Estimating the Degree of Expert s Agency Problem: The Case of Medical Malpractice Lawyers

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1 Estimating the Degree of Exert s Agency Problem: The Case of Medical Malractice Lawyers Yasutora Watanabe Northwestern University March 2007 Very Preliminary and Incomlete - Comments Welcome! Abstract I emirically study the exert s agency roblem in the context of lawyers and their clients. Incentives of lawyers and clients are misaligned in disute resolution under contingency fee arrangement with which lawyers receive a fraction of recovered ayment as comensation while bearing the legal cost. Lawyers refer to ursue a case less than their clients refer as they incur all the legal costs and receive smaller fraction of ayment. In this aer, I measure the degree to which lawyers work in the interest of their clients. To do so, I construct a bargaining model of disute resolution that nests two secial cases as well as their convex combinations in which lawyers work in their clients best interests and in which they work in their own interests, and estimate the model using data of medical malractice disutes. The timing of droed cases identi es the nesting arameter because cases are droed more frequently and at earlier timings if lawyers work in their own interests. I nd that lawyers work almost erfectly in their own interest. Then, I comute the cost of agency roblem resulting from misaligned incentives by simulating the rst-best outcome using the estimated model. Finally, I evaluate the imact of tort reform on contingency fee, and show that limitation of contingency fee lower the joint surlus further. Deartment of Management and Strategy, Kellogg School of Management, Northwestern University, 2001 Sheridan Road, Evanston, IL

2 1 Introduction Exert agents can mislead their clients in their own interests due to lack of exertise on the side of their clients. Exerts such as lawyers, hysicians, auto reairers, and real estate agents are better informed of the services they rovide than their clients. Due to this asymmetry of information, clients make decision based on advices of exerts or even delegate their decision to the exerts, and clients may not know or nd it very costly to verify if the exerts worked in their best interests. As a result, exerts have incentive to work in their own interests instead of their clients interests. On the other hand, factors such as reutational concern, rofessional liability, caacity constraint, and/or rofessional ethics mitigates the exerts agency roblem to some extent. For examle, a dentist may suggest an exensive orcelain lling instead of a quick comosite resin lling to treat a simle dental cavity, but she may refrain from doing so because of reutational concern or due to caacity constraints (lack of time for another aointment to lace orcelain lling). The degree to which exerts work in the interest of their clients deends on relative strength of these factors and is an oen emirical question. Existing studies either con rms the existence of the exerts agency roblem or countering e ects of the mitigating factor, but have not measured the relative strength. In this aer I estimate the degree to which exerts work in their client s interest. Seci cally, I consider the exerts agency roblem for the case of lawyers and their clients in resolution of medical malractice disutes. A lainti in medical malractice disute signs contingency fee contract with a lawyer. The incentives of the lawyer and the client are misaligned in disute resolution under contingency fee arrangement with which the lawyer receives a fraction of recovered ayment as comensation while bearing all the costs of legal services. Since the client incur no cost of the lawyer s services, the lawyer refer to settle the case earlier than the client refers, and the lawyer has stronger incentive to dro a case. 1 Using data on timing of settlements and droing, I answer the rimary question of this aer: What is the degree to which a lawyer work in the interest of a client? Contingency fee is also an imortant olicy issue in tort reform discussion, esecially on medical malractice. Regulation on contingency fees are adoted by 15 states as of and are currently under consideration at the federal level. 3 In site of its olicy imortance, 1 In this aer, "dro" means that the side of the lainti stos ursuing the case, which includes voluntary dismissal, settlement with zero ayment without ling of a lawsuit, and settlement with zero ayment after ling of a lawsuit.. 2 See the database of state tort law reforms constructed by Avraham (2006). These regulations tyically imose a limit on the fraction lawyers can receive as contingency fee such as 40% and 33%. 3 The House of Reresentatives assed the Hel E cient, Accessible, Low-Cost, Timely Healthcare (HEALTH) Act of 2003 in the 108th Congress, while the Senate voted against it. The bill has been reintroduced in 2004 and 2005 and is currently ending. The Medical Care Access Protection Act of 2006 stalled in the Senate also includes regulation on contingency fee for lainti s attorneys on medical malractice 2

