Attorneys' Compensation in Litigation with Bilateral Delegation

Transcription

1 Attorneys' Compensaton n Ltgaton wth Blateral Delegaton by Kyung Hwan Bak * Department of Economcs, Sungkyunkwan Unversty, Seoul , South Korea and Department of Economcs, Vrgna Polytechnc Insttute and State Unversty, Blacksburg, VA 24061, U. S. A. Revsed October 2005 Abstract We consder ltgaton between a plantff and a defendant n whch each player hres an attorney who expends hs effort on her behalf. We examne the equlbrum fxed fees and contngent fees for the attorneys n two legal systems: the system wth the nonnegatve-fxed-fee constrant and the system wth the contngent-fee cap. We show that the fxed fees are always zero n the system wth the nonnegatve-fxed-fee constrant, and the contngent fees are always equal to the cap n the system wth the contngent-fee cap. We also examne the equlbrum expected payoffs for the attorneys and the payoffs for the players n the two systems. By comparng these expected payoffs, we show that the attorneys prefer the system wth the nonnegatve-fxed-fee constrant, whle the players prefer the system wth the contngent-fee cap. Keywords: Fxed fee; Contngent fee; Ltgaton; Contest; Delegaton JEL classfcaton: K41; D72 * I am grateful to Evan Osborne, Amy Bak, Iljoong Km, In-Gyu Km, Va-Lam Mu, Tm Perr, two anonymous referees, and semnar partcpants at Sungkyunkwan Unversty for ther helpful comments and suggestons. Earler versons of ths paper were presented at the 77th Annual Conference of the Western Economc Assocaton Internatonal, Seattle, WA, June 2002, and the 2002 Annual Conference of the Korean Law and Economcs Assocaton, Seoul, Korea, ovember Ths work was supported by the Korea Research Foundaton under Grant o. KRF B Tel.: ; fax: E-mal address: khbak@vt.edu (K.H. Bak).

2 1 1. Introducton The compensaton structure whch comprses a fxed fee and a contngent fee, s the standard form of contract between clents and attorneys n personal njury and medcal malpractce 1 ltgaton n the Unted States. A fxed fee for an attorney s the fee whch s pad to hm regardless of the outcome of the lawsut, and hs contngent fee s the fee whch s pad to hm only f he wns the lawsut. A contngent fee s set as a fxed percentage or fracton of the clent's recovery. Ths compensaton structure s attractve to clents because t s based on the observables and gves them lttle fnancal rsk of partcpatng n a lawsut. Many scholars have studed compensaton for attorneys or related ssues n dfferent contexts. Examples nclude Danzon (1983), Dana and Sper (1993), Gravelle and Waterson (1993), Rubnfeld and Scotchmer (1993), Mcel (1994), Hay (1996, 1997), Emons (2000), and Santore and Vard (2001). What are the equlbrum or optmal fee arrangements for attorneys? Do contngent fees promote excessve ltgaton? Are attorneys pad more than what they should be pad? Addressng these mportant questons, they obtan many nterestng results. For example, Rubnfeld and Scotchmer (1993) show that a clent wth a hgh-qualty case sgnals that her case s hgh qualty by her wllngness to pay a relatvely hgh fxed fee and a relatvely low contngency percentage, and also show that a hgh-qualty attorney sgnals hs qualty by hs wllngness to take a relatvely low fxed fee and a relatvely hgh contngency percentage. Santore and Vard (2001) show that the nonnegatvty constrant on fxed fees can create economc rents for attorneys. The purpose of ths paper s to consder ltgaton between a plantff and a defendant n whch each "player" hres an attorney who expends hs effort on her behalf focusng on the equlbrum fxed fees and contngent fees for the attorneys. The novelty of ths paper s that, unlke the prevous papers, we examne the equlbrum fxed fees and contngent fees for the attorneys by modelng the ltgaton as the two-player contest wth blateral delegaton. 2 A contest s defned as a stuaton n whch players compete wth one another by expendng rreversble effort to wn a prze. Ltgaton fts ths defnton very well. In ltgaton, a plantff and a defendant compete aganst each other; both partes expend rreversble ltgaton effort whch

3 2 nfluences the outcome of the lawsut; the wnner wns a prze, whch s equal to her valuaton for wnnng the lawsut. Ths ndcates that we can consder ltgaton as a contest. Among others, an mportant advantage we get by modelng ltgaton as a contest s that t allows us to look at the strategc aspects of ltgaton, especally those of the attorneys' compensaton schemes. The basc model conssts of the followng. The players are rsk-neutral, and have the same valuaton for wnnng the lawsut. They bear ther own legal costs regardless of the outcome of the 3 lawsut. The attorneys are rsk-neutral. They have the same nonnegatve reservaton wage, and have equal ablty for the lawsut. We set up the followng two-stage game. In the frst stage, each player hres an attorney and wrtes a contract wth hm. The contract specfes how much the attorney wll be pad f he wns the lawsut and how much f he loses t and thus t sets the attorney's fxed fee and hs contngent fee. The contract satsfes the attorney's partcpaton constrant based on the reservaton wage. Then the players smultaneously announce the contracts wrtten ndependently. In the second stage, after knowng both contracts, the attorneys choose ther effort levels smultaneously and ndependently. At the end of the second stage, the wnner s determned and each player pays compensaton to her attorney accordng to the contract wrtten n the frst stage. We examne the equlbrum fxed fees and contngent fees for the attorneys n two legal systems: the system wth the nonnegatve-fxed-fee constrant and the system wth the contngentfee cap. We also examne the attorneys' effort levels, ther expected payoffs, and the players' expected payoffs n equlbrum. The frst legal system s defned as the basc model plus the 4 constrant that fxed fees should be nonnegatve. The second legal system s defned as the basc model plus the exogenously gven cap on contngent fees. 5 In the case of the legal system wth the nonnegatve-fxed-fee constrant, we show that each player chooses zero fxed fees for her attorney regardless of the sze of the reservaton wage. We explan ths as follows. By choosng zero fxed fees, each player can make her attorney's contngent fee as hgh as possble, so that she can most strongly motvate her attorney to wn the lawsut. Another nterestng result s that, when the reservaton wage s low, each attorney's equlbrum

