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Estimating the Degree of the Experts’Agency Problem: The Case of Medical Malpractice Lawyers Yasutora Watanabe Northwestern University December 2010 Abstract I empirically study the experts’agency problem in the context of lawyers and their clients. Incentives for lawyers and clients are misaligned in dispute resolution under contingency-fee arrangements where lawyers receive a fraction of the recovered payment as compensation while bearing the legal costs. Lawyers are less inclined to pursue a case than their clients because they incur all the legal costs and receive a smaller fraction of the payment. In this paper, I measure the degree to which lawyers work in the best interest of their clients. To do so, I construct a bargaining model, which is a convex combination of two extreme cases – lawyers working in their clients’best interest and lawyers working in their own best interest – as well as their convex combinations, and estimate the model using data of medical malpractice disputes. The timing of dropped cases identi…es the weighting parameter because cases are dropped more frequently and at earlier stages if lawyers work in their own best interest. I …nd that lawyers work s closer to their own best interest than to their client’ best interest. Then, I compute the cost of the agency problem resulting from misaligned incentives by simulating the …rst-best outcome using the estimated model. Finally, I evaluate the impact of tort reform on contingency fees, and show that limitation of contingency fees further lowers the joint surplus. Department of Management and Strategy, Kellogg School of Management, Northwestern University, 2001 Sheridan Road, Evanston, IL 60208. Email: y-watanabe@kellogg.northwestern.edu 1 1 Introduction Expert agents can mislead their clients in their own best interests due to their clients’lack of expertise. Experts such as lawyers, physicians, auto repairers, and real estate agents are better informed on the services they provide than their clients. Due to this asymmetry of information, clients make decisions based on the advice of the experts or even delegate their decisions to the experts, and clients may not know, or …nd it very costly to verify, whether the experts worked in the clients’best interest. As a result, experts have incentive to work in their own best interest instead of that of their clients, a phenomenon I call the experts’ agency problem. On the other hand, factors such as reputational concern, professional liability, capacity constraint, and/or professional ethics mitigate the experts’ agency problem. The degree to which experts work in the best interest of their clients depends on the relative strength of these factors against the experts’ agency problem and is an open empirical question. Existing empirical studies either present the existence of the experts’ agency problem or show the countering e¤ects of the mitigating factor, but do not focused on measuring the relative strength of these factors. In this paper I estimate the relative strength of these factors, i.e., the degree to which s experts work in their client’ best interest. Speci…cally, I consider the experts’agency prob- lem in the case of lawyers and clients engaged in medical malpractice dispute resolution. A plainti¤ in a medical malpractice dispute signs a contingency-fee contract with a lawyer. The incentives for the lawyer and the client are misaligned under a contingency-fee arrange- ment where the lawyer receives a fraction of the recovered payment as compensation while bearing all the costs of legal services. Because the lawyer fails to internalize the return from incurring the legal costs due to the contingency-fee arrangement, the lawyer prefers to settle the case earlier than the client, and the lawyer has a stronger incentive to drop a case.1 Using data on the timing of settlements and the dropping of medical malpractice claims, I answer the primary question of this paper: What is the degree to which a lawyer works in the best interest of the client? Contingency fees are also an important policy issue in tort-reform discussion, especially with respect to medical malpractice. Regulation of contingency fees has been adopted by 15 states as of 2005.2 In spite of its policy importance, little is known empirically about how regulation on contingency fees a¤ects the medical-malpractice-dispute outcomes, such as legal costs, settlement payments, and the probability of lawsuits, as well as how it a¤ects 1 In this paper, “drop” indicates that the side of the plainti¤ stops pursuing the case, through voluntary dismissal, settlement with zero payment without …ling a lawsuit, or settlement with zero payment after …ling a lawsuit. 2 See the database of state tort law reforms constructed by Avraham (2006). These regulations typically impose a limit on the fraction lawyers can receive as contingency fees (e.g., 40% and 33%). 2 the agency problem facing lawyers and their clients. That is another question I ask in this paper. In order to address these questions, I construct a bargaining model where the plainti¤ s side’ objective function is written as a convex combination of two cases, one in which lawyers work in the best interest of their clients, and one in which lawyers work in their own best interest. I estimate the model using micro-level data on medical malpractice disputes. The weighting parameter measures the degree to which incentives are misaligned. Using the estimated model, I simulate the …rst-best outcome and measure the cost of the agency problem resulting from the misaligned incentives. Finally, I conduct a counterfactual policy experiment of limiting contingency fees and assess the impact of the policy on outcomes and welfare. In the vast majority of medical malpractice disputes,3 a plainti¤ (i.e. a patient) retains a lawyer and adopts contingency-fee arrangement. A claim …led by the plainti¤ against a s healthcare provider initiates the dispute. The plainti¤’ lawyer and the defendant engage in negotiations over the terms of settlement in anticipation of the court ruling.4 If the plainti¤ …les a lawsuit and the parties do not reach an agreement (and the plainti¤ does not drop the case), the court determines whether the defendant is liable and, if so, the damages awarded to the plainti¤. The plainti¤ can drop the case at any time during the process. s To study this process, I construct a dynamic bargaining model, in which the plainti¤’ 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 s settled nor dropped, the plainti¤’ lawyer has the option of …ling a lawsuit that would initiate the litigation phase. If the case is neither settled nor dropped during the litigation phase, it is resolved in court, where a jury verdict determines whether the defendant is liable and, if so, the damage awarded to the plainti¤. In each period prior to the termination of s a dispute, the plainti¤’ lawyer and the defendant must pay the legal costs, which I allow to di¤er between the two sides and depending on whether a lawsuit is …led. In particular, the legal costs are typically higher during the litigation phase, which entails more legal procedures than the pre-litigation phase. I do not consider the experts’ agency problem for the side of the defendant because medical liability insurance companies, which are well- informed on the legal issues and procedures, make decisions for the defendant. A weighting parameter 2 [0; 1] in the model re‡ects the degree to which the lawyer s works in the client’ best interest. The model is a convex combination (with the weighting 3 Sloan, Githens, Clayton, Hickson, Gentile, and Partlett (1993) reports, for example, that plainti¤s retained lawyers in 99.4 percent of the cases and that a contingency-fee arrangement was adopted in 99.4 percent of the cases in Florida. 4 s I assume throughout the paper that the plainti¤’ lawyer is the decision maker for the side of the plainti¤. This is a common assumption in the literature, and surveys for individual plainti¤s in medical malpractice and other disputes supports this assumption (See, e.g. Sloan et al.,1993, and Kritzer, 1998.). 