3 little is known emirically about how regulation on contingency fee a ects the outcomes of medical malractice disutes such as legal costs, settlement ayments, and robability of lawsuits, as well as how it a ects the agency roblem of lawyers and their clients. This is another question I ask in this aer. In order to answer these questions, I construct a bargaining model of disute resolution that nests two secial cases as well as their convex combinations; one in which lawyers work in the best interests of their clients, and one in which lawyers work for their own interests. I estimate the model using a unique micro-level data on medical malractice disutes. The nesting arameter measures the degree to which incentives are misalignement. Using the estimated model, I simulate the rst-best outcome and measure the cost of agency roblem resulting from the misaligned incentive. Finally, I conduct a counterfactual olicy exeriment of limiting contingency fees and assess the imact of the olicy on outcomes and welfare. A lainti (i.e. a atient) in a medical malractice disute retain a lawyer and adot contingency fee arrangement for vast majority of cases. 4 A claim by the side of lainti against a healthcare rovider initiates a medical malractice disute. The lainti s lawyer 5 and the defendant engage in negotiations over the terms of settlement in the shadow of court judgment. If the side of lainti les a lawsuit and the arties do not reach an agreement or the side of the lainti do not dro the case, they will face a judgment by the court, which determines whether the defendant is liable and, if so, the award to the lainti. The side of the lainti can dro a case at any time during the rocess. To study this rocess, I construct a dynamic bargaining model in which the lainti s lawyer and the defendant bargain over a settlement following Yildiz (2003, 2004) and Watanabe (2006). At any time during the negotiation, as long as the case has neither been settled nor droed, the lainti s lawyer has the otion of ling a lawsuit that would initiate the litigation hase. If the case is neither settle nor droed during the litigation hase, the case is resolved in court, where a jury verdict determines whether the defendant is liable and, if so the award to the side of the lainti. In any eriod rior to the termination of a disute, the lainti s lawyer and the defendant must ay the legal costs which I allow to di er across the sides of the lainti and the defendant and deending on whether or not a lawsuit is led. In articular, the legal costs are tyically higher during the litigation hase, which entails additional legal rocedures with resect to the re-litigation hase. I do not consider exert s agency roblem for the side of the defendant because medical liacases. 4 For examle, Sloan et al (1993) reorts that lainti s retained lawyers in 99.4 ercent of the cases and that contingency fee arrangement was adoted in 99.4 ercent of the cases in Florida. 5 I assume throughout the aer that the lainti s lawyer is the decision maker for the side of the lainti. This is a common assumtion in the literature that is suorted by surveys for individual clients (see, e.g. Sloan et al (1993) and Kritzer (1998)). 3

4 bility insurance comanies who are well-informed of the legal issues and rocedures makes desicions for the defendant. A arameter 2 [0; 1] in the model re ects the degree to which the lawyer work in the client s interest. Deending on the value of this arameter the model nests two secial cases and their convex combination; one in which the lainti s lawyer work in the best interest of the lainti ( = 0), and one in which the lawyer work in her own interest ( = 1). In the former case, the objective of the lainti s lawyer is to maxmize the ayment from the defendant ignoring any legal cost incurred by the side of the lainti. In the latter case, the objective of the lainti s lawyer is to maximize a contingency fee fraction of ayment from the defendant minus the sum of er-eriod legal cost incurred by the side of the lainti. I characterize the unique subgame-erfect equilibrium of this dynamic bargaining game. Equilibrium outcomes secify (i) the lawyer s decision of whether to le a lawsuit and (ii) if so the time to ling, (iii) whether or not the case is droed by the lainti s lawyer, (iv) whether or not the case is settled out of court, (v) the time to resolution by droing or by settlement, (v) the legal costs incurred, and (vi) the terms of settlement. Delaying agreement is costly because of the er-eriod legal costs. However, the ossibility of learning new information makes delay valuable for both layers. This fundamental trade-o lays an imortant role in the equilibrium characterization and is a key determinant of the time to ling and the time to settlement. Furthermore, I nd that the more misaligned the incentives of the lainti and his or her lawyer are, the shorter the time to settlement as well as to droing and the higher the robability of droing a case. This is because the laini s lawyer takes her er-eriod cost more fully if their incentives are misaligned. This roerty identi es the nesting arameter in estimation. I estimate the model using a unique data set on individual medical malractice disutes. The data set contains detailed information on the time, mode, cost, and terms of settlement as well as the time of ling lawsuit (if a lawsuit is led) and the time of droing (if a case is droed) for all medical malractice disutes in Florida over the eriod The estimate for the nesting aramater is , which imlies that the lawyers do not work very much in the best interest of their clients. I use the estimated structural model to comute the cost of agency roblem resulting from the incentive misalignment. To do so, I simulate the rst-best outcome which corresonds to the case that the lainti s lawyer maximize the ayment from the defendant minus the legal cost incurred. 6 I nd that the joint surlus for the side of the lainti is 6 Socially otimal outcome cannot be achieved in the model we estimate. In case the lainti s lawyer work in his/her interest, the lawyer only considers a fraction of the ayment from defendant minus the legal cost. In this case, the lawyer is giving too much weight for the legal cost. In case the lainti s lawyer work in the lainti s best interest, the lawyer ignores the legal cost she incurs. In this case, the legal cost is underweighted. 4

