7 Bargaiig power ca be uderstood i very geeral terms. I fact, i the model, the price lies betwee the seller s ad the buyer s valuatios. Our comparative statics results are parametrized o the actual price chose by the parties betwee these two boudaries. We refer i the paper to bargaiig power as a determiat of such choice; however, our results are more geeral ad apply irrespective of the otio employed to select the price.

4 174 Public Choice (2009) 139: We further demostrate that a secodary market is usually 8 desirable from the poit of view of low-valuig parties, sice it allows them to resell at a higher price tha their valuatio. High-valuig parties however do ot ecessarily beefit from the prospect of ex post reallocatios. This is because the itroductio of a secodary market gives the party with the highest value a opportuity to purchase the right should he fail i the ret-seekig cotest, but also icreases the competitio that he faces from low-valuig cotestats. O a theoretical level, our aalysis brigs together isights ad results from ret-seekig ad property rights theory, showig that there is a importat lik betwee the iitial acquisitio of a property right (e.g., homesteadig) 9 ad its derivative circulatio (e.g., sale). The ret-seekig literature (Tullock 1967;Krueger1974; Poser 1975) 10 iitially cosidered parties competig for the appropriatio of a commoly valued ret, a settig i which the fial allocatio is efficiet by hypothesis. Asymmetries betwee the parties valuatios have subsequetly bee studied i relatio to the parties efforts, the resultig dissipatio, ad etry i the cotest (Hillma ad Riley 1989;Nti1999). 11 We brig the Coase theorem (Coase 1960) to bear o the aalysis, allowig the parties to egotiate after the iitial phase of retseekig. Combiig Tullock s isight that competitio for rets geerates ret-dissipatio with Coase s argumet that markets correct ay iitial misallocatio, we show that, i some situatios the icrease i ret-dissipatio more tha offsets the allocative gais of resale, so that a secodary market results i a icreased social loss. 12 Our paper brigs to light aother cotradictio with existig literature. Accordig to the Coase theorem, a efficiet allocatio of resources is achievable idepedetly of the iitial allocatio of rights ad distributio of the bargaiig surplus betwee the parties. We show istead that the efficiecy of the outcome depeds o the distributio of the bargaiig 8 This result holds if the price is exogeously determied. 9 A well-kow example is the 1862 Homestead Act, cocerig the ew territories of the west of the Uited States. Most legal systems iclude similar rules of first come, first served regulatig how property o thigs that belog to obody (res ullius) ad abadoed thigs (res derelictae) is to be assiged. It has bee observed that rules of first possessio (such as first come, first served ad fiders, keepers ) geerate ret-seekig icetives i the iitial allocatio of property rights (Barzel 1974) leadig to wasteful expeditures i the appropriatio of such rights also i the form of too fast exploitatio (Barzel 1968; Dasgupta ad Stiglitz 1980; Aderso ad Hill 1990; Lueck 1995). Mortese (1982) shows that the ex post distributio of the surplus affects ex ate icetives to ivest i matig ad racig games but does ot discuss the issue of ex post trade. 10 See also Cogleto et al. (2008); Higgis et al. (1985); Epstei ad Nitza (2007); Lockard ad Tullock (2001). 11 Cotests with asymmetric valuatios are also studied i Leiiger (1993) adbaik(1994). Moreover, Baye et al. (1993), Che ad Gale (1998), Goeree et al. (2005), ad Oderstal (2007) study the effect of asymmetric valuatios i a completely discrimiatory cotest, i.e., the highest bidder wis with probability equal to 1. Noe of these cotributios studies the effects of ex post trade of the ret. For a recet review of the literature o (ret-seekig) cotests see Tolliso (2003) ad Epstei ad Nitza (2007). For aalyses of stadard auctios with resale, see Garratt ad Tröger (2006) ad Hafalir ad Krisha (2008) ad the refereces cotaied therei. 12 Some previous cotributios also emphasize the perverse effects of tradig possibilities. Jackli (1987) shows that the presece of liquid equity markets udermies the role of baks i reducig liquidity risk. Literature o icomplete cotracts (Grossma ad Hart 1986) emphasizes the importace of rights that caot be trasferred by cotract ad the effects that the allocatio of such rights has o the parties ivestmets i the relatioship. Our perspective is differet, sice we aalyze the ivestmets of the parties prior to the allocatio of such rights.