4 3 expected payoff s greater than the reservaton wage, meanng that he s "pad" more than what he should be "pad." Ths gap between the attorney's equlbrum expected payoff and the reservaton wage s the economc rent for the attorney. We argue that ths economc rent s created due to the players' strategc decsons on ther attorneys' compensaton. In the case of the legal system wth the contngent-fee cap, we show the followng results. Frst, the attorneys' equlbrum fxed fees are negatve or zero or postve dependng on the sze of both the reservaton wage and the contngent-fee cap and ther equlbrum contngent fees are always equal to the cap, regardless of the sze of the reservaton wage or that of the contngent-fee cap. Second, the equlbrum expected payoffs for the attorneys are always equal to the reservaton wage n other words, the economc rents for the attorneys never exst. Fnally, other thngs beng equal, as the contngent-fee cap ncreases, each player's equlbrum expected payoff decreases whle the attorneys' equlbrum expected payoffs reman unchanged. By comparng the attorneys' equlbrum expected payoffs n the two legal systems, we argue that the attorneys prefer the system wth the nonnegatve-fxed-fee constrant to the system wth the contngent-fee cap. In addton, by comparng the players' equlbrum expected payoffs n the two systems, we show that the players prefer the system wth the contngent-fee cap. Farmer and Pecorno (1999) and Hrshlefer and Osborne (2001) model ltgaton as contests, and address mportant ssues n law and economcs. However, nether of the papers consders delegaton by attorneys. Farmer and Pecorno (1999) consder three-stage games n whch the plantff and the defendant sequentally decdes whether or not to "partcpate" n the lawsut, and then they compete by expendng rreversble effort to wn the lawsut. They examne the relatonshp between case qualty, legal expendture, and legal technology, under both the Amercan rule and the Englsh rule. Hrshlefer and Osborne (2001) frst propose the ltgaton success functon whch satsfes desrable features that a satsfactory ltgaton success functon should dsplay. Then, usng the ltgaton success functon, they set up the smultaneous-move game and the sequental-move game wth the plantff as the leader, and examne the ltgaton efforts, proportonate success, and values of the lawsut on each sde, n the two games.

5 4 The paper proceeds as follows. Secton 2 develops the basc model and sets up the twostage game. In Secton 3, we consder the legal system wth the nonnegatve-fxed-fee constrant. In Secton 4, we consder the legal system wth the contngent-fee cap. Secton 5 compares the two legal systems n several respects, and hghlghts ther dfferences and smlartes. Fnally, Secton 6 dscusses the legal system wth both the nonnegatve-fxed-fee constrant and the contngent-fee cap, and offers our conclusons. 2. The basc model Consder a lawsut between a plantff and a defendant. For concse exposton, let us call the plantff player 1 and the defendant player 2. If player 1 wns the lawsut, she receves v dollars from player 2. If player 2 wns the lawsut, no money changes hands. Snce the players' valuaton for wnnng the lawsut s v dollars, ths ltgaton can be modeled as the contest n whch the two players each want to wn the prze of legal costs, regardless of the outcome of the lawsut. 6 v dollars. The players are rsk-neutral, and bear ther own Each player hres an attorney who expends hs effort to wn the lawsut on her behalf. Each player desgns her attorney's compensaton scheme: Player chooses W and L. Compensaton of W s pad to attorney f he wns the lawsut, and L f he loses t. Let W œ α v and let L œ " v, where 0 α 1 and α ". Then, snce v s exogenously gven, player desgns her attorney's compensaton scheme by choosng the values of α and ". In ths compensaton structure, " v represents attorney 's fxed fee whch s pad to hm regardless of the outcome of the lawsut, whle ( α ") v s attorney 's contngent fee whch s pad only f he wns the lawsut. The attorneys are rsk-neutral and have a common reservaton wage of R, where R s nonnegatve and s much less than v. Hence, f attorney sgns up for player, hs expected payoff must be greater than or equal to the reservaton wage, gven the compensaton scheme desgned by player. If hs expected payoff falls short of the reservaton wage, attorney prefers not to work for player and accepts alternatve employment nstead.

6 5 We formally consder the followng two-stage game. In the frst stage, each player hres an attorney and wrtes a contract wth hm n other words, player desgns and offers attorney a compensaton scheme whch attorney accepts. The contract specfes how much the attorney wll be pad f he wns the lawsut and how much f he loses t. Then the players smultaneously announce the contracts wrtten ndependently that s, player 1 announces publcly the values of α 1 and " 1, and player 2 announces publcly the values of α 2 and " 2. In the second stage, after knowng both contracts, the attorneys choose ther effort levels smultaneously and ndependently. At the end of the second stage, the wnner s determned and each player pays compensaton to her attorney accordng to the contract wrtten n the frst stage. In the second stage of the game, the attorneys compete wth each other by expendng rreversble effort to wn the lawsut. Let x represent the effort level expended by attorney. Effort levels are nonnegatve and are measured n monetary unts. Let p1( x 1, x2) denote the probablty that attorney 1 wns the lawsut when the attorneys' effort levels are x and x. The probablty-of- wnnng functon for attorney 1 s gven by: 1 2 p ( x, x ) œ x Î( x x ) for x x 0 (1) Î2 for x x œ Let 1 represent the expected payoff for attorney. Then the payoff functon for attorney 1 s œ ( W x ) p ( x, x ) ( L x )(1 p ( x, x )) (2) œ " v ( α " ) vp ( x, x ) x Smlarly, the payoff functon for attorney 2 s œ α v ( α " ) vp ( x, x ) x. (3) ext, consder the players' expected payoffs computed n the frst stage of the game when player beleves that attorney 1 wll expend an effort level of x 1 and attorney 2 wll expend an effort level of n the second stage. Gven player 's contract, (, ), f her attorney wns the lawsut n x W L 2