5 15% less comared to the rt-best outcome. Under the rst-best outcome, more cases are litigated and the time to resolution increases. The legal costs of the both side increases, but the increase in ayment to the side of the lainti from the side of the defendant is larger than the increase in cost. Finally, I conduct couterfactural olicy exeriments and evaluate how regulation on contingency fee arrangement a ects the outcomes of disute resolution as well as the cost of the agency roblem. Seci cally, I consider a limit on contingency fee at 25%. I nd that more cases are droed, frequency of ling of lawsuits decrease, and mean time to resolution decreases. This is because the lainti s lawyer receive less fraction of the ayment as comensation with such limit on fees while the er-eriod legal cost remains the same. The exected join-surlus decreases by about 13% re ecting a large decrease in exected ayment. 1.1 Literature A small but growing emirical literature studies agency roblem in exert services. 7 Levitt and Syverson (2006) comares home sales in which real estate agents are the owner of the house and in which they are not, and found that houses are sold at higher rice and stayed longer on market in the former case. Gruber and Owings (1996) analyzed hysicians likelihood of erforming cesarean section delivery, and show a strong correlation between decline in fertility and increase in cesarean, which they interret as hysician s inducing demand for cesarean by exloiting agency relationshi. Iizuka (2006) investigates Jaanese rescrition market in which hysician can sell drugs as well as rescribing it and nd that their rescritions are in uenced by the marku of the drugs. All these aers resents evidence that agency roblem exists in exert services. Hubbard (1998) studies auto reaireres s emission insection decision, and shows that auto reairers hel to ass by execting customers to return rather than maximizing short run ro ts by selling reairs. This aer shows an evidence of factors that mitigates agency roblem. None of the aer, however, directly estimates the degree of incentive misalignment and quantify the cost of the agency roblem. The aer also adds to the literature on emirical study of re-trial bargaining. 8 Using data on the mode of resolution of civil disutes, Waldfogel (1995) estimates two-eriod bargaining model with heterogenous belief. Sieg (2000) estimates barganing model with 7 Theoretical models of exert sercises includes Dranove (1988), Wolinsky (1993), Emons (1997), and Fong (2005). 8 See the surveys by Kessler and Rubinfeld (2005) and Sier (2007). Theoretical literature on agency roblem regarding contingency fee contract includes Miller (1987), Rubinfeld and Scotchmer (1990), Dana and Sier (1993), Watts (1994), Hay (1996, 1997), Rickman (1999), Choi (2003), and Polinsky and Rubinfeld (2003). These aers emloys two-eriod models which abstracts from learning during rocedure. Hence, droing during re-trial bargaining is not considered. 5

6 asymmetric information to study the mode, cost, and terms of settlement in medical malractice disute using the same data set I use in this aer. Watanabe (2006) further investigate the disute resolution rocess by constructing and estimating a multi-eriod dynamic barganing model to study the timing of settlement and litigation in addition to the mode, cost and terms of settlement. All of these aers, however, abstract from the droing decision and otential agency roblem bewteen lainti and lainti s lawyer, and do not utilize the data of droed cases. Danzon and Lillard (1983) uses data of droing in medical malractice disutes and nd that limits on contingency fees lowers likelihood of droing signi cantly. Helland and Tabarrok (2003) also uses data on medical malractice disutes and nd that limit on contingency fees both lowers likelihood of droing and increases time to settlement. These two aers, however, are not interested in agency roblem and timing of droing, which I address in this aer. The remainder of the aer is organized as follows. In Section 2, I resent the model and characterize the equilibrium. Section 3 describes the data and Section 4 resents the econometric seci cation. Section 5 will contains the results of the emirical analysis. 2 Model I consider a sequential bargaining model of legal disute resolution with erfect-information and stochastic learning. The layers of the bargaining game are a lainti s lawyer () and a defendant (d). 9 The lainti s lawyer and the defendant bargain over the comensation ayment x 2 R + from the defendant to the lainti to resolve the disute. Each layer i 2 fd; g has linear von-neumann-morgenstern references over monetary transfer and legal costs. Both layers know the amount of the otential jury award V 2 R +, but the outcome of the judgement is uncertain, i.e. the layers do not know who will win the case in the event of a trial. 10 Hence, defendant ays V to the side of the lainti if the lainti wins the judgmen, while the defendant does not ay any amount otherwise. I denote the lainti s robability of revailing by. 9 Throughout the aer, I assume the decision maker for the side of the lainti as lainti s lawyer. Sloan et al. (1993) nds that lainti s "almost always followed their lawyer s advice regarding settlement (85)." They reort that lainti s setted 100% of the cases when lawyer s advice favored acceting and 4.8% when lawyer s advice favored rejecting. Another reason for this assumtion is due to the informational asymmetry between lainti s and his/her lawyers. Lawyers are exerienced exerts of the medical liability system, and the lainti s are less likely have better information than the lawyers. However, for the side of the defendant, I do not assume defendants lawyers as decision makers because the defendants are rimarily insurance comanies and are well-informed and have exertise on disute resolution. 10 In the literature the uncertainty of the judgment is caused either (i) by the uncertainty of the winning arty (see e.g., Pries and Klein(1984)) or (ii) by the uncertainty of the award amount (see e.g. Sier (1992)). Imlications of the model do not di er between (i) and (ii) because the exected award is what matters. I take the former assumtion because I can better exlain the data in which a large roortion of cases concludes with no jury award at the court judgment. 6