5 Public Choice (2009) 139: surplus a distributive aspect of the Coase theorem, which is geerally cosidered irrelevat for the efficiet allocatio of resources. 13 We are ot the first to study the effect of a secodary market o welfare. I Zardkoohi adpustay(1989) ad Gahvari (1989), the total social loss is uaffected by ex post resale. This result derives from a assumptio of total ret-dissipatio by the wiig party oly (the losig party icurs o costs). If the low-valuig party secures the ret ad resale is ot allowed, the misallocatio cost is equal to the differece i the parties valuatios ad ret-dissipatio is equal to the low value. With a secodary market, the misallocatio cost disappears, but ow the low-valuig party is willig to ivest as much as the high-valuig party. That is, ret-dissipatio icreases exactly by the differece i the parties valuatios. Thus, a secodary market does ot affect social welfare. I our model, this sceario is a special case of the more geeral situatio i which the parties exert effort but do ot ecessarily dissipate the full ret. Gahvari (1989) further cojectures that whe both parties icur costs or whe dissipatio is less tha full, a secodary market might reduce social welfare. I our paper, we derive ew results that partially cofirm Gahvari s cojecture, idetifyig coditios uder which social welfare decreases. Our aalysis, however, shows that uder some parameter values, social welfare icreases whe secodary markets are itroduced. I additio, we examie the effect of a secodary market o parties participatio choices. Like us, Sue (1989) fids that a secodary market may both icrease ad decrease welfare. He does so i a model of ret-seekig through waitig (essetially a model of ratioig). He assumes that multiple parties compete with oe aother for the appropriatio of a trasferable ret (ratioed commodity). I Sue s model, the variace i waitig costs determies whether a secodary market has a positive or egative effect o welfare. I cotrast, i our model, the key variables are bargaiig power ad the distributio of the parties values, while the cost of ret-seekig effort is the same for all parties. I additio to ex post resale, a corrective tax may realig private ad social icetives i ret-seekig cotests. Like us, Epstei ad Nitza (2002) examie a situatio i which the parties asymmetric valuatios ca geerate misallocatio. They fid that a miimal asymmetry i the parties valuatios is a ecessary coditio for a corrective tax to be effective. I our framework, istead, ex post resale always corrects misallocatio, while ret-dissipatio might icrease or decrease as a result of ex post resale. Moreover, ulike the literature just revised, we allow the price i the secodary market to deped o the efforts exerted by the parties i the prelimiary ret-seekig face of the game. Our paper is orgaized as follows. I Sect. 2, we preset our baselie model: a two-party ret-seekig cotest where the cotestats have differet valuatios of the cotested ret. I Sect. 3, we begi by aalyzig the ret-seekig cotest without a secodary market. Next, we cosider ret-seekig with a secodary market. We assess the effects of a secodary market o icetives ad social costs of ret-seekig by comparig the outcome of retseekig with ad without a secodary market. I Sect. 4, we exted the aalysis to the N-party case. We coclude i Sect. 5 with some qualificatios of our aalysis, possible refiemets, ad ideas for future research. Proofs are relegated to the Appedix. 13 Discussig the use of property rules ad liability rules for the iteralizatio of exteral costs, Bebchuk (2001) observes that the choice of remedy affects the ex post distributio of resources with ex ate effects o the levels of the parties ivestmets i their property. However, he does ot discuss the effects o ex ate icetives to ivest i ret-seekig.

6 176 Public Choice (2009) 139: The model We cosider a ret-seekig cotest betwee two 14 risk-eutral parties L ad H. 15 Let B i 0 deote party i s ret-seekig expeditures ad V i his valuatio of the ret, with i {L,H}. The parties are completely iformed about each others values. Followig the covetioal Tullock (1980) framework, 16 party i s success fuctio is give by Br i S i = Bi r + Bj r with {i, j} ={L,H}. The success fuctio ca be iterpreted either as the probability to wi the etire ret or as the share of the appropriated ret. 17 As usual, r is a idex of the parties margial retur to effort. 18 Accordigly, party i s payoff is give by U i = S i V i B i. (1) Parties differ i their valuatio of the ret V i. For simplicity, we assume that there are two types of parties: a low-valuig party L with value V L 0 ad a high-valuig party H with value V H V L. To simplify otatio, we itroduce a variable measurig the level of asymmetry betwee the parties valuatios, γ V L V H.Thevalueofγ rages 0 γ<1, approachig 1 if the parties have similar valuatios of the ret ad approachig 0 if the parties valuatios differ substatially. I the followig, we will focus o situatios where a equilibrium i pure strategies exists. This is guarateed whe the parties face decreasig, costat or moderately icreasig margial returs to effort: 0 <r<1 + γ r (Nti 1999). 19 It is easy to show that the upper boud for r icreases i γ ad is at least 1 ad at most 2. Whe parties have heterogeeous valuatios of the ret, losses from iefficiet allocatios of the ret should be cosidered i additio to the traditioal problems of retdissipatio. The misallocatio losses ca be viewed as a opportuity cost due to the fact that the ret could have bee put to a higher-value use. Accordigly, we defie two loss 14 I Sect. 4, we cosider a extesio of the model to N parties. 15 See Allard (1988), Nti (1999), Lockard ad Tullock (2001), Epstei ad Nitza (2007), ad Korad (2007) o this model. 16 I reality, the splittig of a prize betwee two cotestats may ot follow Tullock s sharig rule ad may take up a whole rage of possible fuctioal forms. We follow Tullock s traditioal framework sice it allows for a easier compariso of our results with those of the existig ret-seekig literature ad for a explicit calculatio of equilibrium values, but see also Hirshleifer (1989). Most of the results preseted i this paper could also be obtaied with a more geeral success fuctio. 17 Assumig risk eutrality, these two iterpretatios are fuctioally equivalet. Thus, hereiafter, we refer idifferetly to either iterpretatio. If the success fuctio is iterpreted as a share of the prize, it allows for multiple, partial wiers ad may geerate additioal social loss due to property fragmetatio (Parisi 2002). 18 Alteratively, r ca be see as the discrimiative power of the auctio. 19 Nti (1999) writes this coditio as V r L + V r H >rvr H.