7 6 the second stage, player 's net payoff wll be v W; otherwse, player wll gan nothng, but should pay L to her attorney. Let G represent the expected payoff for player. Then the payoff functon for player 1 s G œ ( v W ) p ( x, x ) ( L )(1 p ( x, x )) (4) œ " v (1 α " ) vp ( x, x ) Smlarly, the payoff functon for player 2 s G œ (1 α ) v (1 α " ) vp ( x, x ). (5) Fnally, we assume that all of the above s common knowledge among the players and attorneys. We employ subgame-perfect equlbrum as the soluton concept The second stage of the game To solve for a subgame-perfect equlbrum of the game, we work backward. We begn by consderng the second stage n whch, after knowng the contracts chosen n the frst stage, ( α 1, " 1) and ( α 2, " 2), attorney seeks to maxmze hs expected payoff over hs effort level, gven the other attorney's effort level. Gven a postve effort level of attorney 2, the frst-order condton for maxmzng attorney 1's expected payoff,, yelds 1 1 ( α " ) v( `p ( x, x ) Î`x ) œ 1. (6) Gven a postve effort level of attorney 1, the frst-order condton for maxmzng attorney 2's expected payoff,, yelds 1 2 ( α " ) v( `p ( x, x ) Î`x ) œ 1. (7) Attorney 's payoff functon s strctly concave n hs effort level. Thus the second-order condton for maxmzng 1 s satsfed, and attorney 's best response s unque.

8 7 We obtan a unque ash equlbrum of the second-stage subgame usng the attorneys' 1 2 reacton functons, whch are derved from condtons (6) and (7). Let ( x, x ) denote the ash equlbrum. Lemma 1. At the ash equlbrum of the second-stage subgame, the effort levels of the attorneys are x œ ( α " ) ( α " ) v Î{( α " ) ( α " )} and x œ ( α " )( α " ) vî{( α " ) ( α " )} The frst stage of the game Consder now the frst stage n whch the players choose ther contracts, ( α 1, " 1) and ( α2, " 2 ), smultaneously and ndependently. The players have perfect foresght about the second-stage competton more specfcally, the ash equlbrum of each second-stage subgame. Let p ( x, x 2 ) be the probablty that attorney 1 wns the lawsut at the ash equlbrum of the second-stage subgame, gven contracts, ( α 1, " 1) and ( α 2, " 2). Then, usng payoff functons (4) and (5), we obtan the players' payoff functons whch take nto account the ash equlbrum of the second-stage subgame: 1 1 and G œ " v (1 α " ) vp ( x, x ) (8) G œ (1 α ) v (1 α " ) vp ( x, x ), (9) where p ( x, x ) œ ( α " )/{( α " ) ( α " )}, whch are obtaned usng functon (1) and Lemma When choosng a contract for her attorney, each player should consder her attorney's partcpaton constrant. Havng perfect foresght about the ash equlbrum of each second-stage subgame, the players and attorneys can compute, n the frst stage, the attorneys' expected payoffs. Usng payoff functons (2) and (3), we obtan the attorneys' payoff functons whch are assocated wth the ash equlbrum of the second-stage subgame, gven contracts, (, ) and (, ): α " α "

9 8 and 1 œ " v ( α " ) vp ( x, x ) x œ α v ( α " ) vp ( x, x ) x. Attorney 's partcpaton constrant s then R. ow player faces the followng constraned-maxmzaton problem: 1 Max G (10) α, " subject to 1 R. That s, takng the opponent's contract as gven, player seeks to maxmze her expected payoff over ( α, "), subject to her attorney's partcpaton constrant. By dong so, she obtans her best response to the gven contract of her opponent. We wll frst look at the unconstraned-maxmzaton problem: Maxmze G over ( α, " ). Then, we wll look at each attorney's partcpaton constrant. Maxmzng player 's expected payoff n the absence of her attorney's partcpaton constrant, we obtan Lemma 2. Lemma 2. ( a) Gven player j's contract, ( α, "), and gven α, player 's expected payoff s always j j decreasng n ": In terms of the symbols, we have `G / `" 0. 8 ( b) Gven player j's contract, ( α, "), and gven ", player 's expected payoff s maxmzed at α œ " k, where j j k œ α j " j α j " j α j " j ( ) {( ) ( )}. Part (a) can be explaned as follows. As " decreases, attorney 's contngent fee ncreases. A larger contngent fee, n turn, gves attorney more ncentves to wn the lawsut and makes hm exert more effort. A hgher effort level of attorney then yelds a hgher probablty that attorney wns the lawsut n second-stage equlbrum. Therefore, a hgher probablty of wnnng and less compensaton n the case of losng lead to an ncrease n player 's expected payoff.

10 9 In part (b), k s determned by the gven contract of player j, and s treated as a constant. It s easy to see that k s postve but less than a half. ote that k s ndependent of the players' valuaton for wnnng the lawsut, v, and the reservaton wage, R. Then, n the "α-space of Fgure 1, the graph of α œ " k s a straght lne wth a vertcal ntercept of k and a slope of unty. ext, we look at attorney 's partcpaton constrant. Consder frst the graph of 1 œ R n the "α -space, whch s represented as ( α " ) 3 v œ ( R " v){( α " ) ( α " )} 2. (11) j j 9 Let us call t attorney 's partcpaton constrant curve. Snce R s constant, we have ( `1 Î`α) dα ( `1 Î`" ) d" œ 0 along the curve. Ths equaton yelds 2 2 j j j j j j dα Îd" œ ( α " ) {3( α " ) ( α " )} Î( α " ) {( α " ) 3( α " )}. (12) It s mmedate from expresson (12) that dα Îd" s negatve, whch means that attorney 's partcpaton constrant curve slopes downward from left to rght n the "α -space. The curve meets the 45 o lne when " œ α œ Rv Î, and has a vertcal ntercept of m, where m satsfes 3 j j 2 mvœ Rm { ( α ")}. (13) It s easy to see that the vertcal ntercept, m, s equal to zero when R œ 0, and ncreases n R. Fgure 1 llustrates attorney 's partcpaton constrant curve. Player 's contracts whch satsfy her attorney's partcpaton constrant, 1 R, le on or above her attorney's partcpaton constrant curve. Thus they are located n the shaded area of Fgure 1. Lemma 3 descrbes a useful property of the curve. Lemma 3. Gven player j's contract, ( α, "), as " decreases along attorney 's partcpaton j j constrant curve, player 's expected payoff, G, ncreases f and only f ( α " ) ( α " ) 2( α " )( α " ). j j j j