3 s parameter ) of two special cases; one in which the plainti¤’ lawyer works in the best interest of the plainti¤ ( = 0) and one in which the lawyer works in her own best interest s ( = 1). In the former case, the objective of the plainti¤’ lawyer is to maximize the payment from the defendant regardless of the legal costs incurred by the side of the plainti¤. In the s latter case, the objective of the plainti¤’ lawyer is to maximize the contingency-fee fraction of the payment from the defendant minus the sum of the per-period legal cost incurred by the side of the plainti¤. I characterize the unique subgame-perfect equilibrium of this dynamic bargaining game. Equilibrium outcomes specify (i) whether the lawyer decides to …le a lawsuit, and, if so (ii) the time it takes to …le it, (iii) whether the case is dropped by the plainti¤, (iv) whether the case is settled out of court, (v) the time it takes to reach a resolution by dropping or by settlement, (vi) the legal costs incurred, and (vii) the terms of the resolution. Delaying an agreement is costly because of the per-period legal costs. However, the possibility of learning new information makes delay valuable for both players. This fundamental tradeo¤ plays an important role in the equilibrium characterization and is a key determinant of the time it takes to …le a lawsuit and the time it takes to reach a resolution. Furthermore, I …nd that the more misaligned are the incentives of the plainti¤ and his lawyer, the less time it takes to reach a resolution or drop a case and the higher is the probability that the case s will be dropped. This is because the plainti¤’ lawyer incurs more of her per-period cost if the incentives are misaligned. This property identi…es the weighting parameter . I estimate the model using a data set on individual medical malpractice disputes. The data set contains detailed information on the time, mode, cost, and terms of resolution as well as the time of …ling lawsuit (if a lawsuit is …led) and the time of dropping (if a case is dropped) for all medical malpractice disputes in Florida over the period 1985-1999. The estimate for the weighting parameter is 0.8128, which implies that lawyers generally do not work in the best interest of their clients. I use the estimated structural model to compute the cost of the agency problem resulting from the incentive misalignment. To do so, I simulate the …rst-best outcome, which corre- s sponds to the case where the plainti¤’ lawyer maximizes the payment from the defendant minus the legal costs incurred.5 I …nd that the joint surplus for the side of the plainti¤ is 15% less compared to the …rst-best outcome. Under the …rst-best outcome, more cases are litigated and the time to resolve a dispute increases. The legal costs on both sides increase, but the increase in the payment to the plainti¤ is larger than the increase in the 5 Socially optimal outcome cannot be achieved in the model we estimate. In case the plainti¤’ lawyers work in his/her interest, the lawyer only considers a fraction of the payment from defendant minus the legal s cost. In this case, the lawyer is giving too much weight for the legal cost. In case the plainti¤’ lawyer s work in the plainti¤’ best interest, the lawyer ignores the legal cost she incurs. In this case, the legal cost is underweighted. 4 cost incurred. Finally, I conduct counterfactual policy experiments and evaluate how regulation on contingency-fee arrangements a¤ects the outcomes of dispute resolution as well as the cost of the agency problem.6 Speci…cally, I consider a 20% limit on contingency fees. I …nd that, with that limit, more cases are dropped, the frequency of lawsuit …ling decreases, and the mean time it takes to resolve a dispute decreases. This is because, with such a limit, s the plainti¤’ lawyer receives a smaller fraction of the payment as compensation while the per-period legal costs remain the same. The expected joint surplus decreases by about 13%, re‡ecting a large decrease in the expected payment. 1.1 Related Literature A small but growing empirical literature studies the agency problem in expert services.7 Gruber and Owings (1996) analyze the likelihood of physicians performing a cesarean- section delivery, and show a strong correlation between the decline in fertility and the increase in cesarean deliveries, which they interpret as physicians inducing demand for cesarean sections by exploiting the agency relationship. Iizuka (2007) investigates the Japanese prescription market, which allows physicians to sell drugs as well as prescribe them, and …nds that the markup of the drugs in‡uences what physicians prescribe if they sell drugs. Levitt and Syverson (2008) compare the sale of homes owned by the real estate agents selling them and of those not owned by the agent, and …nd that that houses are sold at a higher price and stay on market longer in the former case. Using a …eld experiment of bringing in cars with defects to auto repairers, Schneider (2007) …nds a large fraction of data with unnecessary repairs and under-treatment. All these papers present evidence that an agency problem exists in expert services. On the other hand, Hubbard (1998) reports the mitigating factor to the experts’agency problem by studying auto repairers’emission- inspection decisions. He shows that auto repairers help clients pass, so that those clients are more likely to return, rather than maximizing short-run pro…ts by selling repairs. This paper takes a di¤erent approach: it directly measures the relative strength of these factors by estimating the objective function of the experts, i.e., the degree to which the experts work in the best interest of their clients. The paper also adds to the literature on the empirical study of pre-trial bargaining.8 6 The results of the counterfacutal policy experiments are limited in the sense that the e¤ects of policy changes only a¤ect the behavior through the model. Given that the estimated parameter is a reduced form s of the various factors that a¤ect the incentive of plainti¤’ lawyers, the results is a benchmark for what would happen if policy change would a¤ect factors only in the way the model considers. 7 For example, theoretical models of expert servises includes those by Dranove (1988), Wolinsky (1993), Emons (1997), and Fong (2005). 8 See the surveys by Kessler and Rubinfeld (2005) and Spier (2007). Theoretical literature on the agency problem regarding contingency-fee contracts includes Miller (1987), Rubinfeld and Scotchmer (1990), Dana 5 Using data on the mode of resolution in civil disputes, Waldfogel (1995) estimates a two- period bargaining model with heterogenous beliefs. Sieg (2000) estimates a bargaining model with asymmetric information to study the mode, cost, and terms of resolution in medical malpractice disputes using the same data set I use in this paper. Watanabe (2006) further investigates the dispute-resolution process by constructing and estimating a multi- period dynamic bargaining model to study the timing of settlement and litigation in addition to the mode, cost and terms of resolution. All of these papers, however, abstract from the decision to drop a case and does not address the potential agency problem between the plainti¤ and his lawyer. Danzon and Lillard (1983) use data on dropped cases in medical malpractice disputes and …nd that limits on contingency fees signi…cantly lower likelihood of cases being dropped. Helland and Tabarrok (2003) also use data on medical malpractice disputes and …nd that limits on contingency fees both lower likelihood of a case being dropped and increase the time it takes to reach a settlement. These two papers, however, are not interested in the agency problem and time it takes to drop a case, both of which I address in this paper. The remainder of the paper is organized as follows. Section 2 presents the model and characterizes the equilibrium. Section 3 describes the data and Section 4 presents the econometric speci…cation. Section 5 contains the results of the empirical analysis. 2 Model I consider a sequential bargaining model of legal dispute resolution with perfect information and stochastic learning that extends Yildiz (2003) and Watanabe (2006). The players of the bargaining game are the plainti¤’ lawyer (p) and the defendant (d).9 The plainti¤’ lawyer s s and the defendant bargain over the compensation payment x 2 R+ from the defendant to the plainti¤ to resolve the dispute. Each player i 2 fd; pg has linear von-Neumann- Morgenstern preferences over the monetary transfer and legal costs.10 Both players know and Spier (1993), Watts (1994), Hay (1996, 1997), Rickman (1999), Choi (2003), and Polinsky and Rubinfeld (2003). These papers employ two-period models, which abstracts from learning during the dispute-resolution process. Hence, dropping a case during pre-trial bargaining is not considered. 9 s Throughout the paper, I assume that the plainti¤’ decision maker is his lawyer. Sloan et al. (1993) s …nd that plainti¤s “almost always [follow] their lawyer’ advice regarding settlement” (p.85). They report s that plainti¤s settled 100% of the cases when the lawyer’ advice favored accepting the settlement o¤er and s 4.8% when the lawyer’ advice favored rejecting it. Another reason for this assumption is the informational asymmetry between plainti¤s and their lawyers. Lawyers are experts on medical liability, and the plainti¤s are less likely to have better information than them. However, I do not assume that the defense lawyers are the decision makers because the defendants are primarily well-informed insurance companies with expertise on dispute resolution, and they generally have their own in-house council. 