7 Public Choice (2009) 139: variables: 20 the ret-dissipatio, give by the sum of the parties efforts D = B L + B H (2) ad the ret-misallocatio, give by the share of the ret appropriated by the low-valuig party scaled by the valuatio differece M = S L (V H V L ). (3) Obviously, i the special case of parties with equal valuatios of the ret, the misallocatio is equal to zero. 3 Aalysis 3.1 Ret-seekig without a secodary market The equilibrium for the ret-seekig cotest with asymmetric parties is as follows: Propositio 1 (Nti 1999) The two-party ret-seekig cotest without a secodary market has a uique pure Nash equilibrium, i which the parties efforts are γ r BL = r (1 + γ r ) V L, ad 2 BH = r (1 + γ r ) V 2 H. The parties expected payoffs are γ r U L = γ 2r + (1 r)γ r (1 + γ r ) 2 V L, ad U H = 1 + (1 r)γr (1 + γ r ) 2 V H. The social cost of ret-seekig is give by the sum of ret-dissipatio ad retmisallocatio. Usig the results of the previous propositio, it is possible to calculate these costs, expressed i (2) ad(3) Note that we assume throughout that the parties ret-seekig efforts have purely redistributive effects. For a aalysis of ret-seekig cotests where ret-seekig effort is socially productive see Dari-Mattiacci ad Parisi (2005) ad Dari-Mattiacci et al. (2007) ad the coclusios at the ed of this paper. The otio of cotest efficiecy (CE) itroduced by Hurley (1998, p. 291) ca be writte i terms of dissipatio ad misallocatio as follows: CE = 1 D+M V. The reader ca easily apply this trasformatio to compare our H results. As from Hurley s aalysis, it follows from our aalysis that oly miimizig ret dissipatio may ot miimize total social costs i the presece of ret misallocatio. 21 It is show i Nti (1999) that the low-valuig party s effort icreases i his ow valuatio ad decreases i the valuatio of the other party. I cotrast, effort by the high-valuig party icreases i both parties valuatios. This ca be easily uderstood. A icrease i the valuatio of the low-valuig party makes the cotest more symmetric, iducig both parties to exert more effort. Cosequetly, the wiig probabilities coverge towards 1 2. I cotrast, a icrease i the valuatio of the high-valuig party makes the cotest more asymmetric, to the advatage of the high-valuig party, who icreases his effort. Istead, the low-valuig party decreases his effort ad the wiig probabilities diverge.

8 178 Public Choice (2009) 139: Corollary 1 I the two-party ret-seekig cotest without a secodary market, the retdissipatio is ad the ret-misallocatio is γ r D = r (1 + γ r ) (1 + γ)v 2 H (4) M = 3.2 Ret-seekig with a secodary market γ r 1 + γ r (1 γ)v H. (5) The presece of asymmetric ret valuatios geerates allocative iefficiecies. I some situatios, possible misallocatios of the ret ca however be corrected through ex post resale. Low-valuig wiers ca sell the ret to their high-valuig oppoets. We cosider the geeral case i which the wier of the ret ad the party who wishes to buy it bargai with oe aother ad reallocate the ret, splittig the cotractual surplus accordig to their respective bargaiig power. Wheever a reallocatio of the ret takes place, the price paid by the high-valuig party to the low-valuig party is V L P V H. The resale price P ca be read as a idex of the parties relative bargaiig power: the higher the price, the greater the bargaiig power of the low-valuig party ad vice versa. We assume that the relative bargaiig power depeds o the parties efforts i the precedig ret-seekig cotest. More i particular, we take the resale price to be a fuctio of the low-valuig party s wiig probability P = p(s L ), 22 where p( ) is a twice differetiable fuctio ad P = p (S L ) = P S L. Sice we focus o situatios where a equilibrium i pure strategies exists, we assume that P is sufficietly small. Note that the case where P = 0 correspods to parties cosiderig the ivestmets i the cotest as suk costs oce they start bargaiig. I that case, a uique outcome will be reached, idepedet of efforts i the cotest, similarly to the outcome of stadard bargaiig models, such as the oes by Nash (1950) ad Rubistei (1982). I cotrast, if the resale price depeds o the parties efforts, 23 P < 0 implies that the price decreases i the low-valuig party s effort ad icreases i the high-valuig party s effort. This situatio could represet a case where larger expeditures i ret-seekig i the first stage weake the low-valuig party s bargaiig power or where a high-valuig party, who has spet large resources i the first stage of the game ad failed to appropriate the ret, may be willig to pay a higher price to secure the ret i the secodary market. Fially, P > 0 implies that the price icreases i the low-valuig party s effort ad decreases i the high-valuig party s effort. This settig could describe a situatio where a 22 Note that this formulatio captures the idea that the price depeds o the divergece betwee the parties efforts. Note also that, sice S H = 1 S L, our model captures situatios where the price is a fuctio of both the high ad low party s wiig probabilities. 23 The margial effects of the parties efforts o price are as follows: sice S L B L sig P > 0, we have sig P B L = sig P S L. Likewise, sice P B H P B L = P S L S L B H ad S L B H = S P S L L B ad L < 0, we have = sig( P S L ). Note that i our aalysis the price varies i differet directios as the parties B H efforts icrease. This is due to the choice of the probability of wiig as a argumet of the price fuctio. A more geeral model should ecompass the possibility for the parties efforts to affect the price i the same directio, such as i situatios where the willigess-to-pay ad willigess-to-accept for the ret is edogeously determied by effort expeditures i the ret-seekig game.