11 10 We obtan Lemma 3 by utlzng the fact that dg Îd" œ ( `G Î`α)( dα Îd" ) `G Î`" along the curve, where dα Îd" s gven by expresson (12). It follows from Lemma 3 that, gven player j 's contract, ( α, "), where ( α " ) 1, as " j j j j decreases along attorney 's partcpaton constrant curve, player 's expected payoff, G, ncreases as long as ( α ") s less than unty along the curve. Ths mples that, as " decreases along attorney 's partcpaton constrant curve n the frst quadrant of the payoff always ncreases. "α -space, player 's expected 3. The nonnegatve-fxed-fee constrant In ths secton, we consder the legal system wth the nonnegatve-fxed-fee constrant the constrant that fxed fees should be nonnegatve. More specfcally, we consder the model whch conssts of the basc model plus the nonnegatve-fxed-fee constrant. We frst obtan the equlbrum fxed fees and contngent fees for the attorneys, and then examne the attorneys' effort levels, ther expected payoffs, and the players' expected payoffs n equlbrum. To obtan the equlbrum contracts chosen by the players n the frst stage, we begn by solvng constraned-maxmzaton problem (10) subject to the addtonal constrant of " The best response of each player Gven player j's contract, ( α, "), player 's best response to ( α, ") s defned as a contract j j j j whch maxmzes her expected payoff, G, subject to attorney 's partcpaton constrant, 1 R, and the nonnegatve-fxed-fee constrant, 0. Denote t by ( α, " ). Usng Fgure 1 and b b b " b Lemmas 2 and 3, we obtan: α œ max{ k, m} and " œ 0. Because m ncreases n Rwhle k s ndependent of R, we have two dfferent cases dependng on the sze of the reservaton wage, R. w Let R be the value of the reservaton wage at whch m s equal to k. From equaton (13), we obtan then R w œ k 3 vî{( α " ) 2 ( α " )}. Lemma 4 descrbes the two cases. j j j j

12 11 Lemma 4. ( a) In the case where 0 Ÿ R R w, attorney 's partcpaton constrant s not bndng: b b j j k m. Player 's best response to player j's contract, ( α, "), s then: ( α, " ) œ ( k, 0). ( b) In w the case where R R, attorney 's partcpaton constrant s bndng: k Ÿ m. Player 's best b b response s then: ( α, " ) œ ( m, 0). Lemma 4 says that, gven player j's contract, player 's optmal choce of the fxed fee for her attorney s 0. Ths can be explaned as follows. Wthout the nonnegatve-fxed-fee constrant, her 10 optmal choce of the fxed fee would be negatve (see Lemmas 2 and 3). But she cannot choose a negatve fxed fee due to the constrant. Because she wants to make her attorney's contngent fee as hgh as possble n order to most strongly motvate hm to wn the lawsut, faced wth the constrant, she chooses 0 for the fxed fee. Part (a) says that, when the reservaton wage s low, player 's best response to a gven contract of her opponent s just the contract whch maxmzes her expected payoff, G, n the absence of her attorney's partcpaton constrant. It also says that player chooses a contract whch gves attorney an expected payoff hgher than hs reservaton wage. The explanaton for ths follows. Attorney wll compete aganst attorney j to wn the lawsut n the second stage. Player wants to nduce attorney to exert the "optmal" effort the optmal effort for player by choosng the best contract, gven player j's contract, ( α j, " j). In ths case, the best contract that s, player 's best response happens to yeld attorney 's expected payoff greater than hs reservaton wage, because hs reservaton wage s low. Part (b) says that, when the reservaton wage s hgh, the contract whch solves the maxmzaton problem wthout attorney 's partcpaton constrant yelds attorney 's expected payoff less than hs reservaton wage. Hence, to take care of her attorney's partcpaton constrant, player chooses a contract whch les on attorney 's partcpaton constrant curve The equlbrum contracts of the players

13 12 * * Let ( α, " ) represent player 's contract whch s specfed n the subgame-perfect 1 * 2 * equlbrum of the two-stage game. We frst obtan from Lemma 4 that " œ " * * œ 0. In order to obtan α and α, we utlze the players' reacton curves n the α α -space. It follows from Lemma * j 1 2 b o o 4 that, gven " œ 0, player 's reacton curve n the α α -space s the graph of α œ max{ k, m }, o 2 1/2 o o 3 o 2 j j j j where k œ α ( α α) and m satsfes ( m ) v œ R( m α), whch are based on Lemma 2 and equaton (13), respectvely. The ntersecton of these two reacton curves determnes α * * 1 2 and α. * * 1 2 k o Fgure 2 s useful n obtanng α and α. For concse exposton, we draw the graphs of o b o o and m separately rather than draw the graph of α œ max{ k, m }, whch s player 's reacton curve. Lemma 5 descrbes propertes of the graphs n Fgure 2. o Lemma 5. ( a) k s ncreasng n α at a decreasng rate. ( b) m s ncreasng n α at a j o o 1 2 decreasng rate. ( c) The ntersecton of the graphs of k and k occurs at pont Q on the 45 lne. o o 1 2 ( d) The ntersecton of the graphs of m and m always occurs on the 45 lne. ( e) As the o o 1 2 reservaton wage, R, ncreases, the graph of m shfts to the rght whle the graph of m shfts upward. o o j o o 2 Lemma 5 says that, as the reservaton wage ncreases, the ntersecton of the graphs of m o 1 o and m moves up along the 45 lne also called straght lne OS. However, note that, as the o o o 1 2 reservaton wage changes, the graphs of k and k reman unchanged because k s ndependent of the reservaton wage. * * 1 2 Usng Lemma 5 and Fgure 2, we obtan α and α, and report them n Lemma 6. o o Lemma 6. ( a) If the ntersecton of the graphs of m 1 and m 2 les on lne segment OQ, or * * * * equvalently, f 0 Ÿ R v Î12, then ( α, α ) occurs at pont Q: ( α, α ) œ (1Î3, 1Î3). ( b) If the o o ntersecton of the graphs of m 1 and m 2 les on lne segment QS, or equvalently, f v/12 Ÿ R v /4, * * * * 11 then ( α1, α2) occurs at ths very ntersecton: ( α1, α2) œ (4 RÎv, 4 RÎv).