10 As in the most empirical work structurally estimating a dynamic bargaining model in the literature, I am limiting my attention to the linear utility (see, e.g., Merlo, 1997, Diermeier et. al. 2003, Merlo and Tang, 2010). This is mainly because introducing risk-aversion with di¤erent risk pereference makes the model computationally intractable. 6 the amount of the potential jury award V 2 R+ , but the outcome of the court ruling is uncertain, i.e., the players do not know who will win the case in the event of a trial.11 Hence, the defendant pays V to the plainti¤ if the court decides in favor of the plainti¤, s while the defendant does not pay any amount otherwise. I denote the plainti¤’ probability of prevailing in court by . Plainti¤s and their lawyers adopt a contingency-fee arrangement in the vast majority of medical malpractice cases in Florida, while defendants’lawyers charge an hourly legal fee.12 s The contingency-fee arrangement entitles the plainti¤’ lawyers to a fraction of the money received from the defendant only if a positive payment is received. I denote this fraction by s . This means that once a case is concluded, the plainti¤’ lawyer receives fraction of the s defendant’ payment to the plainti¤ as compensation for the legal services provided, and the plainti¤ receives the remaining fraction 1 of the transfer from the defendant. Timing and Phases I model the bargaining game to have two multi-period phases, the second being conditional on the …ling of a lawsuit in the …rst: the pre-litigation phase (Phase O) and the litigation phase (Phase L). The game starts with Phase O at period t = 0: The players bargain in each period of each phase until they reach an agreement. Phase O has a …nite number of periods T < 1, due to the statute of limitation at period t = T + 1 < 1, s after which the plainti¤’ claim to recover damages is barred by law. The plainti¤ has the option to …le a lawsuit in Phase O as long as no agreement has been reached and the case has not been dropped. The …ling of a lawsuit moves the game to Phase L. Thus, in Phase O, a case may either be …led (leading to Phase L), dropped, settled, or terminated by the statute of limitations. s The plainti¤’ endogenous decision to …le a lawsuit initiates Phase L. Let tL 2 f0; :::; T g denote the date of lawsuit …ling. Once the plainti¤ …les a lawsuit, the case is processed in court, moving towards the ruling, scheduled for the date T + 1 periods after the …ling date tL + T + 1 < 1. While the case is processed in court, it can be dropped by the plainti¤ or settled by both parties at anytime prior to tL + T . Failure to reach a settlement agreement by tL + T results in resolution by court ruling on tL + T + 1. Information and Beliefs The model is a game of perfect information. Each player can observe all the actions of the other player. The information revealed is also commonly ob- served by both players. Hence, there is no asymmetric information. The players, however, 11 In the literature the uncertainty regarding the ruling is caused either (i) by the uncertainty about the winning party (see, e.g., Priest and Klein, 1984) or (ii) by the uncertainty about the award amount (see, e.g., Spier, 1992). Implications of the model do not di¤er between (i) and (ii) because the expected award is what matters. I take the former assumption because of model and computational tractability. 12 In their Survey of Medical Malpractice Claimants conducted in Florida during 1989-1990, Sloan et al. (1993) report that 99.4% of the cases adpoted a contingency-fee arrangement. 7 pre-litigation phase (length T ) statute of limitation (T+1 ) filing lawsuit (tL ) judgment (tL+T+1 ) litigation phase (length T ) Figure 1: Diagram of the Timing. The plainti¤ and the defendant start bargaining over a settlement in the pre-litigation stage. At any time during the pre-litigatioin phase, as long as the plainti¤ has not dropped the case or no agreement has been reached, the plainti¤ has the option of …ling a lawsuit which endogenously determines tL and initiates the litigation phase. If a lawsuit is not …led, the case is not dropped, or a settlement is not reached by T in the pre-litigation phase, the case is no longer valid due to the statute of limitation. If agreement is not reached or the case is not dropped during the litigation stage, the case is resolved in court on tL + T + 1: do not have a common prior over the probability that the plainti¤ will win the case ( ). This s asymmetry in initial beliefs may be due, for example, to di¤erences in each party’ percep- tion of the relative ability of his lawyer or to di¤erences in opinion about the predisposition of potential juries. In fact, self-serving bias can lead players to interpret the same information in di¤erent ways. In experiments using actual civil case materials, Babcock, Loewenstein, Issacharo¤, and Camerer (1995) and Babcock and Pogarsky (1999) report that subjects involved in pre- trial bargaining interpret the same information di¤erently and come up with very di¤erent estimates depending on their role they are assigned to, i.e., whether they are the plainti¤ or the defendant in the experiment. I use Beta distributions to describe the players’beliefs on the probability of the plainti¤ prevailing s 2 [0; 1]. Player i’ initial belief, denoted by bi ; 0 is represented by Beta( i ; i ); where 0 < d < p < :13 A common parameter 13 See Yildiz (2003, 2004) for a similar learning mechanism. The arrival of information is deterministic in his model, while in this paper it is stochastic. 8 represents the …rmness of the belief as explained later. On the initial date, the players have the following expected probability of the plainti¤ prevailing: E(bp ) = 0 p for the plainti¤ and E(bd ) 0 = d for the defendant. New information arrives stochastically at the beginning of each period and players up- dates their beliefs. At the beginning of each period t in Phase J 2 fO; Lg, information related to the winning probability (such as the result of a third-party medical examination or testimony by an expert witness) arrives with probability J. I denote the presence of new information in period t by nt 2 f0; 1g, where nt = 1 denotes the arrival of new infor- mation while nt = 0 represents no arrival. The cumulated amount of information in period t is denoted by a state variable nt 2 f0; :::; tg, which can be written as, nt = n t 1 + nt : The informational content of the news is either in favor of or against the plainti¤. I denote the content of the new information, nt , by mt 2 f0; 1g. The content in favor of the plainti¤ is denoted by mt = 1, while mt = 0 denotes the information against the plainti¤. The content, mt , is drawn from a Bernoulli distribution with a true winning probability, , as the parameter, i.e., ( t 0 with probability 1 m = 1 with probability ; where is not known to the players. Thus, the cumulated information in favor of the plainti¤ denoted by mt 2 f0; :::; nt g, which is the second state variable in this model, is written as mt = mt 1 + nt mt ; where mt is multiplied by nt because the content of the information matters only if infor- mation arrives. Players update their beliefs according to Bayesian updating. The beliefs in period t, denoted by bp and bd , follow Beta( p +mt ; t t p +nt mt ) and Beta( d +mt ; d +nt mt ), respectively. Hence, in period t; the expectations on the probability of the plainti¤’s prevailing in the court ruling are p + mt E(bp ) = t , and + nt d + mt E(bd ) = t : + nt The players use these expectations as their estimates of . The …rmer are the beliefs (i.e., the higher is the …rmness of the belief parameter ), the lower is the impact of the 9 information obtained during the legal process. By modeling beliefs this way, I can capture the relative impact of new learning on prior beliefs. The information environment I have described above is common knowledge to both players. For notational convenience, I let kt = (nt ; mt ) 2 f0; tg f0; tg denote the two state variables (on the state of the information): the amount of information and its content in period t: Stage Games In Phase O, players play the following stage game in every period t 2 f0; :::; T g. 1. At the beginning of each period, information arrives with probability Ot and does not arrive with probability 1 Ot . The information is such that it a¤ects beliefs on the outcome of the ruling (e.g., the result of a third-party medical examination) and is commonly observed by both players. s 2. The plainti¤’ lawyer chooses whether to drop or continue the case. If she chooses to drop the case, the case is resolved without any payments from the defendant to the plainti¤. 