Reaearch Paper By Park, Sung-hoon And Myunghoon Lee

━━━━━━━━━━━━━━━ ─────────────── 經濟硏究 第26卷 第3號 韓國經濟通商學會 2008. 9. 133面～144面

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Bilateral Delegation in Contests with *1)

Contingent-Fee Contracts

Sung-Hoon Park** and Myunghoon Lee*** (접수일 : 2008. 4. 29 / 수정일 : 2008. 8. 5 / 게재확정일 : 2008. 8. 21)

Abstract

The primary goal of this paper is to show that the variation in risk aversion plays a key role in the bilateral-delegation contests. We first characterize the comparative statics of the two-period game in which the delegates of risk-averse players work on a pure contingent fee. We show that the more risk-averse player offers the larger contingent fee, thus raising his delegate’s effort level. We also show that the more risk-averse player has a higher-than-50 percent chance of winning. We next consider the case where the players recognize the delegates' non-negative reservation wages.

*

We are grateful for the valuable comments made by an anonymous referee.

** Research Fellow, Department of Industry and Economy(first author) / E-mail: [email protected] / ☎ +82-31-250-3552 / 󰂕 Gyeonggi Research Institute, 179 Pajang-dong, Jangan-gu Suwon, Gyeonggi Province, Republic of Korea *** Professor, Department of Economics(corresponding author) / E-mail: [email protected] / ☎ +82-41-860-1514 / 󰂕 Korea University, Jochiwon Chungnam, Republic of Korea

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『경제연구』제26권 제3호

Assuming equal degree of risk-aversion for the players and identical magnitude of reservation wages for the delegates, it is shown that a higher degree of risk-aversion results in overcompensation to the delegates when the delegates' reservation wages are sufficiently low. Key Words : bilateral delegation, contest, contingent-fee, reservation wage, risk-aversion JEL Classification: D72, D74

Ⅰ. Introduction Ⅱ. Framework of Analysis Ⅲ. Bilateral Delegation for Two Risk-averse Players Ⅳ. Conclusion

I. Introduction Delegation contests have been studied with the assumption that both principals and agents are risk-neutral (Park and Lee, 2007a; Park and Lee, 2007b; Baik and Kim, 1997; Wärneyard, 2000; Baik and Kim, 2007a; Baik and Kim, 2007b; Baik, 2007). Schoonbeek (2002) explored a situation where the risk-neutral player represents himself for a contest while the risk-averse player hires a delegate who works on a pure contingent fee contract. Our analysis extends the unilateral delegation model of Schoonbeek (2002) to a bilateral-delegation model, where both players(or, principals) are risk-averse while both delegates(or, agents) are risk-neutral. Both players offer pure contingent fee contracts to their delegates. The contingent fee is paid to a delegate if the delegate wins the contest. Our

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Bilateral Delegation in Contests with Contingent-Fee Contracts

objective in this paper is to investigate how variation in the degree of risk-aversion

affects

the

fees

structure

and

effort

levels

in

a

bilateral-delegation contest. The following section delineates the framework of analysis. In Section Ⅲ, we describe the development of a bilateral-delegation model with two risk-averse players. Conclusions are presented in Section Ⅳ.

Ⅱ. Framework of Analysis The bilateral-delegation contest, in which two risk-averse players compete with each other for an indivisible prize(  ), has the following general structure: 1. Player  hires delegate  under a pure contingent-fee contract (  = 1, 2). The delegates are assumed risk-neutral. 2. Our Cournot-based contest is of a two-stage game. In stage 1, delegate  ’s compensation has the following structure:  if delegate  wins the prize and 0 otherwise, where 0 <   <  . Player  offers   to delegate  . Once both delegates accept the contract offers, the contest goes to stage 2. 3. In stage 2, recognizing  chosen in stage 1, delegate  competes on behalf of player  for the prize and chooses her effort level,  . The contest is characterized by a “lottery auction” where delegate  ’s probability of wining,     , is determined by a unit-logit function such that          for  ≠  . In this structural framework, we work backward to solve for the subgame-perfect equilibrium. For this we first solve for the Nash

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『경제연구』제26권 제3호

equilibrium in stage 2. Then, by reflecting such equilibrium results back in stage 1, we solve for the subgame-perfect equilibrium of the contest.

Ⅲ. Bilateral Delegation for Two Risk-averse Players Let   represent the expected payoff of delegate  such that

                        

(1)

Player  ’s expected utility,  , is then            ,

(2)

where (1 -  ) is the Arrow-Pratt measure of (relative) risk-aversion. We first consider stage 2 in which each delegate seeks to maximize her expected payoff over her effort level, given the other delegate’s effort level. Standard calculations provide the equilibrium outcomes as summarized in Lemma 1.

Lemma 1. At the Nash equilibrium in stage 2, the effort levels of delegates      1 and 2 are, respectively,               and             .

   The probabilities of winning for players 1 and 2 are             

       

and

                       

The

expected

     payoffs of the delegates are             and             .