14 13 Lemma 6 mples that there are two types of the equlbrum-contracts pars: the pars of contracts at whch nether of the attorneys' partcpaton constrants s bndng, and the pars of contracts at whch both attorneys' partcpaton constrants are bndng. The frst type s assocated wth part (a) of Lemma 6, and the second type s assocated wth part (b). Because we have dentcal players and dentcal attorneys, the players choose the same * * * * contract n equlbrum: α œ α and " œ " œ 0. Therefore, the equlbrum contract of player * * * * specfes that attorney earns W œ α v f he wns the lawsut, and L œ " v œ 0 f he loses t. Ths means that attorney 's equlbrum fxed fee s zero, and hs equlbrum contngent fee s v. α * 3.3. The fxed fees, contngent fees, and expected payoffs Let x * represent the effort level of attorney whch s specfed n the subgame-perfect equlbrum. Let and G represent the equlbrum expected payoff for attorney and that for 1 * * player, respectvely. Then, based on Lemma 6 and the result that expressons (1) through (5) and Lemma 1, we obtan Proposton 1. 1 * 2 * " œ " œ 0, and usng * * 1 2 Proposton 1. ( a) In the case where 0 Ÿ R v Î12, the attorneys' fxed fees are: " v œ " v œ 0. * * * * Ther contngent fees are: α v œ α v œ v Î3. Ther effort levels are: x œ x œ vî12. Ther * * * * expected payoffs are: 1 œ 1 œ vî12 R. The players' expected payoffs are: G œ G œ vî3. * * 1 2 ( b) In the case where vî12 Ÿ R v Î4, the attorneys' fxed fees are: " v œ " v œ 0. Ther * * * * contngent fees are: α v œ α v œ 4 R. Ther effort levels are: x œ x œ R. Ther expected payoffs * * * * are: 1 œ 1 œ R. The players' expected payoffs are: G œ G œ ( v 4 R) Î Proposton 1 says that, n the subgame-perfect equlbrum, each player chooses zero fxed fees for her attorney, regardless of the sze of the reservaton wage mplyng that each player pays nothng to her attorney f he loses the lawsut. Ths s because, by choosng zero fxed fees, each player can make her attorney's contngent fee as hgh as possble, and therefore most strongly

15 14 motvate her attorney to wn the lawsut. Proposton 1 also says that, when the reservaton wage s low, each attorney's contngent fee s equal to one thrd of the players' valuaton for wnnng the lawsut. 12 The players have the same valuaton for wnnng the lawsut. The attorneys have the same reservaton wage and have equal ablty for the lawsut. Therefore, n equlbrum, the players choose the same contngent fees for ther attorneys. Ths, n turn, leads to the same effort level expended by the attorneys, the same expected payoff for the attorneys, and the same expected payoff for the players. Ths s natural because the attorneys are motvated equally by the same contngent fee to wn the lawsut. Another nterestng fndng n Proposton 1 s that, when the reservaton wage s low, the 13 attorneys' equlbrum expected payoffs are greater than the reservaton wage. Ths gap consttutng the economc rents for the attorneys s totally due to the players' strategc decsons on ther attorneys' compensaton. Competng aganst the other player, each player needs to "overcompensate" her attorney n order to motvate hm to work harder. In ths case, whle the players look benevolent, they are actually pursung ther self-nterest. It s obvous that the economc rents for the attorneys exst due to the nonnegatve-fxed-fee constrant. 14 Fnally, when the reservaton wage s hgh, the players must choose the contngent fees that guarantee ther attorneys the reservaton wage n order to hre them. Ths means that the attorneys' equlbrum expected payoffs are equal to the reservaton wage Comparatve statcs We examne the effects of ncreasng the reservaton wage on the attorneys' contngent fees, ther effort levels, ther expected payoffs, and the players' expected payoffs n equlbrum. Usng Proposton 1, we obtan Proposton 2. Proposton 2. ( a) As the reservaton wage ncreases from zero, the attorneys' contngent fees, ther effort levels, ther expected payoffs, and the players' expected payoffs reman unchanged. Ths

16 15 s true untl the reservaton wage s equal to v/12. ( b ) As the reservaton wage ncreases beyond v Î12, the attorneys' contngent fees, ther effort levels, and ther expected payoffs ncrease whle the players' expected payoffs decrease. Part (a) comes mmedately from the fact that, for 0 Ÿ R vî12, nether of the attorneys' partcpaton constrants s bndng n equlbrum, and thus the equlbrum contracts of the players * * * * are ndependent of the reservaton wage: ( α, " ) œ (1Î3, 0) and ( α, " ) œ (1Î3, 0). As the reservaton wage ncreases beyond vî12, the attorneys' contngent fees ncrease. Ths can be explaned as follows. Frst, for vî12 Ÿ R vî4, both attorneys' partcpaton constrants are bndng, and thus gven the equlbrum fxed fees of zero, the players must offer ther attorneys hgher contngent fees n order to hre them as the reservaton wage ncreases. Second, when the opponent offers a hgher contngent fee to her attorney, each player has an ncentve to follow sut. Facng a more aggressve attorney of the opponent, each player must make her attorney more aggressve by ncreasng hs contngent fee. In part (b), larger contngent fees n turn gve the attorneys more ncentves to wn the lawsut and make them exert more effort. ote, however, that the probablty that each attorney wns the lawsut n second-stage equlbrum remans constant. Therefore, the same probablty of wnnng and larger contngent fees as well as constant fxed fees lead to smaller expected payoffs for the players. Because the attorneys' equlbrum expected payoffs are equal to the reservaton wage, they ncrease as the reservaton wage ncreases. 4. The contngent-fee cap In ths secton, we consder the legal system wth the contngent-fee cap that s, the model whch conssts of the basc model plus the contngent-fee cap. The cap s exogenously gven. We frst obtan the equlbrum fxed fees and contngent fees for the attorneys, and then examne the attorneys' effort levels, ther expected payoffs, and the players' expected payoffs n equlbrum.