9 Public Choice (2009) 139: low-valuig party has expeded large amouts of effort to appropriate the ret ad is uwillig to resell the ret at a low price or could refer to a high-valuig party who, havig expeded large outlays of effort i the first stage, may be uable to exped additioal large amouts for purchasig the ret i the secodary market (e.g., due to budget costraits or icreasig margial costs of capital). I this settig, the low-valuig party will choose his effort level cosiderig the price that he could extract from the high-valuig cotestat i the ex post resale. The possibility of ex post resale affects the payoffs as follows: U L = S L P B L, ad U H = S H V H + S L (V H P) B H = S H P B H + (V H P). The above formulatios of the parties payoffs highlight the iterestig features of retseekig with a secodary market. The parties payoffs remai asymmetric, because the parties actual gais from the game deped o their respective bargaiig power. Uless the low-valuig party has full bargaiig power ad is able to extract the etire surplus from the other party (P = V H ), the high-valuig party will still have a larger payoff from the game if both ivest the same amout. This is due to the fact that the high-valuig party appropriates the differece betwee his valuatio ad the price paid to his oppoet. The derivatives of the utilities are as follows: 24 U L B L = S L B L (P + S L P ) 1, (6a) U H B H = S L B H (P + S L P ) 1. (6b) S L B L Note that if B L = B H, = S L B H, 25 so that U L B L = U H B H. This yields that the parties equilibrium levels of efforts are the same ad ca be calculated, as i a traditioal Tullock game, by referece to the price P rather tha to the differet valuatios of the cotested ret. Therefore, the parties icetives to ivest i ret-seekig effort are symmetric, despite the asymmetry persistet i the parties payoffs. The ituitio is that the high-valuig party will always obtai his high-valued ret V H, whether through direct appropriatio i the primary cotest, or through ex post reallocatio i the secodary market. By itroducig a secodary market, parties with asymmetric payoffs ed up playig symmetrically. Holdig V H costat, the differece for a high-valuig party betwee success ad failure oly depeds o the price to be paid to the low-valuig party, if the latter successfully appropriates the ret. If the high-valuig party wis the ret-seekig game, he avoids the paymet of P. Equatig the expressios i (6) to zero, the followig propositio ca be readily derived; a double asterisk deotes equilibrium values with a secodary market: Propositio 2 The two-party ret-seekig cotest with a secodary market has a pure Nash equilibrium, i which the parties efforts are BL = B H = r (P + 12 ) 4 P 24 Note that rearragig the expressio above we have U H = V H S L P B H. 25 We have S i B = S j i B = rs i S j i B, with {i, j}={l,h}. i

10 180 Public Choice (2009) 139: ad the parties expected payoffs are U L U H = 2 r 4 P r 8 P, ad = V H 2 + r 4 P r 8 P where P = p( 1 2 ) is the equilibrium resale price ad P = p ( 1 2 ). Note that, otwithstadig the equivalece i the parties effort, both parties would beefit from the fact that the resale price decreases with the low-valuig party s effort ad icrease with the high-valuig party s effort, as both payoffs icrease as P becomes small ad possibly egative. A cosequece of the equivalece betwee the parties icetives is that the social cost ca be expressed as follows: Corollary 2 I the two-party ret-seekig cotest with a secodary market, the retdissipatio is D = r (P + 12 ) 2 P (7) ad the ret-misallocatio is M = 0. Note that if P = 0, which is the case if the price P is take as exogeous, the parties equilibrium efforts ad payoffs ad the ret-dissipatio are as i the traditioal Tullock game. 3.3 Compariso The possibility of ex post resale alters the parties equilibrium levels of effort ad cosequetly their payoffs. A secodary market trasforms the game i two ways. O the oe had, a secodary market makes the game symmetric i terms of the parties icetives; o the other had, a secodary market chages the magitude of the ret for which the parties compete. The former effect is clear if we examie the ratio betwee the parties efforts. Without a secodary market, the parties ivest i the ret-seekig game i proportio to their valuatios: B L = γ. I cotrast, with a secodary market, the parties efforts are the same B H BL BH = 1. However, as we have observed, also the prize implicitly chages. This gives rise to the followig questios: Will a party icrease or decrease his effort whe ex post resale is allowed? How will the social cost of ret-seekig chage? Likewise, ex post resale affects the parties payoffs. Both with ad without a secodary market, the high-valuig party ears more tha the low-valuig party. However, while without a secodary market this differece depeds o the asymmetry betwee the valuatios <γ), 26 with a secodary market the high-valuig party s advatage ( U L UH < 1) depeds o the parties bargaiig power. This observatio suggests that the parties payoffs are ( U L U H 26 This iequality derives from propositio 1 ad 0 < γ 2r +(1 r)γ r 1+(1 r)γ r < 1.