3. If the case is not dropped, the Nature chooses the proposer, selecting the plainti¤ with probability and the defendant with probability 1 . The chosen party proposes the compensation-payment amount, x, which is then either accepted or rejected by the other party. If the proposal is accepted, the game concludes with the proposed amount of money, x; transferred from the defendant to the plainti¤, and the dispute s is resolved. In this case, the plainti¤’ lawyer receives x and the plainti¤ receives (1 )x. 4. If the proposal is rejected, the plainti¤ chooses whether to …le a lawsuit. The case moves to Phase L if the plainti¤ …les a lawsuit and remains in Phase O, where the same stage game is repeated, if the plainti¤ chooses not to …le. If the case is neither s …led nor settled within T periods, the statute of limitation renders the plainti¤’ claim ine¤ective. After the plainti¤ …les a lawsuit, the game moves to Phase L and the parties play the following stage game in every period t 2 ftL ; :::; tL + T g until the court decides on the case in period t = tL + T + 1. 1’ At the beginning of each period, information related to the outcome of the ruling arrives with probability Lt and does not arrive with probability 1 Lt . 10 s 2’ The plainti¤’ lawyer chooses whether to drop or continue the case. If she chooses to drop the case, the case is resolved without any payments from the defendant to the plainti¤. 3’ If the case is not dropped, the Nature chooses the proposer, selecting the plainti¤ with probability and the defendant with probability 1 . The chosen party proposes the compensation-payment amount, x, which is then either accepted or rejected by the other party. If the proposal is accepted, the game ends with the proposed amount of payment, x, transferred from the defendant to the plainti¤, and the dispute is s resolved. In such a case, the plainti¤’ lawyer receives x and the plainti¤ receives (1 )x. If the proposal is rejected, the case remains in Phase L and the same stage game is repeated until t = tL + T: Information arrival rates are allowed to di¤er across stages. One of the reasons for di¤ering arrival rates stems from the “discovery process,” which follows the …ling of a lawsuit and during which both parties can employ a variety of legal devices to acquire information on the case that follows the …ling of a lawsuit. I also allow the distribution of per-period costs to di¤er depending on whether the plainti¤ …les a lawsuit. The legal costs on both sides are typically considered to be higher in Phase L due to the discovery process as well as other procedures that require court involvement. I denote the per-period legal costs in period t during Phases O and L for player i i 2 fd; pg by COt 2 R+ and CLt 2 R+ , respectively. I assume that these legal costs are drawn independently in each period t from identical distributions in each Phase, which are denoted by GCO ( ) and GCL ( ). The realization of per-period costs only a¤ects the total legal costs, and not the equilibrium stopping timing and settlement terms. This is because players make those decisions only after they have paid the legal costs for that period, and their stopping decision is based on the expected legal costs for subsequent periods. Finally, I consider a common time-discount factor denoted by 2 [0; 1]. A Parameter for Measuring the Degree of Incentive Misalignment A contingency- s fee arrangement entitles the plainti¤’ lawyer to a fraction 2 [0; 1] of the money received from the defendant only if a positive amount of payment is received. By delaying the agree- s ment, the plainti¤’ lawyer incurs additional per-period legal costs, while receiving only a fraction s of the return from the costs. The plainti¤’ incentives and those of his lawyer are not aligned. The plainti¤ prefers to pursue the case much longer than his lawyer. The s plainti¤s’ objective is to maximize the expected payment from the defendant, while the lawyer always considers the cost in addition to the payment. In other words, the plainti¤’s lawyer does not have the incentive to pursue the case the way she would if she were the 11 plainti¤. The …rst-best solution for the side of the plainti¤ would be obtained if = 1, i.e., if the lawyer were the plainti¤ himself. The main objective of this paper is to measure the degree to which the incentives of lawyers and their clients are misaligned. I measure this degree using the following parameter: 2 [0; 1]: For an expositional simplicity, consider a one-period model. If a lawyer works entirely in the best interest of her client, she maximizes the expected payment and does not directly consider per-period legal costs.14 If the payment is x, this lawyer’ payo¤ can be written as s u0 = (1 )x: If, on the other hand, the lawyer works in her own best interest, she maximizes the fraction of the payment minus the legal costs she incurs. Hence, what she obtains is written as u1 = x C; where C corresponds to the legal cost she incurs. My objective is to measure the payo¤ s function for the plainti¤’ lawyer. The above two cases are the special cases in which (i) the lawyer works entirely in the best interest of her client and in which (ii) she works only in her own best interest respectively. Since these two cases are the two extremes, I can consider their convex combination with parameter , which can be written as u = u1 + (1 ) u0 (1) = (1 +2 )x C: While, = 1 implies that the lawyer behaves in her own interest and = 0 implies that she works in the best interest of her client. Hence, measures the degree to which the incentives of the lawyer and the plainti¤ are misaligned. For notational convenience, I de…ne function A( ) as A( ) 1 +2 : Note that = :5 does not imply …rst best in the multi-period model though it implies u = :5(x C) in the one-period example above. This is because the Bellman equations are not as simple as Equation (1) for mutli-period case. The socially optimal outcome corresponds to the case of = 1 and = 1:15 14 Note that even if the lawyer works in the best interest of the client, the per-period legal costs still a¤ect the utility of the plainti¤ indirectly through the equilibrium transfer x in the multi-period model. 15 Here, “socially optimal”means the e¢ cient outcome for the joint surplus of the plainti¤ and the plainti¤’ s lawyer, and the side of the defendant is not considered. This is because we focus on the agency problem s between the plainti¤ and the plainti¤’ lawyer. 12 2.1 Equilibrium Characterization The model is a dynamic game with perfect information and I employ a subgame-perfect equi- librium as the solution concept. Because the model has a …nite number of periods, backward induction provides us with a characterization of the unique subgame-perfect equilibrium. I start the analysis from the last stage in Phase L, and move to Phase O. 2.1.1 Phase L (Litigation Phase) In order to characterize the unique subgame-perfect equilibrium by backward induction, I start my analysis from the date of the court ruling at the end of Phase L. Recall that a Phase L subgame is reached only if the case is litigated at some point during Phase O. Let tL 2 f0; :::; T g denote the lawsuit …ling date, which is determined endogenously. If the players cannot settle after T periods by date tL + T , a court ruling on t = tL + T + 1 determines the outcome of the case. Since two decisions (whether to drop the case and whether to settle) are made in each period during Phase L, we de…ne two continuation values for each player. Let Vti tL (kt ) denote the continuation value for player i 2 fp; dg at i the beginning of period t in Phase L before any decisions have been made, and V t tL (kt ) denote the interim continuation value after the dropping decision, but before the settlement decision has been made at the end of the period t. Note that the subscript is t tL , which is the number of periods in Phase L, so that V1i (ktL ) corresponds to the continuation value in the …rst period of Phase L. The continuation values of the court rulings are EtL +T [VT +1 (ktL +T +1 )] = A( )E[bp +tL ]V , and p p T d d EtL +T [VT +1 (ktL +T +1 )] = E[bd +tL ]V; T s where A( )V is the amount the plainti¤’ lawyer obtains if she wins the case and V is the amount paid by the defendant to the side of the plainti¤. The plainti¤ receives the di¤erence: (1 i A( ))V . The notation of expectation Et [ ] denotes the subjective expectation of player i based on the information in period t;which is know to the plainti¤. The term E[bp +tL ] T is the expected probability of winning based on bd +tL , which is the belief of the plainti¤’ T s i lawyer one period before the court ruling. Note that the subscript for VT +1 is T + 1 because t = tL + T + 1 is T + 1 periods after the …ling period, tL : i i Having EtL +T [VT +1 (ktL +T +1 )] as the …nal values allows me to obtain Vti tL (kt ) and i V t tL (kt ) by applying backward induction. The equilibrium in the Phase L subgame is characterized in Proposition 1 below. Proposition 1 In the unique subgame-perfect equilibrium given kt and tL , 13 1. the payo¤ for the players at t 2 ftL + 1; :::; tL + T g in Phase L is expressed as p Vtp tL (kt ) = max 0; V t tL (kt ) ( p 0 if 0 > V t tL (kt ); Vtd tL (kt ) = d V t tL (kt ) otherwise, where n h i o p d d d p p p Vt tL (kt ) = max A( )Et Vt+1 tL (kt+1 ) CL ; Et Vt+1 tL (kt+1 ) CL p p p +(1 ) Et Vt+1 tL (kt+1 ) CL ; h i d d d d Vt tL (kt ) = Et Vt+1 tL (kt+1 ) CL 1 h i +(1 ) max Ep V p tL (kt+1 ) p d d CL ; Et Vt+1 tL (kt+1 ) p CL ; A( ) t t+1 s 2. the plainti¤ ’ lawyer drops the case at t 2 ftL + 1; :::; tL + T g in Phase L i¤ p 0>Vt tL (kt ); 3. the players settle at t 2 ftL + 1; :::; tL + T g in Phase L i¤ p p p d d d 0 Et [Vt+1 tL (kt+1 ) CL ] + A( )Et [Vt+1 tL (kt+1 ) CL ]; 4. given that players settle at t 2 ftL + 1; :::; tL + T g in Phase L, the payment is ( d d Et Vt+1 d tL (ktS +1 ) CL if the plainti¤ is a proposer xt = 1 p p p A( ) Et Vt+1 tL (ktS +1 ) CL if the defendant is a proposer, Proof. See Appendix A This proposition characterizes the subgame-perfect equilibrium in Phase L. The expres- sion for Vtp tL (kt ) in 1 is the continuation value at the beginning of the period when the s plainti¤’ lawyer is faced with the decision whether to drop the case. As described in 2, p s the plainti¤’ lawyer drops the case if her interim continuation value V t tL (kt ) is below 0. In that situation, the continuation value for both sides are zero. The interim continuation value after the decision whether to drop the case is made and before the decision whether to i settle is made, V t tL (kt ), is obtained by solving for random-proposer bargaining solution. Note that expectations are indexed by player because di¤erent players have di¤erent beliefs over the probability of winning in court. 14 The identity of the proposer does not a¤ect the settlement decision as can be seen in 3. The payment amount depends on the identify of the proposer as in 4. This arises because the players choose to settle if the joint surplus from reaching a settlement that day is larger than the joint surplus from continuing the case. The compensation payment depends on the identity of the proposer because the recognized proposer obtains all of the surplus. s This is true even in the extreme case of the plainti¤’ lawyer working in the best interest of her client (i.e., A( ) = 1 ) and not taking per-period legal costs into consideration. Delaying agreement is still costly for the lawyer because she misses out on the savings s from the defendant’ lower legal costs, which indirectly increase the compensation payment. Therefore, delaying agreement is costly for both players, and a quick settlement is preferable for both sides. However, the possibility of learning new information makes delay valuable because such information enhances the probability of a settlement, which in turn generates a positive surplus for both players. This fundamental tradeo¤ is the key determinant of the timing of settlement. The decision whether to drop a case is an individual rationality constraint for the plain- s ti¤’ lawyer in each period: she simply drops the case if the continuation value is negative. s If there is no learning, the continuation values for the plainti¤’ lawyer increase monotoni- cally over time as can be seen in 1. This monotonicity implies that if the value is negative in period t, the value is also negative in any period t0 < t, i.e., if a case is to be dropped at t, it must be dropped at any earlier date, t0 . Hence, if there is no learning, cases are dropped only in the initial period. Therefore, the dropping of cases in periods other than the initial period only occurs with the possibility of the arrival of new information against the plain- ti¤. In such cases, the continuation value stays positive over the …rst several periods, and becomes negative upon the arrival of unfavorable information. The proposition also shows that decisions whether to drop cases or settle are directly a¤ected by the degree of incentive misalignment . An increase in decreases the contin- s uation value of the plainti¤’ lawyer in two ways: through an increase in per-period costs p and through a decrease in A( ). A decrease in A( ) lowers V t tL (kt ) because the expected p damage to be awarded by the court, Et [A( )V ], decrease in A( ). Thus, the larger is the p parameter and the smaller is the continuation V t tL (kt ), the more likely it is that the case s will be dropped. In other words, the plainti¤’ lawyer prefers to drop a case at an earlier date if her incentives are less aligned with those of his client. In the extreme case of the s lawyer working entirely in her client’ best interest ( = 0), the case is never dropped. Similar intuition works for the e¤ect of incentive alignment and settlement decisions. The expression in 3 can be rewritten as p d p p d d Et [ CL + CL ] Et [Vt+1 tL (kt+1 )] + A( )Et [Vt+1 tL (kt+1 )]; 15 where the left-hand side is the expected per-period legal cost. The right-hand side is the expected surplus from continuation. An increase in increases the value of the left-hand side and decreases the value of the right-hand side.16 Hence, the less aligned are the incentives (the larger s is), the less time it takes on average to reach a settlement. The plainti¤’ lawyer who takes her own legal costs more seriously and focuses more on the payments she will receive, payments that are much lower than those received by the plainti¤, prefer to settle at earlier dates. 2.1.2 Phase O (Pre-Litigation Phase) Given the continuation value of …ling a lawsuit, V1i (kt ), as above and the zero continuation value of the statute of limitation, a similar backward-induction argument can be made for Phase O. Let Wti (kt ) denote the continuation value for player i 2 fp; dg at the beginning of date t in Phase O with information state kt . The maximum number of periods in Phase O is T ; after which the claim by the plainti¤ loses its value due to the statute of limitation. Hence, each player has a continuation payo¤ of 0 at date T + 1, i.e., p d WT +1 (kT +1 ) = WT +1 (kT +1 ) = 0: I compute Wti (kt ) by applying backward induction and having the above as …nal values. In Phase O, the plainti¤ has the option to …le a lawsuit on any date t 2 f0; :::; T g. Solving the value function in Proposition 1 up to the …rst period in Phase L allows me to obtain the continuation value of the Phase L subgame, V1i (kt ). Note that this continuation value of litigation does not depend on the date of litigation, tL , itself, but does depend on the information state kt in period t. s First, I look at the …ling decision by the plainti¤’ lawyer because that decision is the s last one to be made in a period. The plainti¤’ lawyer chooses to litigate at the end of date t if and only if Et V1p (kt+1 ) p CL p Et Wt+1 (kt+1 ) p CO ; where the left-hand side is the payo¤ from …ling a lawsuit and the right-hand side is the payo¤ from staying in the pre-litigation phase. Hence, the interim continuation value for s the plainti¤’ side after the settlement decision and before the …ling decision in period t, Ytp (kt ), is written as Ytp (kt ) = max Et V1p (kt+1 ) p p CL ; Et Wt+1 (kt+1 ) p CO : 16 d d d d Though A( ) increases in , A( )Et [Vt+1 tL (kt+1 )] decreases in because Et [Vt+1 tL (kt+1 )] is always negative and decreases in : 16 s The defendant’ interim continuation value depends on the …ling decision of the plainti¤’s lawyer described above, and is written as ( Et V1d (kt+1 ) d CL if Et V1p (kt+1 ) p CL p p Et Wt+1 (kt+1 ) p CO Ytd (kt ) = d p Et Wt+1 (kt+1 ) CO otherwise. I can now conduct the exact same analysis for settlement and dropping decisions as I did for Phase L; and the subgame-perfect equilibrium for Phase O is characterized in Proposition 2. Proposition 2 In any subgame-perfect equilibrium given kt ; 1. the payo¤ for the players at t 2 f0; :::; T g in Phase O is expressed as p Wtp (kt ) = max 0; W t (kt ) ( p d 0 if 0 > W t (kt ) Wt (kt ) = d W t (kt ) otherwise, where n o p W t (kt ) = max A( )Ytd (kt ); Ytp (kt ) + (1 )Ytp (kt ); d 1 W t (kt ) = Ytd (kt ) + (1 ) max Y p (kt ); Ytd (kt ) ; A( ) t 2. the players settle at t 2 f0; :::; T g in Phase O i¤ Ytp (kt ) + A( )Ytd (kt ) 0; s 3. the plainti¤ ’ lawyer drops the case at t 2 f0; :::; T g in Phase O i¤ p 0 > W t (kt ); s 4. the plainti¤ ’ lawyer litigates at t 2 f0; :::; T g in Phase O i¤ Et [V1p (ktL +1 )] p Et Wt+1 (kt+1 ) ; 5. given that players settle at t 2 f0; :::; T g in Phase O, the payment is ( Ytd (kt ) if the plainti¤ is the proposer xt = Ytp (kt ) if the defendant is the proposer. 