17 16 Recall that attorney 's contngent fee s ( α ") v. Hence, a cap on attorney 's contngent fee means a cap on α ". Let ) represent the cap on α ", where 0 ) 1. To obtan the equlbrum contracts chosen by the players n the frst stage, we begn by solvng constranedmaxmzaton problem (10) subject to the addtonal constrant, α " Ÿ ) The best response of each player Gven player j's contract, ( α, "), player 's best response to ( α, ") s defned as a contract j j j j whch maxmzes her expected payoff, G, subject to attorney 's partcpaton constrant, 1 R, and attorney 's contngent-fee constrant, α " Ÿ ). Fgure 3 llustrates ths new constraned- maxmzaton problem. In the fgure, straght lne AH s the graph of α " œ ). We call t attorney 's contngent-fee constrant lne. Player 's contracts whch satsfy her attorney's partcpaton constrant, R, and her attorney's contngent-fee constrant, α " ), le n the 1 shaded area of Fgure 3. Usng expressons (8) and (9), we fnd that, as 's contngent-fee constrant lne, player 's expected payoff, G " Ÿ decreases along attorney, ncreases. Usng Lemma 3 and attorney 's contngent-fee constrant, α " Ÿ ) 1, we fnd that, as " decreases along attorney 's partcpaton constrant curve, player 's expected payoff, G, ncreases. It then follows from these fndngs and part (a) of Lemma 2 that, gven player j's contract, ( α, "), player 's best response to ( α j, " j) occurs at pont A that s, at the ntersecton of attorney 's partcpaton constrant curve and attorney 's contngent-fee constrant lne. j j 4.2. The equlbrum contracts of the players ** ** Let ( α, " ) represent player 's contract whch s specfed n the subgame-perfect equlbrum of the new two-stage game. Recall from Secton 4.1 that, gven player j's contract, ( α j, "), player 's best response to ( α, ") occurs at the ntersecton of attorney 's partcpaton j j j constrant curve and attorney 's contngent-fee constrant lne. Attorney 's partcpaton constrant curve s gven by expresson (11), and attorney 's contngent-fee constrant lne s gven by

18 17 α " œ ). Then, the equlbrum contracts of the players, ( α, " ) and ( α, " ), must satsfy the followng four equatons smultaneously: ** ** ** ** and ** ** 1 1 α " œ ) ** 1 1 ** 3 1 ** 1 ** 1 ** 2 ** 2 ** 2 ( α " ) v œ ( R " v){( α " ) ( α " )} ** ** 2 2 α " œ ) ** 2 2 ** 3 2 ** 2 ** 2 ** 1 ** 1 ** 2 ( α " ) v œ ( R " v){( α " ) ( α " )}. ** ** 2 2 The frst two equatons defne player 1's best response to player 2's equlbrum contract, ( α, " ), whle the last two equatons defne player 2's best response to player 1's equlbrum contract, ( α ** 1, " ** 1 ). Lemma 7 reports the players' equlbrum contracts. Lemma 7. The equlbrum contracts of the players are Rv Î ) Î4). ** ** ** ** ( α, " ) œ ( α, " ) œ ( RÎv 3) Î4, ote that each player's equlbrum contract always les on her attorney's partcpaton constrant curve that s, both attorneys' partcpaton constrants are bndng n equlbrum, regardless of the sze of the reservaton wage or that of the contngent-fee cap. ote also that each ** ** attorney's contngent-fee constrant s always bndng n equlbrum: α " œ ) The fxed fees, contngent fees, and expected payoffs Let x ** represent the effort level of attorney whch s specfed n the subgame-perfect equlbrum. Let and G represent the equlbrum expected payoff for attorney and that for 1 ** ** player, respectvely. Then, from Lemma 7, and usng expressons (1) through (5) and Lemma 1, we obtan Proposton 3.

19 18 ** ** ** ** Proposton 3. The attorneys' contngent fees are: ( α " ) v œ ( α " ) v œ ) v. Ther fxed ** ** ** ** fees are: " v œ " v œ R ) vî4. Ther effort levels are: x œ x œ ) v/4. Ther expected ** ** ** ** payoffs are: 1 œ 1 œ R. The players' expected payoffs are: G œ G œ (2 ) ) vî4 R Frst of all, as n the system wth the nonnegatve-fxed-fee constrant, because the players have the same valuaton for wnnng the lawsut and because the attorneys have the same reservaton wage and have equal ablty for the lawsut, the players choose the same fxed fee and the same contngent fee for ther attorneys n equlbrum, regardless of the sze of the contngentfee cap or that of the reservaton wage, R, where 0 Ÿ R (2 ) vî4. The same contngent fee ) 15 for the attorneys n turn lead to the same effort level expended by the attorneys, the same expected payoff for the attorneys, and the same expected payoff for the players. Proposton 3 says that the equlbrum contngent fees are equal to the cap, regardless of the 16 sze of the reservaton wage or that of the contngent-fee cap. It also says that the equlbrum fxed fees are negatve for 0 Ÿ R ) vî4; they are zero for R œ ) vî4; they are postve for 17 ) vî4 R (2 ) ) vî4. In order to most strongly motvate her attorney to wn the lawsut, each player frst chooses her attorney's contngent fee as hgh as possble, and then chooses hs fxed fee (wth whch she cannot motvate hm) so that both the contngent fee and the fxed fee can yeld the attorney's equlbrum expected payoff equal to the reservaton wage. Ths means that the economc rents for the attorneys never exst. When the reservaton wage s low, the equlbrum fxed fees are negatve. In ths case, attorney s requred to pay the absolute value of v to player regardless of the outcome of the lawsut. To put t dfferently, each attorney s requred to purchase from hs employer (or hs clent) by payng the "employment fee" both the rght to compete n the ltgaton and the rght to share the prze of v dollars wth hs employer when he wns the lawsut. Another way to say ths s that each attorney s requred to post the "up-front performance bond" to secure the rghts. " ** 4.4. Comparatve statcs