11 Public Choice (2009) 139: affected i differet ways by a secodary market. Will both parties beefit from the itroductio of trade? I the followig we address these questios. Let us start with the effort ad payoff of the low-valuig party ad defie two threshold levels of P : P BL 4γ r (1 + γ r ) 2 V L 1 2 P, ad P UL 4γ r 1 r + γ r V (1 + γ r ) 2 L + r 2 r 4 2r P. Propositio 3 I the two-party ret-seekig cotest, the low-valuig party exerts more (less) effort with a secodary market tha without if ad oly if the equilibrium price is higher (lower) tha P BL : B L > (<) B L iff P > (<) P BL. The low-valuig party obtais a larger (smaller) payoff with a secodary market tha without if ad oly if equilibrium price is higher (lower) tha P UL : U L > (<) U L iff P > (<) P UL. The followig corollary shows that i the case of a costat resale price, a secodary market uambiguously improves the positio of the low-valuig party, who exerts more effort ad is able to appropriate a larger share of the ret through ex post egotiatios with his high-valuig oppoet. A variatio i the relative bargaiig power of the parties, P, will oly quatitatively affect this result i the sese that a larger P further improves the positio of the low-valuig party. Corollary 3 I the two-party ret-seekig cotest, if P = 0, the low-valuig party exerts more effort ad obtais a larger payoff with a secodary market tha without: BL >B L, ad UL >U L. This suggests that the positio of the low-valuig party is egatively affected by the possible effect of the high-valuig party s effort o price. The positio of the high-valuig party, istead, depeds o the parties relative bargaiig power, eve if the resale price is costat. Let us defie two additioal threshold levels of P : P BH 4γ r (1 + γ r ) 2 V H 1 2 P, ad P UH 4γ r 1 + r + γ r V (1 + γ r ) 2 H r 2 + r 4 + 2r P. Propositio 4 I the two-party ret-seekig cotest, the high-valuig party exerts more (less) effort with a secodary market tha without if ad oly if the equilibrium price is higher (lower) tha P BH : B H > (<) B H iff P > (<) P BH.

12 182 Public Choice (2009) 139: The high-valuig party obtais a smaller (larger) payoff with a secodary market tha without if ad oly if the equilibrium price is higher (lower) tha P UH : U H < (>) U H iff P > (<) P UH. The effect of a secodary market o the high-valuig party s payoff may vary due to the presece of two coutervailig effects. First, a secodary market makes it possible for the ret to be purchased ex post by the high-valuig party. This lowers the high-valuig party s eed to ivest i the ret-seekig cotest for a direct appropriatio of the ret. Secod, the possibility of ex post resale makes the low-valuig party behave more aggressively. The et impact of these two coutervailig effects depeds o the parties bargaiig power. If the high-valuig party has a sufficietly strog bargaiig positio (low P ), the first effect will domiate over the secod, reducig his icetives to compete ad improvig his payoff. If the low-valuig party has strog bargaiig power (high P ), the secod effect prevails ad we have the paradoxical result that givig the high-valuig party the opportuity to buy the ret i the secodary market worses his positio. By the same toke, a secodary market may icrease or decrease ret-dissipatio, depedig o the parties bargaiig power. Cosider two limit cases: Whe the high-valuig party has full bargaiig power (P = V L ), the two cotestats compete as if they were both low-valuig parties. Hece, the opportuity for ex post reallocatio of the ret reduces the equilibrium level of dissipatio. I cotrast, whe the low-valuig party has full bargaiig power (P = V H ), both parties compete for the same higher value. Evidetly, i this case the dissipatio is larger with a secodary market. As above, let us defie a ew threshold level of P : P D 2γ r (1 + γ) (1 + γ r ) V 2 H 1 2 P. The followig propositio follows by comparig (4) ad(7). Propositio 5 I the two-party ret-seekig cotest, ret dissipatio is larger (smaller) with a secodary market tha without if ad oly if the equilibrium price is higher (lower) tha P D : D > (<) D iff P > (<) P D. Although misallocatio is totally elimiated by a secodary market, the possibility for ex post reallocatio may actually icrease the total social cost of ret-seekig if the probability of misallocatio is relatively small. The followig propositio shows that this ca ideed be the case if the resale price is sufficietly large ad ca be easily derived by comparig (4), (5), ad (7). The ew threshold is: P D+M 2γ r ( (1 + γ) 1 + (1 + γ r ) )(1 γ) V (1 + γ r ) 2 H 1 r(1 + γ) 2 P. Propositio 6 I the two-party ret-seekig cotest, the total social loss is larger (smaller) with a secodary market tha without if ad oly if the equilibrium price is higher (lower) tha P D+M : D + M > (<) D + M iff P > (<) P D+M.