17 Proof. See Appendix B. This proposition characterizes the subgame-perfect equilibrium in Phase O. The me- chanics of the settlement and dropping decisions are exactly the same as those used to characterize Phase L. The only di¤erence is that the cost of delay and the possibility of s learning in the subsequent period depend on the plainti¤’ decision whether to …le a lawsuit at the end of each period. s The plainti¤’ decision whether to …le a lawsuit is also derived from a tradeo¤ between the cost of delay and the possibility of learning. The cost of delaying …ling has two compo- nents. The …rst component is the per-period legal cost of the pre-litigation phase, because the total length of the underlying game becomes one period longer if the …ling is delayed by one period. Even though the plainti¤ incurs no legal cost per period, this delay still impacts s him because minimizing the defendant’ legal cost may result in a higher compensation pay- ment to the plainti¤. The second component is the cost of delay due to discounting. The plainti¤ prefers to obtain the continuation value of the Phase L subgame earlier, since a one-period delay in …ling costs him (1 )V1p ( ) when no information arrives. The bene…t of delaying …ling by one period is that the parties have one more period to obtain new information and hence reach an agreement. Thus, the plainti¤ prefers to stay in Phase O i longer if CO are small and is large, while low provides an incentive for the plainti¤ Ot to …le early. 3 Data I use a detailed micro-level data set on medical malpractice disputes in Florida. The data are collected by the Florida Department of Financial Services, an insurance regulator of the Florida state government. In Florida, a statute on professional-liability claims requires medical malpractice insurers to …le a report to the Department of Financial Services on every closed claim once the claim is resolved. The report contains detailed information on the dispute-resolution process as well as on individual case characteristics. The information on the dispute-resolution process includes important calendar dates (such as the date of: the incident, the …ling of the initial claim, and the resolution, reached either by dropping the case, settlement, or court ruling), settlement-payment amounts (or amount of award by the court), and the total legal costs incurred by the defendant. The information on the s case characteristics includes the patient’ characteristics (e.g., age and sex), the defendant’s characteristics (e.g., defendant type of license, specialty, and insurance policy), and the characteristics of the injury (e.g., severity and place of occurrence). Hence, this data set contains detailed information on the variables of interest, i.e., if and when a lawsuit is …led, 18 Number of Resolution Mean Mean Legal Costs Observations Probability Compensation for Defense Dropped without 0 7; 330 783 0.146 Lawsuit (0) (23; 023) Dropped after 0 40; 370 751 0.140 Lawsuit (0) (44; 978) Settled 314; 266 9; 047 472 0.088 without Lawsuit (303; 917) (15; 014) Settled 303; 402 53; 989 2,887 0.537 after Lawsuit (379; 909) (88; 887) Court Ruling 541; 832 127; 966 127 0.024 with Positive Award (620; 722) (113; 147) Court Ruling 0 83; 182 359 0.067 with No Award (0) (87; 268) 203; 210 45; 047 Total 5,379 1.000 (343; 922) (77; 794) Table 1: Descriptive Statistics I –Resolution Probability, Payments and Legal Costs. Com- pensation payments and legal costs are measured in 2000 dollars. The numbers in paren- theses are standard deviations. whether or not the case is settled out of court or dropped, the time it takes to reach a resolution, the terms of that resolution, and the legal costs incurred. The observations I use consist of 5,379 claims against physicians.17 All of the claims were resolved between October 1985 and July 199918 and each defendant’ legal costs exceeded s $1,000.19 I restrict my attention to the cases in which injuries resulted in permanent major damage or the death of the patient. The observations used in this paper include all those used in Watanabe (2006) and all of the cases dropped by the plainti¤s during the legal process. Tables 1 and 2 report the descriptive statistics of the variables used for the estimation. Regarding the modes of resolution, 28.6% of the cases were dropped or settled with zero payment, 62.5% of the cases were settled out of court, and the remaining 9.1% of the cases were resolved by a court ruling. Regarding the dropped and settled cases, only one-seventh 17 In order to control for the heterogeneity of the plainti¤s, I excluded observations with claims against hospitals, HMOs, dentists, ambulance surgical centers, and crisis stabilization units. 18 No major changes in state law pretaining to the resolution of medical malpractice disputes took place during this period. 19 I apply this criterion following Sieg (2000) in order to restrict my attention to non-trivial cases. 19 Time to Time to Resolution Total Time to Filing after Filing Resolution Dropped 4:81 without Filing (3:22) Dropped 2:30 7:36 9:66 after Filing (2:31) (4:51) (5:09) Settled 3:23 without Filing (2:41) Settled 2:48 7:45 9:93 after Filing (2:26) (4:55) (5:16) Court Ruling 2:69 11:41 14:10 with Positive Award (2:67) (6:61) (7:17) Court Ruling 2:50 9:69 12:19 with No Award (3:21) (5:17) (6:24) 2:45 7:75 8:81 Total (2:38) (4:76) (5:57) Table 2: Descriptive Statistics II – Timings: Numbers are in quarters of a year. The numbers in parentheses are standard deviations. of the settled cases were settled without a lawsuit being …led, and the majority of the settled cases were settled after a lawsuit was …led. The proportion of the cases dropped without …ling a lawsuit and after …ling a lawsuit is about the same at 14%. For the cases resolved by court ruling, the defendants were three times more likely to win (i.e., no damages were awarded to the plainti¤). The mean compensation payments in cases settled after a lawsuit was …led were similar to those in cases settled without a lawsuit being …led (around $300,000). The defendants’ legal costs substantially di¤ered across modes of resolution. Defense lawyers usually charge based on the amount of time they spend on the case. It is then not surprising to learn that the mean of the defendants’legal costs correlates strongly with the mean time it takes to reach a resolution, as displayed in Table 2. The mean of defendants’ legal costs for cases settled without a lawsuit is about one-…fth of that for cases settled after a lawsuit. This di¤erence correlates with the shorter mean time to resolution, as outlined in Table 2. The cases settled after a lawsuit have a signi…cantly higher mean cost ($53,989) compared to those settled without a lawsuit ($9,047). Similarly, the cases dropped after …ling a lawsuit have a signi…cantly higher mean cost ($40,370) compared to those dropped 20 0.35 Time to dropping without a lawsuit 0.3 Time to settlement without a lawsuit 0.25 Time to filing a lawsuit fraction 0.2 0.15 0.1 0.05 0 1 2 3 4 5 6 7 8 9 10 11 12 quarters Figure 2: Histogram of Time to Dropping a Case, Reaching a Settlement and Filing a Lawsuit in Pre-litigation Phase. Note that the timing fractions across three di¤erent modes add up to one. without a lawsuit ($7,330). Among the cases resolved by court ruling, the defendants’ mean costs are more than 50% higher for cases won by the plainti¤s than those won by the defendants, which again corresponds to the longer time to resolution. The mean time to …ling a lawsuit, which corresponds to the periods spent in the pre- litigation phase, is similar across the dropped cases, the settled cases, and the cases resolved by court ruling. The time it takes to reach a resolution di¤ers between the cases settled after a lawsuit is …led and those resolved by the court. This di¤erence results from the di¤erence in the time it takes to reach a resolution after a lawsuit is …led. Cases settled without a lawsuit, i.e., those settled during the pre-litigation phase, on average, spend more time in the pre-litigation phase than …led cases, and cases dropped without a lawsuit spend, on average, even more time (about 4.81 quarters) in the pre-litigation phase. Figure 2 provides the histogram of …ling, dropping, and settling cases in the pre-litigation phase. Note that the fractions for …ling, dropping, and settling cases add up to one because each case is either …led, dropped without a lawsuit, or settled without a lawsuit. In Florida, the statute of limitation regarding medical malpractice cases is two years. Hence, more 21 0.1 Time to dropping