Using expression (2) and Lemma 1, we obtain player  ’s utility function,   . 

        



             

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(3)

Bilateral Delegation in Contests with Contingent-Fee Contracts

We then consider stage 1 in which player  chooses the optimal value of   to maximize his expected utility, which yields the following first-order condition: 

                    

(4)

The second-order condition holds as follows:                               

(5)

   where        and         . In order to explore the effects

on the contingent-fees of  ,  , and  , we differentiate (4) for  = 1, 2.                                                                               

= 0

(6)

                                                                             

= 0

(7)

Prior to analyzing the results, we first examine     as follows:                                        >0 ÷                  

(8)

Proof. Since the numerator proves positive by the second-order condition (5),

the

sign

of

(8)

relies

on

the

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denominator

{     

『경제연구』제26권 제3호             }. Using the first-order condition (4), we obtain

                and thus the denominator can be rearranged

such that          . From this we obtain Proposition 1.

Proposition 1. If a player puts up more contingent fees, his opponent

follows suit. Consider now the effects on the contingent-fees of  ,  , and  . Expressions (6) and (7) can be rearranged:                                                            ÷                        

(6a)                                                            ÷                        

(7a)

Putting expressions (6a) and (7a) into expressions (6) and (7), we obtain:                                                                ∙                               ∙                  

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Bilateral Delegation in Contests with Contingent-Fee Contracts   ÷                ∙                        

(9)                                           ∙                                                  ÷                       ∙        

(10)

                                         ∙                                                  ∙                               ÷                                       ∙                         



(11)

Proof. The sign of numerator in expressions (9), (10), and (11) depends      on sgn [                                                                      

By the first-order condition (4), the bracket above is rearranged to {                  }/       > 0. Now follows consideration of the signs of denominators: (a) The denominator’s sign in expression (9) is determined by the second-order condition (5) such that       {   } < 0.                      

(b) The sign of denominator in expression (10) depends on sgn    } = sgn {          } < 0. {                   

(c) By (a) and (b) above, the sign of denominator in expression (11) is found positive.

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『경제연구』제26권 제3호

Using expressions (9), (10), and (11), we obtain Proposition 2.

Proposition 2. In any subgame-perfect equilibrium, the following hold: (a)

the more risk-averse player puts up contingent fees; (b) the more risk-averse player influences his opponent to put up contingent fees; (c) the greater prize causes both players to put in more contingent fees. Next, we examine who qualifies as the contest’s favorite between the two risk-averse players1). Using expression (4), we obtain Proposition 3:

Proposition

3.

In

any

subgame-perfect

equilibrium,

(a)

the

more

risk-averse player puts in more contingent fees as compared with that of the less risk-averse player, and (b) the delegate of the more risk-averse player puts in more effort as compared with that of the less risk-averse player thus qualifying the more risk-averse player as the contest’s favorite. Proof. (a) If    , the first-order condition (4) dictates:                       . In case      , we obtain             . Otherwise, we obtain             , which contradicts the condition [     ]. This establishes that, if    , then      . (b) Recalling Lemma 1, Proposition 3(b) is derivable as a corollary of part (a). Let us now consider the case where the players have to consider delegates’ nonnegative reservation wages,  and  . We assume, for simplicity, that      . We further assume that both players are of equal degree of risk-aversion, i.e.,     . As evident in Lemma 1, the equilibrium outcomes in stage 2 are unaffected by the introduction of reservation wages. In stage 1, player  chooses the level of  that maximizes        , considering the equilibrium outcomes in stage 1) Define the favorite as the player with greater than 50% a chance of winning, while his opponent is named the underdog (Dixit, 1987).

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Bilateral Delegation in Contests with Contingent-Fee Contracts

2. Here, the influence of reservation wage hinges on its magnitude. That is, player  ’s maximization problem is unconstrained if  ≤      , while it comes to be constrained if    ≤    . Key results in the contest’s sub-game perfect equilibrium are summarized in Lemma 2.

Lemma 2. (a) In case  ≤      , the delegates’ effort levels are:           . Their contingent fees are:          .

Their expected payoffs are:   =   =    . The players’ expected utilities are:      = {    }   , (b) In case    ≤    , the delegates’ effort levels are:          .

Their

contingent

fees

are:

         .

Their

expected payoffs are:    =    =  . The players’ expected utilities are:              . Implications from Lemma 2 are as follows: (a) If the reservation wage is sufficiently low; the more risk-averse players prefer the larger contingent fee, thus raising their expected utilities as well as their delegates’ effort levels and expected payoffs. Note that the contingent fee rises faster than the

effort

level

as

a

result

of

decreasing

,

thus

resulting

in

overcompensation to the delegates; (b) If the reservation wage is relatively

high,

the

varying

degrees

of

risk-aversion

appear

inconsequential. This is due to our assumption that the delegates are risk-neutral. Assuming otherwise may yield different results.