13 Public Choice (2009) 139: Table 1 Effort, payoffs ad social cost for the various threshold levels of P for P = 0 It ca be easily show that for all 0 <γ <1, P BL <P D <P BH <P D+M. The rakig of P UL ad P UH relative to the other thresholds are ambiguous i the geeral case. However, if P = 0, it ca be show that P UH falls betwee P D ad P BH, while, from Corollary 3, the threshold P UL is always satisfied. Table 1 illustrates the previous results, showig how the parties effort ad payoff ad the social loss chage at the various threshold levels of P if P = 0. The low-valuig party always icreases his effort whe trade is allowed. For values of P less tha P D, the icrease i the low-valuig party s effort is offset by a decrease i the high-valuig party s effort. As a result, a secodary market reduces ret dissipatio. For values of P greater tha P D, dissipatio icreases with a secodary market as the low-valuig party icreases his effort more tha the high-valuig party reduces his ow. For eve higher values of P (P >P BH ), also the higher-valuig party icreases his effort further icreasig ret-dissipatio ad evetually brigig it to a poit (whe P >P D+M )where the allocative advatage of a secodary market is offset by the icreased ret-dissipatio. Note that P UH <P BH : the high-valuig party exerts more effort i order compesate for the smaller payoff eared i compariso with the situatio without secodary market. Figure 1 illustrates how the threshold levels of P chage whe γ chages for r = 1ad P = 0. Note that oly values above or o the forty-five-degree lie are admissible, sice P V L. The figure shows that the parties relative bargaiig power ad the asymmetry i their valuatios are somewhat complemetary. If there is little asymmetry (large γ ), the, eve if the low-valuig party has strog bargaiig power, the presece of a secodary market is likely to improve welfare. I cotrast, if the parties valuatios are very asymmetric (small γ ), the the secodary market worses social welfare eve for relatively low values of the low-valuig party s bargaiig power. The figure also illustrates that three outcomes are possible. First, for values of the parties bargaiig power P ad asymmetry i the valuatio γ i the area to the right of the curve P D, the secodary market reduces both ret dissipatio ad ret misallocatio. Secod, for values of P betwee P D ad P D+M, the secodary market icreases ret dissipatio but ot eough to offset the reductio i ret misallocatio. This area is further divided ito three parts by the curves for P UH ad P BH. I the regio above [below] P UH, the high-valuig party ears less [more] with a secodary market tha without. I tur, i the area above P BH,

14 184 Public Choice (2009) 139: Fig. 1 Threshold levels of P for differet values of gamma, for r = 1adP = 0 the high-valuig party ivests more whe a secodary market is operatig. Third, i the area above P D+M, the icrease i the high-valuig party s effort brigs dissipatio to a poit at which the allocative advatage of trade is completely lost ad social welfare decreases. If the parties margial retur to effort icreases, the curves depicted i Fig. 1 shift dowwards,as the followig corollary shows. Corollary 4 I the two-party ret-seekig cotest: P BL r, P D r, P B H, P D+M < 0. r r For P = 0 we also have P U L r < 0 ad P U H r < 0. I other words, icreasig r makes the threshold levels of P that we have idetified more easily satisfiable. This i tur implies that for larger r it is more likely that the presece of a secodary market will worse social welfare. This fidig ca be explaied by observig that whe r is large, competitio i the ret-seekig game is already fierce without a secodary market. The itroductio of a secodary market is likely to make it eve more fierce. This is because large margial returs to effort make a low-valuig party icrease his expediture more whe trade is allowed. I tur, the high-valuig party is more likely to icrease his ow effort whe facig a already tough competitor. The result is that icreased competitio (higher r) geerates ret dissipatio losses that are more likely to offset the allocative gais. This result implies that secodary markets are more desirable whe the parties competig i a ret-seekig cotest are weak, that is, they have small margial retur to effort. Corollary 5 I the two-party ret-seekig cotest: P BL P, P D P, P U H P, P B H P, P D+M P < 0, P UL > 0. P