7 Plainti s and his/her lawyers adot contingency fee arrangement for vast majority of the medical malractice cases in Florida 11, while defendants legal councils charge an hourly legal fee. The contingency fee arrangement entitles the lainti s lawyers to a fraction of the money received from the defendant only if a ositive ayment is received. I denote the fraction by ; hence a lainti s lawyer receives fraction of the defendant s ayment to the side of the lainti as comensation for the legal services they rovide, and the alinti receives the remaining fraction 1. Timing and Phases The bargaining game has two multi-eriod hases deending on whether the lainti s lawyer has led a lawsuit or not: the re-litigation hase (Phase O) and the litigation hase (Phase L). The game starts with Phase O at eriod t = 0: Players bargain every eriod until they reach an agreement. Phase O has a nite number of eriods T < 1, due to the statute of limitation at eriod t = T + 1 < 1, after which the lainti s claim to recover is barred by law. The lainti has an otion of ling a lawsuit in Phase O as long as no agreement has been reached and the case has not been droed. The ling of a lawsuit moves the game to Phase L. Thus, in Phase O, a case may be either led (leading to Phase L), droed (without ling a lawsuit), settled (without ling a lawsuit), or terminated by the statute of limitations. The lainti s endogenous decision to le a lawsuit initiates Phase L. Let t L 2 f0; :::; T g denote the date of the ling of lawsuit. Once the lainti les a lawsuit, the case is rocessed in court towards the judgment scheduled T + 1 eriods after the date of ling, that is date t = t L + T + 1 < 1. While the case is rocessed in court, it can always be droed by the lainti or settled by both arties until t = t L + T: Failure to reach a settlement agreement by t = t L + T results in the resolution by the court judgment at t = t L + T + 1. Information and Beliefs The model is a game of erfect information. As described above, the layers can observe all the actions of the other layer. The information revealed is also commonly observed by both layers. Hence, there is no asymmetric information. The layers, however, do not have a common rior over the robability that the lainti will win the case (). This asymmetry in initial beliefs may be due for examle to di erences in each arty s ercetion of the relative ability of his or her lawyer or to di erences in oinion about the redisosition of otential juries. I assume that the layers beliefs of the robability of the lainti s revailing 2 [0; 1] follow beta distributions, a exible as well as tractable distribution with suort [0; 1] that is widely used in statistical learning models on Bernoulli trial rocess. Player i s initial 11 Sloan et al. (1993) reorts that 99.4% of the cases adoted contingency fee arrangement in their Survey of Medical Malractice Claimants conducted in Florida during

8 re litigation hase (length T ) filing lawsuit (tl ) statute of limitation (T+1 ) judgment (tl+t+1 ) litigation hase (length T ) Figure 1: Diagram of Model Structure. The lainti and the defendant start bargaining over a settlement in re-litigation stage. At any time in re-litigatioin stage, as long as the lainti has not droed the case or no agreement has been reached, the lainti has the otion of ling a lawsuit that endogenously determine t L and would initiate the litigation stage. If neither a lawsuit is led, the case is droed, nor a settlement is reached by T in re-litigation hase, the case is no longer valid due to statute of limitation. If neither agreement is reached nor the case is droed during the litigation stage, the case is resolved by court judgment at t L + T + 1: belief, denoted by b i 0 ; are reresented by Beta( i; i ); where 0 < d < < : 12 A common arameter reresents the rmness of belief as exlained later. Thus, at the initial date, the layers have exected robability of lainti s revailing as E(b 0 ) = E(b d 0) = d for the lainti, and for the defendant. At the beginning of each eriod t in Phase J 2 fo; Lg, information related to winning robability (such as the result of a third arty medical examination or testimony by an exert witness) arrives with robability J. I denote an arrival of information at t by n t 2 f0; 1g, where n t = 1 means arrival of information while n t = 0 reresents no arrival. 12 See Yildiz (2003, 2004) for a similar learning mechanism. Arrival of information is deterministic in his model, while it is stochastic in my model. 8

9 The cumulated amount of information at eriod t is denoted by n t 2 f0; :::; tg, and therefore n t = n t 1 + n t : Information is either in favor of or against the lainti. I denote the content of the arrived information n t by m t 2 f0; 1g: The content is for the lainti if m t = 1;while m t = 0 means the arrived information is against the lainti, and ( m t 0 with robability 1 = 1 with robability ; where is not known to the layers. The cumulated information in favor of the lainti is denoted by m t 2 f0; :::; n t g, leading to m t = m t 1 + n t m t where m t is multilied by n t since the content of information matters only if information arrives. Players udate their beliefs according to Bayesian udating as follows: The beliefs at eriod t, denoted by b t and bd t, follow Beta( + m t ; + n t m t ) and Beta( d + m t ; d + n t m t ) resectively. Hence, at eriod t; the exectations on the robability of lainti s revailing on the judgement are and E(b t ) = + m t + n t E(b d t ) = d + m t + n t : The layers use these exectations as their estimates of. The rmer the beliefs (i.e., the higher the rmness of belief arameter ), the less is the imact of the information obtained in the legal rocess. By modelling beliefs this way, I can cature the relative imact of new learning on rior beliefs. The information environment I have described above is common knowledge to both layers. For notational convenience, I let k t = (n t ; m t ) 2 f0; tg f0; tg denote the information state. Stage Games In Phase O, layers lay the following stage game in every eriod t 2 f0; :::; T g. At the beginning of each eriod, information arrives with robability Ot and does not arrive with robability 1 Ot. The information is such that it a ects the outcome of the judgment (e.g., the result of third-arty medical examination) and is commonly 9