after filing 0.08 fraction 0.06 Time to settlement after filing 0.04 0.02 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 quarters Figure 3: Histogram of Time to Dropping and Settlement after Filing. Note that the frac- tions of the two di¤erent resolution modes add up to one. Time is counted, in quarters, starting on the date the lawsuit is …led. Cases exceeding 6 years are included in the bin of the 24th quarter. than 95% of the cases are …led or settled without a lawsuit before the ninth quarter.20 A fraction of the cases in which a lawsuit is …led during the pre-litigation phase and a fraction of the cases settled without a lawsuit have a common pattern. In both, the hazard rate increases signi…cantly in the second quarter and then continues to increase gradually over time. Corresponding to the high hazard rate after the second quarter, the histogram shows that more than 30% of the cases are …led in the second quarter before the fraction declines rapidly over time. Regarding the time to reaching a settlement without a lawsuit, the decline after the second quarter is much slower as a result of a lower hazard rate. Compared to the …led and settled cases, the hazard rate for dropping a case is lower and dropped cases spend more time in the pre-litigation phase. Figure 3 presents more details on the time it takes to reach a settlement or drop a case after a lawsuit is …led. The hazard rates for reaching a settlement and dropping a case are almost the same and they increase over time until ninth quarter, as evident from the very similar subsequent patterns. 90% of the settled cases are settled in 13 quarters and less than 5% of them are settled after 16 quarters. Similarly, 90% of the dropped cases are dropped in 13 quarters and less than 5% of them are settled after 17 quarters. Compensation payments have very high variance. Figure 4 presents the distribution of (log) compensation payments from the defendant to the plainti¤. Of the total number of 20 The remaining cases may involve, for example, invervening medical complications that entail an auto- matic extension of the statute of limitations. 22 0.3 Settled without filing 0.25 Settled after filing Judged by the court 0.2 Dropped fraction 0.15 0.1 0.05 0 7 - 3 7 .4 4 .3 0 6. 8. 2. 3. 7 29 68 2. -3 -9 8 -1 ,9 -7 -2 98 -1 .4 .3 0 1, 3 .7 13 36 8. 4 98 9. 26 72 log of payment (in thousand dollars) Figure 4: Histogram of Payments. The fractions of the four di¤erent modes of resolution add up to one. cases, about 6.7% concluded with a court ruling in favor of the defendant and 28.6% of them were dropped and, thus, resulted in zero payments. The amount of positive payments has a very high variance, and the shape of the distribution is close to a log-normal distribution. The distributions of the cases settled without a lawsuit and of those settled after a lawsuit are similar. More than 40% of the cases involve payments between $98,700 and $729,400. The distribution for the cases with positive damages awarded by the court is also similar to the log-normal distribution, while the mean and variance are much larger. Figure 5 presents the distribution of legal costs for …ve di¤erent modes of resolution. Similar to that of compensation payments, the distribution of legal costs is close to a log- normal distribution for each mode of resolution. The distribution shifts from left to right in the following order: cases dropped without a lawsuit, cases settled without a lawsuit, cases dropped after a lawsuit, cases settled after a lawsuit, and the cases ruled by the court. The cases dropped without a lawsuit tend to have much lower costs compared to the other cases, while those brought before the court tend to have much higher costs than the other cases. 23 0.14 Dropped without filing 0.12 Settled without filing 0.1 Dropped after filing Settled after filing fraction 0.08 0.06 Judged by the court 0.04 0.02 0 .9 .1 4 .9 0 3 9 7 .8 - 2. 6. 9. 8. 3. 8 -4 -8 -2 62 2. -2 -3 -5 -9 -1 -1 16 9 9 0 2. 4. .4 .0 .3 .9 1 8. .7 13 22 36 59 log of legal cost (in thousand dollars) 98 Figure 5: Histogram of the Defense Legal Costs. The fractions of the …ve di¤erent modes of resolution add up to one. 4 Estimation The add several dimensions of unobserved heterogeneity and period-speci…c legal costs for the …rst two periods to the model for estimation. I estimate the model using a maxi- mum likelihood estimator. The likelihood is constructed from the conditional probabilities of dropping a case, reaching a settlement, and …ling a lawsuit as well as the conditional distributions of the legal costs and the settlement amount. Speci…cation The model I estimate allows for unobserved heterogeneity of cases on sev- eral dimensions. First, I consider unobserved heterogeneity in T , the time it takes for a court to rule the case after a lawsuit is …led. For any two cases that are resolved by court ruling, the time it takes to reach that decision, T , can vary signi…cantly. For example, T can be a¤ected by congestion in the legal system in a particular jurisdiction. I denote the cumulative distribution function of T by FT ( ). I can estimate FT nonparametrically using the data of the cases that are judged by the court because there is no reason to believe that 24 settled cases would have a systematically di¤erent distribution of the time leading up to the court ruling compared with case resolved by court.21 This estimation is outside of the model, and it is simply a histogram of time to judgment. The statute of limitation period, T , which is also exogenous in the model, also di¤ers across cases, though for medical malpractice litigation in Florida, it is set at two years. The legally determined statute of limitation period is not equal to the actual length of time players can bargain without …ling a lawsuit, T . The statute of limitation is legally counted from the date the incident occurs, but the bargaining does not necessarily begin on the date of occurrence. For example, a plainti¤ may begin bargaining eight months after the incident occurs, which leaves him 16 months to …le a lawsuit if the legal statute of limitation period is 24 months. I assume T to follow a negative binomial distribution, and denote its cdf by FT ( ) with parameters 0T and 1T : Another dimension of unobserved heterogeneity are on jury awards. Cases that are resolved by court ruling may have very di¤erent potential jury awards, V , depending on the unobserved characteristics of the case, the composition of the jury, and other factors. Hence, I consider unobserved heterogeneity in V , and assume that V follows a log-normal distribution FV ( ) with the mean and variance denoted by and 2. V V Per-period legal costs are independently and identically drawn in each period from Gamma distributions GCO ( ) and GCL ( )22 , and I also consider unobserved heterogeneity in the mean of these distributions. These distributions could di¤er across cases due to factors such as the law …rms the plainti¤s and defendants employ or other unobserved characteristics of the case. I denote the scale parameters of GCO ( ) and GCL ( ) by ! O and ! L , respectively, and assume a common shape parameter denoted by !. Regarding the heterogeneity of these distributions, I assume the mean of distribution GCO ( ) (i.e., expected per-period legal cost in Phase O), ! O !, to follow a log-normal distribution FC ( ) with parameters and 2. With respect to the mean of distribution GCL ( ), I reparametrize ! L with a C C new parameter i i so that E[CL ] = (1 + )E[CO ]; or ! L ! = (1 + )! O !: I do so because per-period legal costs after …ling a lawsuit are most likely to increase due to the increase in the hours worked by the lawyers to prepare more documentation, and there is signi…cant heterogeneity in the degree to which they increases after …ling. I assume parameter to follow a log-normal distribution F ( ) with parameters and 2. Note that there is no need to estimate ! O and ! L because those parameters are drawn from distributions FC and F , which are estimated. For the plainti¤ and the defendant, I draw the parameters of the 21 If time to trial were a¤ected by the selection of cases and unsettled and undropped cases takes longer time to trial, estimate of FT could be biased upward. 22 The legal cost is assumed to be i.i.d. across periods in the same phase because there is no particular reason or evidence available to have a particular functional form. Even if there is it would be hard to identify unless I make a very strong functional form assumptions across periods in the same phase. 