15 Public Choice (2009) 139: This corollary suggests that with the exceptio of P UL, which moves i the opposite directio P is more likely to be above the thresholds the more the effort by the low-valuig party icreases the price while effort by the high-valuig party decreases it. The fact that the low-valuig party s effort has a positive effect o price, while the impact of the high-valuig party s effort is egative, makes the ret-seekig stage more fierce as both parties have a reaso to exert more effort. This makes the equilibrium level of effort with a secodary market go up, alog with ret dissipatio ad the total social loss. The payoffs of the parties go dow accordigly, due to the icreased outlays of effort. 4 N parties I this sectio, we exted our aalysis to a cotest with N > 2 parties, labeled i = 1,...,N, such that V 1 V 2 V N 0. We assume that V 2 > 0, so that there are at least two parties with a geuie iterest i the ret. For simplicity, i this sectio we restrict the aalysis to the special case of costat margial retur to effort r = 1 ad costat exogeous price P.Partyi s payoff is give by U i = B i N j=1 B j V i B i, with i = 1,...,N. As before, we will study the complete iformatio case i which parties are commoly iformed about each others values. With N parties, ret-dissipatio ad ret-misallocatio are defied as follows: D M N B i, i=1 N i=1 4.1 Ret-seekig without a secodary market ad B i N j=1 B (V 1 V i ). j We will first examie the equilibrium properties of the ret-seekig cotest i a situatio without a secodary market: fial allocatios of the ret are determied by the ret-seekig cotest, without a opportuity for ex post trasfers. Let ˆV k deote the harmoic mea of the highest k values: ˆV k k k j=1 Defie as the smallest iteger for which, 1 V j k= 1,...,N. V +1 1 ˆV. Hillma ad Riley (1989) show that oly the parties with the highest values will be active i equilibrium. A immediate cosequece of this observatio is that, i cotrast to the two-party case, some parties may decide to exped o effort i equilibrium. Stei (2002) derives equilibrium efforts, ad Matros (2006) shows that the equilibrium is uique.

16 186 Public Choice (2009) 139: Propositio 7 (Matros 2006) The N-party ret-seekig cotest without a secodary market has a uique pure Nash equilibrium, i which the parties efforts are B i = { 1 ad the parties expected payoffs are ˆV 1 V i ( 1 ˆV ) 2 if i = 1,...,, 0 if i = + 1,...,N { 1 Ui Vi = (V i 1 ˆV ) 2 if i = 1,...,, 0 if i = + 1,...,N. (8) From this, we ca readily derive ret-dissipatio ad misallocatio losses i equilibrium. Corollary 6 I the N-party ret-seekig cotest without a secodary market, the retdissipatio is D = 1 ˆV ad the ret-misallocatio is M = ( 1) ˆV 4.2 Ret-seekig with a secodary market V i. Now, cosider the situatio where parties ca trasfer the ret i the secodary market after the iitial ret-seekig cotest. Assume that the secodary market always reallocates rets i the hads of the highest valuig cotestat, party 1. We assume that all sellig parties have the same bargaiig power, that the resale price is exogeous ad lies somewhere betwee the highest ad the secod-highest valuatios of the cotestats, P [V 2,V 1 ]. The sellig price is i fact always at least V 2, give that party 2 would be willig to buy or keep the ret if the price is less tha V 2. Similar to the two-party case, the possibility of ex post resale trasforms a situatio with asymmetric icetives ito oe with symmetric icetives. I particular, the possibility of resale affects the parties payoffs as follows: U i = U 1 = i=2 B i N j=1 B j P B i, for i = 2,...,N, ad ( ) B 1 B 1 N j=1 B V N j j=1 B (V 1 P) B 1 (9) j = B 1 N j=1 B j P B 1 + (V 1 P). The first term o the right-had-side of (9) referstothesituatiowhereparty1wisithe ret-seekig stage, while the secod term refers to the bargaiig stage. As i the two-party case, icetives ad equilibrium strategies are symmetric. We have the followig result.

17 Public Choice (2009) 139: Propositio 8 The N-party ret-seekig cotest with a secodary market has a uique pure Nash equilibrium, i which the parties efforts are Bi = N 1 P, for i = 1,...,N N 2 ad the parties expected payoffs are U i = 1 N 2 P, for i = 2,...,N, ad (10) U 1 = 1 N 2 P + (V 1 P). Corollary 7 I the N-party ret-seekig cotest with a secodary market, the retdissipatio is D = N 1 N P ad the ret-misallocatio is M = 0. Observe that with a secodary market, all parties are active i equilibrium, i cotrast to what we observed i the absece of a secodary market. 4.3 Compariso How does the availability of a secodary market affect efforts, expected payoffs, misallocatio ad ret-dissipatio? As i the two-party case, low-valuig parties beefit from a secodary market. More precisely, we observe that some of the low-valuig parties that remaied iactive whe there were o resale opportuities joi the cotest whe resale opportuities are itroduced. These are cotestats that do ot have a sufficietly geuie iterest i the ret to eter the cotest i the first case, who become attracted by resale opportuities, eterig the market i the secod case. With a secodary market these parties exert strictly positive efforts ad obtai positive payoffs. For active parties with a geuie iterest i the ret, the effects of a secodary market are ambiguous. For parties 2,...,, this ca be immediately see by comparig (8)ad(10). If N icreases, the payoffs with a secodary market coverge to zero, while they are strictly bouded away from zero without a secodary market for costat. I other words, if sufficietly may iactive parties become active with a secodary market, a secodary market worses the positio of those parties who were already active. The opposite result obtais if the sellig party has strog bargaiig power (P is close to V 1 )adv 1 is large. Aalogously, party 1 is worse off with a secodary market if N is large relative to ad P is close to V 1, while a secodary market improves party 1 s payoff if both P ad the values of all other parties are close to V 2. Thus, as i the two-party case, the relative performace of a secodary market for the payoffs of the parties depeds o the bargaiig power of the sellig party, reflected i the resale price P. The effect of a secodary market o ret-dissipatio also depeds o the bargaiig power of the sellig party. As i the two-party case, we ca defie a threshold for the sellig