20 19 We examne the effects of ncreasng the reservaton wage on the attorneys' fxed fees, ther contngent fees, ther effort levels, ther expected payoffs, and the players' expected payoffs n equlbrum. We also examne the effects of ncreasng the contngent-fee cap. Usng Proposton 3, we obtan Proposton 4. Proposton 4. ( a) As the reservaton wage ncreases, the attorneys' contngent fees and ther effort levels reman unchanged; the attorneys' fxed fees and ther expected payoffs ncrease; the players' expected payoffs decrease. ( b ) As the contngent-fee cap ncreases or, equvalently, as ) ncreases the attorneys' contngent fees and ther effort levels ncrease; the attorneys' fxed fees and the players' expected payoffs decrease; the attorneys' expected payoffs reman unchanged. As the reservaton wage ncreases, the attorneys' equlbrum contngent fees reman unchanged because the gven contngent-fee constrants are always bndng n equlbrum. The players must then ncrease the fxed fees to guarantee the attorneys the ncreased reservaton wage. Constant contngent fees result n constant effort levels expended by the attorneys whch, n turn, make constant the probablty that each attorney wns the lawsut n second-stage equlbrum. ext, constant contngent fees, constant probabltes of wnnng, and larger fxed fees lead to smaller expected payoffs for the players. Fnally, because the attorneys' equlbrum expected payoffs are equal to the reservaton wage, they ncrease as the reservaton wage ncreases. As the contngent-fee cap ncreases, the attorneys' contngent fees ncrease because they are always equal to the cap n equlbrum. Hgher contngent fees, n turn, gve the attorneys more ncentves to wn the lawsut, and thus make them exert more effort. Gven the reservaton wage, as the contngent-fee cap ncreases, the attorneys' equlbrum expected payoffs reman unchanged because they are always equal to the reservaton wage; ther fxed fees decrease because ther contngent fees ncrease. To guarantee the attorneys the gven reservaton wage, less fxed fees are needed due to hgher contngent fees. Fnally, gven that the probablty that each attorney wns the lawsut n second-stage equlbrum remans unchanged, a decrease n the fxed fee tends to ncrease

21 20 each player's equlbrum expected payoff whle an ncrease n the contngent fee tends to decrease her expected payoff. The former s more than offset by the latter, so that each player's expected payoff decreases as the contngent-fee cap ncreases. 5. Comparng the two legal systems We have separately consdered the legal system wth the nonnegatve-fxed-fee constrant, and the system wth the contngent-fee cap. We now compare the two systems n several respects, and hghlght ther dfferences and smlartes. Frst, the attorneys' fxed fees are always zero wth the nonnegatve-fxed-fee constrant, whle they are negatve or zero or postve wth the contngent-fee cap dependng on the sze of both the reservaton wage and the contngent-fee cap. Second, n the system wth the nonnegatvefxed-fee constrant, the attorneys' contngent fees are one thrd of the players' valuaton for wnnng the lawsut when the reservaton wage s low; they are four tmes the reservaton wage when the reservaton wage s hgh. In the system wth the contngent-fee cap, the contngent fees are always equal to the cap, regardless of the sze of the reservaton wage or that of the contngent-fee cap. ote that the nonnegatve-fxed-fee constrants and the contngent-fee constrants are always bndng that s, the fxed fees are always the lowest possble ones n the system wth the nonnegatve-fxed-fee constrant, and the contngent fees are always the hghest possble ones n the system wth the contngent-fee cap. Thrd, n both systems, each attorney's effort level s one fourth of hs contngent fee, whch results from employng the smplest logt-form probablty-of-wnnng functons. Fourth, n the system wth the nonnegatve-fxed-fee constrant, each attorney's expected payoff s greater than hs reservaton wage when the reservaton wage s low; t s equal to the reservaton wage when the reservaton wage s hgh. By contrast, the payoff s always equal to the reservaton wage n the system wth the contngent-fee cap. In other words, the economc rents for the attorneys may exst n the system wth the nonnegatve-fxed-fee constrant, whereas they never exst n the system wth the contngent-fee cap. Fnally, each player's expected payoff s greater n

22 21 the system wth the contngent-fee cap than n the system wth the nonnegatve-fxed-fee constrant, regardless of the sze of the reservaton wage or the sze of the contngent-fee cap. Proposton 5 hghlghts these nterestng results. Proposton 5. ( a) The economc rents for the attorneys may exst n the system wth the nonnegatve-fxed-fee constrant, whereas they never exst n the system wth the contngent-fee cap. Ths mples that the attorneys prefer the system wth the nonnegatve-fxed-fee constrant to the system wth the contngent-fee cap. ( b) The players prefer the system wth the contngent-fee cap to the system wth the nonnegatve-fxed-fee constrant. Proposton 5 makes ntutve sense. Wth the nonnegatve-fxed-fee constrant, the players are prohbted from choosng negatve fxed fees, and thus may not be able to reduce fxed fees so that the attorneys' expected payoffs equal the reservaton wage. Ths s ndeed the case, when the reservaton wage s low. By contrast, wth the contngent-fee cap, the players beneft from two thngs. Frstly, due to the cap, the players are forced to reduce ther competton on contngent fee. Secondly, the players are allowed to choose negatve fxed fees, whch enables them to reduce ther attorneys' expected payoffs to the reservaton wage. Both contrbute to ncreasng the players' expected payoffs relatve to the system wth the nonnegatve-fxed-fee constrant. 6. Conclusons We have consdered ltgaton between a plantff and a defendant n whch each player hres an attorney who expends hs effort on her behalf. We have examned the equlbrum fxed fees and contngent fees for the attorneys n the two legal systems: the system wth the nonnegatve-fxed-fee constrant and the system wth the contngent-fee cap. We have also examned the attorneys' effort levels, ther expected payoffs, and the players' expected payoffs n equlbrum. We have consdered the legal systems whch have ether the nonnegatve-fxed-fee constrant or the contngent-fee cap, but not both. What happens n the system wth both the