10 observed by both layers. Then, the lainti s lawyer chooses whether to dro the case or continue it. If she chooses to dro the case, the case is resolved with no ayments being made from defendant to the side of the lainti. Otherwise, nature chooses the rooser, with robability for the lainti and 1 for the defendant. The chosen arty rooses the amount of comensation ayment x, which will be either acceted or rejected by the other arty. If acceted, the game concludes with the roosed amount of money x being transferred from the defendant to the side of the lainti, and the disute is resolved. In such a case, the lainti s lawyer receives x and the lainti receives (1 )x. If the roosal is rejected, the lainti chooses whether or not to le a lawsuit. The case moves to Phase L if the lainti les a lawsuit, while it remains in Phase O and the same stage game is reeated if the lainti chooses not to le. If the case is neither led nor settled during the T eriods, the statute of limitation renders the claim by the lainti to be ine ective. After the side of the lainti les a lawsuit, the arties lay the following stage game in every eriod t 2 ft L ; :::; t L + T g until the court judges on the case at t = t L + T + 1. At the beginning of each eriod, information that a ects the outcome of the judgment arrives with robability Lt and does not arrive with robability 1 Lt. Then, the lainti s lawyer chooses whether to dro the case or continue it. If she chooses to dro the case, the case is resolved with no ayments being made from defendant to the side of the lainti. Otherwise, nature again chooses the rooser, with robability for the lainti and 1 for the defendant. The chosen arty rooses the amount of comensation ayment x, which will be either acceted or rejected by the other arty. If acceted, the game terminates with the roosed amount of money being transferred from the defendant to the side of the lainti, and the disute is resolved. In such a case,, the lainti s lawyer receives x and the lainti receives (1 )x. If rejected, the case remains in Phase L and the same stage game is reeated until t = t L + T: I allow the rates of information arrival to di er across hases. One of the reasons for di ering rates stems from the discovery rocess. In this rocess, both arties can emloy a variety of legal devices to acquire information on the case that follows the ling of a lawsuit. I denote er-eriod legal costs at eriod t in Phases O and L for layer 2 fd; g by COt i 2 R+ and CLt i 2 R+ resectively. I assume that these legal costs are drawn indeendently in each eriod t from identical distributions. I allow the distribution of er-eriod costs to di er deending on whether or not the lainti has led a lawsuit. In articular, the legal costs of both sides are tyically higher in the litigation hase, which entails additional legal rocedures with resect to the re-litigation hase. The distributions from which er-eriod costs are drawn in Phases O and L are denoted by G CO () and G CL (). The realizations of the er-eriod costs only a ect the total legal costs, 10

11 and they do not a ect the equilibrium stoing timing and settlement terms because layers make desicions based on exeted legal cost of the subsquent eriods. I consider a common time-discount factor denoted by 2 [0; 1]. A Parameter to Measure the Degree of Incentive Misalignment As discussed above, lainti s lawyers use a contingency fee arrangement, while defendants legal councils charge an hourly legal fee. The contingency fee arrangement entitles the lainti s lawyers to a fraction 2 [0; 0:5) 13 of the money received from the defendant only if a ositive ayment is received. Because of this deferment, the lainti incurs no additional legal cost by delaying agreement because the lainti s lawyer incurs the legal cost. Hence, the lainti and his/her lawyer s incentive are not aligned. The lainti refer to ursue a case much longer than her lawyer refer. The lainti s s objective is to maximize the exected ayment from the defendant, while the lawyer always considers the cost as well as the ayment. The main objective of this aer is to measure the degree to which lawyers and their clients incentives are misaligned, and I will measure this with the following arameter 2 [0; 1]: If a lawyer behaves erfectly in the interst of her lainti, she maximizes the exected ayment and does not consider er-eriod legal cost. If ayment is x, such lawyer s ayo can be written as u 0 = (1 )x: If a laywer behaves in her best interest, she considers the legal cost and only fraction of the ayment. Hence, we can write it as u 1 = x C j : where j 2 fo; Lg. My interest is to measure what the true ayo system is for the lainti s lawyer, and these two cases are the secial cases in which the lawyer are erfectly working in the interest of her client and in which she work in her own interest. Since these two cases are the extreme ones, I can consider a convex combination of these two cases with a arameter, which can be written as u = u 1 + (1 ) u 0 (1) = (1 + 2)x C j Now, = 1 imlies that a lawyer behaves in her interest and = 0 imlies that a lawyer behaves in the lainti s interest. Hence, mesures the degree to which the incentive of 13 I imose this assumtion because contingency fee are below 50% for vast majority of cases. 11

12 the lawyers and the lainti are misaligned. For notational convenience, I de ne function A() as A() 1 + 2: 2.1 Equilibrium Characterization The model is a dynamic game with erfect information. Thus, I emloy subgame-erfect equilibrium as the equilibrium concet. Because the model has a nite number of eriods, backward induction rovides us with a characterization of the unique subgame-erfect equilibrium. I start the analysis from the last stage in Phase L, and move to Phase O Phase L (Litigation Phase) In order to characterize the unique subgame erfect equilibrium by backward induction, I start my analysis from the date of the judgment by the court at the end of Phase L. Recall that a Phase L subgame is reached only if the case is litigated at some eriod during Phase O. Let t L 2 f0; :::; T g denote the date of ling the lawsuit which is an endogenously determined. If the layers cannot settle by date t L + T, the judgement by the court at t = t L + T + 1 determines the outcome of the last stage. Since two decisions are made in each eriod in Phase L, we de ne two continuation values for each layer. Let V i t t L (k t ) denote the continuation value for layer i 2 f; dg at the beginning of date t in Phase L with information state k t, and V i t t L (k t ) the continuatoin value after droing decision by lainti s lawyer and before settlement decision at the end of the eriod t. Note that the subscrit is t t L, which is the number of eriods in Phase L so that V1 i(k t L ) corresonds to the continuation value at the rst eriod in Phase L that will be on the right hand side of the Bellman equation in Phase O. where A()V V T +1 (k t L +T +1) = V d T +1(k tl +T +1) = ( ( The continuation value of the judgment is A()V V 0 0 if the lainti revails otherwise. if the lainti revails otherwise. is the amount lainti s laywer obtains if she win the case and V is the amount aid the defendant to the side of the lainti. The lainti will have the di erence (1 A())V: Having above V i T +1 (k t L +T +1) as nal values, I can obtain V i t t L (k t ) and V i t t L (k t ) by alying backward induction. The equilibrium in Phase L subgame is characterized in the Proosition 1 below. 12