18 188 Public Choice (2009) 139: party s bargaiig power: P D 1 N N 1 ˆV. Propositio 9 I the N-party ret-seekig cotest, retdissipatio is larger (smaller) with a secodary market tha without if ad oly if the equilibrium price is higher (lower) tha P D : D > (<) D iff P > (<) P D. The followig corollary provides sufficiet coditios for a secodary market to icrease or decrease ret-dissipatio. If the bargaiig power of the seller is weak (reflected i a low P ), ad most parties are already active i the situatio without a secodary market, a secodary market has a egative effect o dissipatio. I other words, the higher the relative value of parties 2,...,N (reflected i a high umber of active parties ad a high ˆV N )the more likely it is that the opportuity to resell reduces dissipatio. Corollary 8 I the N-party ret-seekig cotest, ret-dissipatio is larger with a secodary market tha without if the resale price is higher tha ˆV : D >D if P> ˆV. Ret-dissipatio is smaller with a secodary market tha without if the resale price is lower tha ˆV N ad ˆV N is larger tha V 2 : D <D if N =, P < ˆV N ad ˆV N >V 2. Let us ow cosider the relative performace of the situatios with ad without a secodary market i terms of the total social costs D + M. Recall that ad D + M = 2 1 ˆV D + M = D = N 1 N P. Thus, the threshold value for P ca be expressed as follows: P D+M N ( 2 1 ˆV N 1 i=2 V i V i ). Propositio 10 I the N-party ret-seekig cotest, the total social loss is larger (smaller) with a secodary market tha without if ad oly if the resale price is higher (lower) tha P D+M : D + M > (<) D + M iff P > (<) P D+M. The expressio for P D+M ad the above propositio provide us with a idicatio of what are the likely situatios where total social loss is actually icreased by the itroductio of a secodary market: is small relative to N (oly a small fractio of geuie cotestats i=2

19 Public Choice (2009) 139: who are active without a secodary market), ad P is close to V 1 (strog bargaiig power of the lower-valuig cotestats/sellers). The followig corollary makes this ituitio more precise. Corollary 9 I the N-party ret-seekig cotest, the total social loss is greater with a secodary market tha without oe, i.e., D + M >D + M, if at least oe of the followig coditios holds: 1. P> N2 N 1 V 2; 2. P = V 1, ad = 2; 3. P> N N 1 (4 2 3)V 1, ad = 2; 4. V 1 = V 2, ad = 2; 5. V 1 = V 2 ad N> 2 i=1 V i. The above corollary shows that i additio to the bargaiig power of the seller, i the N-party case also the umber of parties who would be iactive without a secodary market becomes relevat. Note that coditios 1, 2, ad 3 poit to situatios where the seller has strog bargaiig power. A substatial umber of iactive parties without a secodary market is reflected i coditios 2 through 5 above (where a substatial umber could be as little as oe). I spite of the potetial etry of parties without a geuie iterest i the N-party cotest, the result obtaied for the two-party case geeralizes to the N-party case: total social loss may decrease with a secodary market. The followig result establishes that a secodary market decreases social loss if V i is close to V 2 for all i = 3,...,N (little dispersio at the bottom) ad P is close to V 2 (little bargaiig power of the seller). The ituitio behid the first coditio is that little dispersio at the bottom implies that all parties are geuie cotestats who would be active ret-seekers also i the absece of a secodary market. Corollary 10 I the N-party ret-seekig cotest, for all N, V 1, ad V 2 <V 1, there is a o-zero mass of parameters (P,V 3,...,V N ) i the eighborhood of P = V 2 ad V i = V 2 for all i = 3, 4,...,N, for which total social cost is reduced by itroducig a secodary market: D + M <D + M. 5 Coclusios New dimesios of the ret-seekig problem emerge whe ret-seekers have differet valuatios of the sought-after ret. Ret-seekig cotests may serve as valuable mechaisms to force parties to reveal their prefereces through ivestmet choices. High-valuig parties will fight more aggressively ad will thus have higher probabilities to appropriate the ret. I this sese, ret-seekig cotests may play a valuable allocative role. Give the mechaics of ret-seekig cotests, however, this allocative role is imperfect ad, ulike i all-pay auctios, misallocatio remais possible. I this paper, we have studied the effect of ex post reallocatio of the ret o the parties icetives ad total ret-dissipatio. Accordig to the Coase theorem, ex post cotractual