23 22 nonnegatve-fxed-fee constrant and the contngent-fee cap? Frst, the attorneys' equlbrum fxed fees are zero or postve dependng on the sze of both the reservaton wage and the contngent- fee cap. Second, the contngent fees may be smaller than the cap. Ths occurs when the cap s relatvely hgh. Thrd, the attorneys' expected payoffs may be greater than the reservaton wage that s, the economc rents for the attorneys may exst. However, these rents are smaller than or equal to those n the system wth the nonnegatve-fxed-fee constrant. Fnally, the players' expected payoffs are greater than or equal to those n the system wth the nonnegatve-fxed-fee constrant, and are smaller than or equal to those n the system wth the contngent-fee cap dependng on the sze of both the reservaton wage and the contngent-fee cap. Based on these and the prevous results, we argue that, among the three legal systems, the attorneys prefer the system wth the nonnegatve-fxed-fee constrant whle the players prefer the system wth the contngent-fee cap. The hybrd system s ranked second for both the attorneys and the players. In Secton 3, we had the constrant that fxed fees should be nonnegatve n other words, we set zero as the lower bound of fxed fees. Instead, f we set a postve or negatve number as the lower bound, we wll obtan smlar qualtatve results for example, the equlbrum fxed fees wll be equal to the lower bound of fxed fees. However, wth a suffcently small negatve number as the lower bound, there wll be no economc rents for the attorneys n equlbrum, regardless of the sze of the reservaton wage. We have assumed that the players have the same valuaton for wnnng the lawsut. By consderng ltgaton n whch players have dfferent valuatons and attorneys have dfferent reservaton wages dependng on ther ablty for the ltgaton we may be able to address the queston of who hres whom. Other extensons of ths paper nclude a model whch ncorporates the possblty of settlement and a model whch explctly ntroduces the objectve merts of the case nto the ltgaton success functons. We leave all these consderatons for future research.

24 23 Footnotes 1. See, for example, Danzon (1983), Rubnfeld and Scotchmer (1993), and Santore and Vard (2001), for detals. 2. Many economsts have studed the theory of contests: Appelbaum and Katz (1987), Dxt (1987), Hllman and Rley (1989), Ellngsen (1991), tzan (1991), Bak and Shogren (1992), Baye et al. (1993), Che and Gale (1998), and Konrad (2000), to name a few. Bak and Km (1997),.. Warneryd (2000), and Konrad et al. (2004) study delegaton n contests. 3. Ths s the so-called Amercan rule of fee allocaton. Under the Englsh rule, the loser must pay the compensaton of the wnner's attorney as well as her own attorney's. 4. Ths s justfed by the fact that the Amercan Bar Assocaton Model Rules of Professonal Conduct requre that fxed fees n tort ltgaton should not be negatve (see Santore and Vard (2001)). 5. Ths can be justfed by the fact that many states n the Unted States have upper lmts on contngent fees more precsely, upper lmts on contngent-fee percentages for tort cases. See, for example, Danzon (1983), Rubnfeld and Scotchmer (1993), Hay (1996), and Emons (2000). 6. Imagne that, pror to tral, the judgment v s taken from the defendant and held n escrow. An alternatve way to model ths ltgaton s that the plantff s tryng to maxmze an expected gan at tral, whle the defendant s tryng to mnmze an expected loss. These two models, however, result n the same outcomes. 7. ote that, gven zero effort of hs opponent, each attorney's best response s to expend nfntesmal effort. 8. Throughout the paper, when we use and j at the same tme, we mean that Á j j j j j 9. ote that k s less than a half f and only f {( α " ) ( α " )} s less than 0.5 ( α "). The latter s proven easly. j j

25 A negatve fxed fee means that attorney s requred to pay the amount to player regardless of the outcome of the lawsut. To put t dfferently, attorney s requred to purchase from player, by payng the amount, both the rght to compete n the ltgaton and the rght to share the prze of v dollars wth player when he wns the lawsut. 11. We must assume that the reservaton wage s less than vî4. Otherwse that s, f R vî4 the players end up wth nonpostve expected payoffs, whch mples that the players each have no ncentve to hre an attorney at the begnnng (see Proposton 1). 12. In the Unted States, the contngent fee for the plantff's attorney s typcally 25 percent to 42 percent of the reward obtaned by the attorney. See, for example, Hay (1996). 13. In ths case, each player's best reply to the other player's equlbrum contract s the contract whch maxmzes her expected payoff n the absence of her attorney's partcpaton constrant. 14. Santore and Vard (2001) also show n a dfferent framework that the nonnegatve-fxed-fee constrant can create economc rents for attorneys, and argue that the constrant can be understood as a means of mantanng the attorneys' economc rents. 15. We must assume that the reservaton wage s less than (2 ) ) vî4. Otherwse, as we see n Proposton 3, the players end up wth nonpostve expected payoffs, whch mples that the players each have no ncentve to hre an attorney at the begnnng. 16. Recall that the cap on attorney 's contngent fee s assumed to be less than the players' valuaton for wnnng the lawsut: 0 ) 1. Recall also that the reservaton wage s assumed to be less than (2 ) ) vî When the fxed fees are postve, the players may be confronted by the moral hazard problem: Each attorney takes many cases, but expends no effort n each case. If ths problem occurs and the players have no way of dealng wth t, then the players each have no ncentve to hre an attorney at the begnnng. I thank one of the referees for pontng out ths possblty.

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28 Fgure 1. Player s constraned-maxmzaton problem.