13 Proosition 1 In the unique subgame erfect equilibrium given k t and t L, 1. the ayo of the layers at t 2 ft L + 1; :::; t L + T g in Phase L are exressed as where V n V t t L (k t ) = max t t L (k t ) = max 0; V t t L (k t ) ( Vt d 0 if 0 > V t L (k t ) = V d t t L (k t ) t t L (k t ) otherwise. A()Et d +(1 )E t V (k t+1 ) CL ; h i V d V d t t L (k t ) = Et d (k t+1 ) CL d +(1 ) max h i Vt+1 d t L (k t+1 ) CL d ; E t V (k t+1 ) C o L 1 A() E t V (k t+1 ) C L ; E d t hvt+1 d t L (k t+1 ) CLi : 2. the lainti s lawyer dros the case at t 2 ft L + 1; :::; t L + T g in Phase L i 0 > V t t L (k t ) 3. the layers settle at t 2 ft L + 1; :::; t L + T g in Phase L i 0 E t [V (k t+1 ) C L ] + A()Ed t [V d (k t+1 ) C d L]: 4. given that layers settle at t 2 ft L + 1; :::; t L + T g in Phase L, the ayment is x t = ( Et d 1 A() E t Proof. See Aendix A V d t+1 tl (k ts +1) CL d V (k ts +1) C L if the lainti is a rooser if the defendant is a rooser, This roosition characterizes the subgame erfect equilibrium in Phase L: The exression for V t t L (k t ) in 1 is the continuation value at the beginning of the eriod when lainti s lawyer faces the droing decsion. As described in 2, lainti s lawyer dros a case if her interim continuation value V t t L (k t ) is below 0. In such a situation, both sides have 0 as their continuation value. Interim continuation value after droing decision and before settelment decision V i t t L (k t ) is obtained by solving for random-rooser barganing. Note that exectations are indexed by layer because two layers have di erent beliefs over the robability of winning at judgment. 13

14 The identify of the rooser do not a ect the settlement decision as can be seen in 3. The amount of ayment deends on the identify of the rooser as in 4. This arises because the layers choose to settle if the joint surlus of settlement today is larger than the joint surlus of continuing the case. The comensation ayment deends on the identity of the rooser because the recognized rooser obtains all of the surlus. This is true even in the extreme case that lainti s lawyer work in the interest of her client (i.e. A() = 1 ) and do not take er-eriod legal costs into consideration. Delaying agreement is still costly for her because she misses the bene ts of the saving of the defendant s legal costs that indirectly increases comensation ayment. Therefore, delaying agreement is costly for both layers, and quick settlement is referable. However, the ossibility of learning new information makes delay valuable, since this information enhances the robability of a settlement, which in turn generates ositive surlus for both layers. This fundamental trade-o lays an imortant role in the equilibrium characterization and is a key determinant of the timing of settlement. Droing decision is simly an individual rationality constraint for the lainti s lawyer at each eriod. If there is no learning, the continuation values for the lainti s lawyer increases monotonically over time as can be seen from 1. This monotonicity imlies that if the value is negative at eriod t, the value is also negative for any eriod t 0 < t, i.e. if a case is to be droed at t, it must be droed at any earlier date t 0. Hence, all droing occurs only in the initial eriod if there is no learning. Therefore, droing in eriods other than the initial eriod only occurs with learning at arrival of new information against the lainti s side. In such case, continuation value stays ositive for rst several eriods, and it become negative at the arrival of unfavorable information. The roosition also shows that droing and settlement decisions are directly a ected by the degree of incentive misalignment. Increase in decreases the continuation value of the lainti s lawyer in two ways; through increase in er-eriod cost and through decrease in A(). Decrease of A() lowers V t t L (k t ) because the exected award at judgement E t [A()V ] decreases corrsoding to change in A(). Thus, the larger is, the smaller V t t L (k t ) is, and the more likely tha the case is droed. In other words, lainti s lawyer refer to dro a case at earlier date if her incentives is less aligned. In the extreme case that the lawyer works erfectly in her client s interest ( = 0), the case is never droed. Similar intuition works for the e ect of incentive alignment and settlement decision. The exression in 3 can be rewritten as E t [C L + Cd L] E t [V (k t+1 )] + A()E d t [V d (k t+1 )]; where the left-hand side is exected er-eriod legal cost which is not indexed by layer 14