Ⅳ. Conclusion In this paper we explore how risk-averse players design contingent fees for their delegates. It is shown that variation in the degree of risk-aversion plays a key role in bilateral-delegation contests. For this we examine the

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『경제연구』제26권 제3호

process by which variation in the degree of risk-aversion affects the size of contingent fees and, consequently, the delegates’ effort levels. Major findings are as follows: (i) If a player puts in larger contingent fees, his opponent follows suit; (ii) The more risk-averse player offers the larger contingent fee, thus raising his delegate’s effort level; (iii) The more risk-averse player enjoys a higher-than-50 percent chance of winning. In case the players recognize the delegates' reservation wages, under simplifying assumptions, it is shown that higher degree of risk-aversion results in overcompensation to the delegates when the delegates' reservation wages are sufficiently low. In this paper, we assumed a situation where both risk-averse players hire delegates for a given prize. A couple questions might ensue: (A) Is the more, or the less, risk-averse of the two players more likely to hire a delegate? (B) If only one of the two risk-averse players hired a delegate, would the results vary much? To answer (A), we would have to consider the following four scenarios: (i) Both players hire delegates as shown in this paper; (ii) Only the more risk-averse player hires a delegate; (iii) Only the less risk-averse player hires a delegate; (iv) Both players represent themselves. For each scenario we could figure out the expected payoff for the more risk-averse player, and likewise for her less risk-averse counterpart. Each player would naturally desire the scenario that promises the highest expected payoff to her/him. With regard to (B), we presume that the alternative model's equilibrium outcomes could be obtained and compared with this paper, if the players' degrees of risk-aversion are assumed identical or close enough to each other. The paper's extension in line with (A) and (B) above is saved for future analysis.

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Bilateral Delegation in Contests with Contingent-Fee Contracts

References Baik, Kyung Hwan and In-Gyu Kim, 1997, “Delegation in Contests”, European Journal of Political Economy, Vol.13, No. 2, 281-298. ___________________________________________, 2007a, “Contingent Fees versus Legal Expenses Insurance”, International Review of Law and

Economics, Vol.27, No.3, 351-361. ___________________________________________, 2007b, “Strategic Decisions on Lawyers' Compensations in Civil Disputes”, Economic Inquiry, Vol.45, No.4, 854-863. Baik, Kyung Hwan, 2007, “Equilibrium Contingent Compensation in Contests with Delegation”, Southern Economic Journal, Vol.73, No.4, 986-1002. Dixit, Avinash, 1987, “Strategic Behavior in Contests”, American Economic

Review, Vol.77, No.5, 891-898. Lim, Byung In and Jason F. Shogren, 2004, “Unilateral Delegation and Reimbursement Systems in an Environmental Conflict”, Applied Economics Letters, Vol.11, No.8, 489-493. Park, Sung-Hoon and Myunghoon Lee, 2007a, “A Bilateral Delegate Model with Asymmetric Reimbursement in Environmental Conflicts”,

Environmental and Resource Economics Review, Vol.16, No.1, Korean Resource Economics Association, 3-26. (in Korean). _________________________________________, 2007b, “Policy Implications of the Asymmetric Reimbursement Rule in a Unilateral Delegate Model of Environmental Conflicts”, Journal of Environmental Policy and

Administration, Vol.13, No.1, Korea Environmental Policy and Administration Society, 65-88. (in Korean). Schoonbeek, Lambert, 2002, “A Delegate Agent in a Winner-takes-all Contest”, Applied Economics Letters, Vol.9, No.1, 21-23. Wärneryd, Karl, 2000, “In Defense of Lawyers: Moral Hazard as an Aid to Cooperation”, Games and Economic Behavior, Vol.33, No.1, 145~158.

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『경제연구』제26권 제3호

성공보수와 쌍방대리인 콘테스트

박성훈⋅이명훈

국문요약 본 연구는 쌍방대리인 콘테스트 모형에서 당사자의 위험기피가 콘테스트에 미치는 영향을 분석함을 목적으로 한다. 분석모형에서 각 당사자는 자신의 대리인과 성공보수 계약을 맺는 것으로 가정된다. 2단계 게임의 비교정태분석으로부터 얻어지는 주요 결론은 다음과 같다: 당사자의 위험기피도가 높아짐에 따라 성공보수는 증가하며 이는 당해 대리인의 노력수준을 증가시킨다. 따라서 위험기피도가 높은 당사자는 50% 이상의 승소확률을 갖게 된다. 본 연 구는 또한 당사자들이 대리인들의 유보보수에 어떻게 반응하는지에 대해 분석을 진행한다. 당사자들의 위험기피도가 동일하며 대리인들의 유보보수가 동일하다는 가정 하에서 다음의 함의가 도출된다: 즉, 유보보수의 수준이 충분히 낮은 경우에는, 당사자들의 위험기피도가 높아질수록 대리인들은 유보보수를 상회하는 높은 성공보수를 지급받게 된다. 핵심 주제어: 성공보수, 쌍방대리인, 유보보수, 위험기피, 콘테스트

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