15 because layers have common belief over the distribution of cost. The right-hand side is the exected surlus from continuation. Increase in increase the value of the left-hand side, and decreases the value of the right hand side 14. Hence, the less aligned the incentives are (the larger is), the shorter it takes to reach a settlement on average. The lainti s lawyer who considers her own legal cost more seriously and focusing more on ayments to herself, which is much smaller than the ayment to the lainti, refer to settle at earlier dates Phase O (Pre-Litigation Phase) Let W i t (k t ) denote the continuation value for layer i 2 f; dg at the beginning of date t in Phase O with information state k t. Again, I start from the last stage of Phase O subgame. The maximum number of eriods in Phase O is T ; after which the claim by the lainti loses its value due to the statute of limitation. of 0 at date T + 1, i.e., W T +1 (k T +1 ) = W d T +1 (k T +1 ) = 0: Hence, each layer has continuation ayo I comute W i t (k t ) by alying backward induction and having the above as nal values. In Phase O, the lainti has an otion of litigating a case at any date t 2 f0; :::; T g. Solving the value function in Proosition 2 u to the rst eriod in Phase L, I can obtain the continuation value of the Phase L subgame, V i 1 (k t). Note that this continuation value of litigation does not deend on the date of litigation t L itself but does deend on the information state k t at eriod t. First, I will look at the ling decision by the lainti s lawyer because ling decision is the last decsion to be made in a eriod. The lainti s lawyer chooses to litigate at the end of date t if and only if E t V 1 (k t+1) C L Et W t+1 (k t+1) CO ; where the left-hand side is the ayo from ling a lawsuit and the right-hand side is the ayo from stay in re-litigation hase. Hence, the continuation value for the lainti at the end of date t is written as Y t (k t) = max E t V 1 (k t+1) C L ; Et W t+1 (k t+1) CO ; where Y t (k t) denotes an interim continuation value for the lainti s lawyer after the settlment decision and before the ling decision in eriod t in Phase O given state k t. The 14 Though A() increase in, A()Et d [Vt+1 d t L (k t+1)] decrease in because Et d [Vt+1 d t L (k t+1)] is always negative and decreases in : 15

16 defendant s continuation value deends on the ling decision of the lainti s lawyer described above, and is written as Y d t (k t ) = ( E t V d 1 (k t+1 ) CL d E t W d t+1 (k t+1 ) C O if E t V 1 (k t+1) C L otherwise, E t W t+1 (k t+1) C O where Y d t (k t ) denotes an interim continuation value for the lainti s lawyer after the settlment decision and before the ling decision in eriod t in Phase O given information state k t. I can then conduct the exact same analysis for settlement and droing decision as I did for Phase L; and the subgame-erfect equilibrium for Phase O is characterized in Proosition 2. Proosition 2 In any subgame erfect equilibrium given k t ; 1. the ayo s of the layers at t 2 f0; :::; T g in Phase O are exressed as W t (k t) = max 0; W t (k t ) ( Wt d 0 if 0 > W (k t ) = W d t (k t ) t (k t ) otherwise. where n W t (k t ) = max W d t (k t ) = Y d t (k t ) + (1 o A()Yt d (k t ); Y t (k t) + (1 )Y t (k t); 1 ) max A() Y t (k t); Yt d (k t ) : 2. the layers settle at t 2 f0; :::; T g in Phase O i Y t (k t) + A()Y d t (k t ) 0: 3. the lainti litigates at t 2 f0; :::; T g in Phase O i E t [V 1 (k t L +1)] E t W t+1 (k t+1) : 4. given that layers settle at t 2 f0; :::; T g in Phase O, the ayment is x t = ( Y d t (k t ) Y t (k t) if the lainti is a rooser if the defendant is a rooser, 16

17 Proof. See Aendix B This roosition characterizes the subgame erfect equilibrium in Phase O: The mechanics of the settlement and droing decisions are exactly the same as for the characterization for Phase L. The only di erence is that the cost of delay and the ossibility of learning in the next eriod deend on the lainti s decision to le or not at the end of each eriod. The decision to le a lawsuit by the lainti is also derived from a trade-o between the cost of delay and the ossibility of learning. The cost of delaying ling has two comonents. One is the er-eriod legal cost of the re-litigation hase, because the total length of the underlying game becomes one eriod longer if ling is delayed by one eriod. Even though the lainti incurs no legal cost er eriod, this delay still imacts him because minimizing the defendant s legal cost may result in a higher comensation ayment to the lainti. The second comonent is the cost of delay due to discounting. The lainti refers to obtain the continuation value of the Phase L subgame earlier, since a one-eriod delay in ling costs him (1 )V 1 () when no information arrives. The bene t of delaying ling by one eriod is that the arties have one more eriod to obtain new information and hence reach an agreement. Thus, the lainti refers to stay in Phase O longer if CO i are small and is large, while low Ot rovides an incentive for the lainti to le early. 3 Data The Florida Deartment of Financial Services, an insurance regulator of the Florida state government, collects a detailed micro-level data set on medical malractice disutes in Florida. A statute on rofessional liability claims requires medical malractice insurers to le a reort to the Deartment on all of their closed claims once the claim is resolved. The reort contains detailed information on the disute resolution rocess, as well as individual case characteristics. The information on the disute resolution rocess includes imortant dates (calender date of occurrence, initial claim, ling of lawsuit, resolution either by droing, settlement, or by court judgement), settlement ayments (or award by the court in case of resolution by court judgment), and total legal costs incurred by the defendants. The information on case characteristics includes atient characteristics (e.g. age and sex), defendant characteristics (e.g. defendant tye, secialty, and insurance olicy), and the characteristics of injury (e.g. severity and lace of occurrence). Hence, this data set contains detailed information on all the variables of interest, i.e., if and when a lawsuit is led, whether or not the case is settled out of court or droed, the time to resolution, the legal costs incurred and the terms of settlement. My samle of observations consists of 5,379 claims against hysicians 15 which were 15 Claims against hositals, HMOs, dentists, ambulance surgical centers, and crisis stabilization units are 17

18 Droed without Lawsuit Droed after Lawsuit Settled without Lawsuit Settled after Lawsuit Resolved by Judgment with Positive Award Resolved by Judgment with No Award Number of Observations Resolution Probability , Total 5, Mean Comensation 0 (0) 0 (0) 314; 266 (303; 917) 303; 402 (379; 909) 541; 832 (620; 722) 0 (0) 203; 210 (343; 922) Mean Defense Legal Cost 7; 330 (23; 023) 40; 370 (44; 978) 9; 047 (15; 014) 53; 989 (88; 887) 127; 966 (113; 147) 83; 182 (87; 268) 45; 047 (77; 794) Comensation ayments and legal costs are measured in 2000 dollars. rovides standard deviations. Numbers in arentheses Table 1: Descritive Statistics I - Resolution Probability, Payments and Legal Costs resolved between October 1985 and July Following Sieg (2000), I restrict attention to cases with the defendant s legal cost exceeding $1,000 and which were not droed during the litigation rocess. 17 Because the timing and disosition of cases di er greatly deending on the severity of the injury, I restrict attention to the cases in which injuries resulted in ermanent major damage to or death of the atient. Tables 1 and 2 show the descritive statistics of all the variables I use for estimation. In my data, 28.6% of the cases are droed or settled with zero ayment, 62.5% of the cases were settled by arties, and the remaining 9.1% of the cases were resolved by judgement by the court. Regarding droed and settled cases, only one senveth of settled cases settled without ling a lawsuit and majority of settled cases are settled after ling a lawsuit, while the roortion of cases droed without ling a lawsuit and after ling a lawsuit are about the same around 14%. For the judged cases, the defendants were three times more likely to win the judgment (i.e., award was zero). Mean comensation ayments were similar for excluded from my samle to control for the heterogeneity of the lainti. 16 During this eriod, there were no major changes in state law retaining to resolution of medical malractice disutes. 17 This rocedure eliminates small cases. 18

19 Time to Filing Time to Resolution after Filing Total Time to Resolution Droed without Filing 4:81 (3:22) Droed after Filing 2:30 (2:31) 7:36 (4:51) 9:66 (5:09) Settled without Filing 3:23 (2:41) Settled after Filing 2:48 (2:26) 7:45 (4:55) 9:93 (5:16) Resolved by Judgment with Positive Award 2:69 (2:67) 11:41 (6:61) 14:10 (7:17) Resolved by Judgment with No Award 2:50 (3:21) 9:69 (5:17) 12:19 (6:24) Total 2:45 (2:38) 7:75 (4:76) 8:81 (5:57) Numbers are in quarters of a year. The numbers in arentheses rovide standard deviations. Table 2: Descritive Statistics II - Timings the cases that were settled after ling a lawsuit and those that were settled without ling a lawsuit (around $300,000). Legal costs for defendants substantially di ered across modes of resolution. Defense lawyers usually charge based on the amount of time they sent. It is not surrising to nd that the mean of the defense legal costs correlates strongly with the mean time to resolution as dislayed in Table 2. Mean of defence legal costs for cases settled without a lawsuit is about one- fth of the cases settled after a lawsuit. This di erence correlates with the shorter mean time to resolution as in Table 2. The cases settled after ling a lawsuit have a signi cantly higher mean cost ($53,989) comared to the settled cases without a lawsuit ($9,047). Similary, the cases droed after ling a lawsuit have a signi cantly higher mean cost ($40,370) comared to the cases droed without a lawsuit ($7,330). Among the cases resolved by court judgement, mean defense costs are more than 50% higher for the cases won by the lainti s than the cases won by the defendants, which again corresonds to the longer time to resolution. Mean time to ling a lawsuit, which corresonds to the eriods sent in the re-litigation hase, is similar across the droed cases, the settled cases, and the cases resolved by court 19

20 fraction Time to droing without a lawsuit Time to settlement without a lawsuit Time to filing a lawsuit quarters Figure 2: Histogram of Time to Droing, Settlement and Filing. Fractions of time to droing without a lawsuit, settlement without a lawsuit, and ling a lawsuit add u to one. This isbecause cases are either 1) droed without a lawsuit, 2) settled without a lawsuit, or 3) resolved after a lawsuit is led. judgment. The time to resolution di er between the cases settled after ling and the cases resolved by the judgment. This di erence results from the di erence in time to resolution after ling lawsuit. The cases settled without a lawsuit, which corresond to settlement in re-litigation hase, send longer eriods in the re-litigation hase than the led cases on average, and the cases droed without a lawsuit sends even longer time on average about 4.81 quarters. Figure 2 rovides the histogram of ling, droing, and settlement in re-litigation hase. Note that the sum of the fractions for ling, droing, and settlement add u to one because each case is either led, droed without a lawsuit, or settled without a lawsuit. In Florida, the statute of limitation regarding a medical malractice cases is two years. Hence, more than 95% of the cases are led or settled without ling a lawsuit before the ninth quarter. 18 A fraction of cases ling lawsuits in re-litigation hase and a fraction of cases settling 18 The remaining cases may be due, for examle, to invervening medical comlications that entail an automatic extension of the statute of limitations. 20

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