25 cost distributions independently from the above distributions, FC ( ) and F ( ), as discussed below as an identi…cation assumption. For notational convenience, I denote the realization of unobserved heterogeneity by Z = f! O ; ; V; T; T g: Finally, I have also introduced the cost of …ling at very early period in order to re‡ect the fact that it can be extremely costly to prepare all documentations at early stage in pre-litigaiton phase. This cost is drawn from log-normal distributions FEF t ( ), which has mean = t and variance 2 . The only reason I include this addtional cost t EF 0 EF 1 EF is to explain low …ling rate in the …rst quarter of pre-litigation phase. The model cannot generate the pattern that less cases are …led in 1st quarter than in 2nd quarter. Note that the players in the model are assumed to have perfect information about realzations Z = f! O ; ; V; T; T g, which are the primitives for the model each players face in playing the bargaining game. As an econometrician, I cannot observe the realization Z = f! O ; ; V; T; T g, hence I need to intergrate over them in order to obtain the distribution of outcomes as below. Estimation I use the equilibrium characterization derived by Propositions 1 and 2 to compute the likelihood contribution of each observation. I construct the likelihood function using the conditional probabilities of equilibrium outcomes as in Appendix C. The contribu- tion to the likelihood function of each observation in the sample is equal to the probability of observing the vector of endogenous events (x; tS ; tL ; s; l) given the vector of the parame- ters =f ; d; p; ; O0 ; O1 ; L0 ; L1 ; ; ; FT ; FT ; FC ; F ; FV ; FEFt g. Because I consider unobserved heterogeneity in Z = f! O ; ; V; T; T g, I conduct a Monte Carlo integration over Z in order to obtain the likelihood, which is written as Z Z L( jx; tS ; s; tL :l) = Pr(x; tS ; s; tL ; ljZ; )dFC dF dFT dFT dFV dFEF t ; where Pr(x; tS ; s; tL ; ljZ; ) is computed as in Appendix C. For parameters (contingency fee), I use = 0:33 following Sieg (2000) and Sloan et al (1993). I use = 0:995 for the time-discount factor for a quarter. Identi…cation The distribution of the time it takes for the court to rule a case from …ling, FT , is nonparametrically identi…ed by the observations on the time it takes to court ruling from …ling date for the cases ruled by the court. The assumption behind this is that the time to trial from …ling date are not systematically di¤erent among ruled, settled, and dropped cases. Similarly, the winning rate for the plainti¤ in court ruling identi…es , which is estimated outside of the model. The distribution of time to statute of limitation FT is identi…ed by the data on the time it takes to …le a lawsuit with the distributional 26 assumption. s Because data on the plainti¤’ legal costs are not available, an identi…cation assumption s must be made. I assume that the parameters for the plainti¤’ cost distributions are drawn p d s from the same distribution as the defendant’ legal costs, i.e., Cjt and Cjt are independently and identically drawn from di¤erent cost distributions whose parameters are independently and identically drawn from the same distributions, FC and F . We make this assumption considering that the market for legal services is competitive. These distributions are identi- …ed based on the observation of the defendants’cumulative legal costs and the observation of time it takes for each case to reach a resolution in Phase O and Phase L. Given that the cost distributions are identi…ed as above, the distribution of awards, FV , the arrival rate of information, and the belief parameters are jointly identi…ed by the fraction of the di¤erent modes of resolution, the resolution timing, and the observed distribution of settlement payments conditional on lawsuit-…ling and settlement timing as well as the court awards for ruled cases. The …rmer are the initial beliefs, the more quickly the case is resolved. The more di¤erent are the initial beliefs, the longer it takes for cases to resolve. The parameters on the cost of …ling lawsuits early, FEF t , are identi…ed by the fractions of cases …ling lawsuits in the …rst three periods. I adopted this speci…cation because the model without this cost could never generate the patterns of …ling for the …rst three periods. The distribution is identi…ed with three parameters, and the information on the fractions of these three periods can identify FEF t given that other parameters of the model are identi…ed. The parameter for relative bargaining power, , is identi…ed based on the settlement payment conditional on …ling and settlement timing. This is because lawsuit-…ling and set- tlement timing determines the surplus given other parameters, which the proposer receives with probability and non-proposer with probability 1 . Finally, as discussed in the model section, the parameter for incentive misalignment, , a¤ects the fraction of cases dropped and the timing of dropping cases and settlement. The more misaligned are the incentives (the higher is ), the less time it takes to reach a settlement or dropping a case and the greater is the number of cases that are dropped. Hence, is identi…ed based on the fraction of the cases that are dropped and the timing of their dropping given the identi…cation of the rest of the parameter as discussed in the model section above. 5 Results 5.1 Parameter Estimates Parameter estimates are reported in Table 3. The estimate of the weighting parameter s 2 [0; 1] is 0.8128. This implies that the plainti¤’ lawyer works more in her own short- 27 Parameter Estimates Parameter Estimates 0.8128 (0.0377) ! 0.7966 (5.1032) 0.0001 (0.0001) 2 0.4219 (0.1475) d C p 0.0021 (0.0001) -0.9013 (0.2372) 0.0066 (0.1075) 2 3.4108 (0.4460) O0 O1 0.0073 (0.0101) V 13.2785 (0.4797) 0.0064 (0.0451) 2 0.9360 (0.2209) L0 V L1 0.0231 (0.0023) EF 0 15.4099 (1.4625) 0.2613 (0.1930) EF 1 9.8190 (0.5432) 0.6053 (0.3549) 2 3.7688 (2.8544) EF 0.0022 (0.0001) 0T 7.5491 (0.3328) C 6.7278 (0.4909) 1T 0.6488 (0.0012) Log-likelihood 40504.60 Table 3: Maximum Likelihood Estimates. Standarad errors are reported in parentheses. run interest than in the best interest of her clients. Though I cannot distinguish among the di¤erent mitigating factors such as reputational concern, professional liability, capacity constraint, or altruism, the results show that these factors do not work very strongly in this context. The arrival rate of information in period t of the pre-litigation phase is Ot = 0:0066 + 0:0073 t, and in the t0 -th period of the litigation phase is Lt0 = 0:0064+ 0:0231 t0 : Thus, information is more likely to arrive as the process moves forward in both Phases. Also, the arrival rate is much higher once a lawsuit is …led. This is a natural result because the …ling of a lawsuit institutionally in‡uences the arrival of new information, which a¤ects the outcome of the court ruling. The di¤erences in beliefs are large at the initial stage, but the initial beliefs are very s weak. The plainti¤’ belief on the probability he will win is p= = 0:9554, while the mean s of defendant’ belief on the probability the plainti¤ will win is d= = 0:0315. These beliefs are very weak because the …rmness parameter, ; is very small at 0:0021. This implies that learning plays an important role. The mean defense per-period legal cost in the pre-litigation phase is $2,762, with an estimated variance of $4,252. On average, the per-period legal cost increases 2.115 times after the …ling of a lawsuit and the mean per-period legal cost in the litigation phase is $5,842. The mean damages, V 23 ; are $905,738. Finally, the relative bargaining power between the plainti¤ and the defendant, captured by , is estimated to be 06053.24 This implies that plainti¤s have relatively stronger bargaining power though the standard error 23 This is the amount the plainti¤ will (potentially) receive if he wins in court 24 In the model, is a probability that the plainti¤ will be recognized as the proposer. This is a measure s of the bargaining power in the model because the proposer always bene…ts from the proposer’ advantage, while the other party is only o¤ered the amout equal to his continuation payo¤. 28 is relatively large. 5.2 Model Fit I …nd that the model …ts all aspects of the data well. I provide the …t of the model with respect to the time it takes to …le a lawsuit and reach a settlement, as well as the compensation payments for the di¤erent modes of resolution. In Figures 6 and 7, I present the …t of the model to the data on the time it takes to …le a lawsuit and drop a case. The model replicates the dynamic patterns of …ling lawsuits and dropping cases in the pre- litigation phase shown in Figure 6. In particular, the model is able to …t the data on the time it takes to …le a lawsuit, which increases sharply in period 2 and decreases gradually thereafter. Regarding settlement in the pre-litigation phase, the magnitude of the fraction is captured correctly. Though I do not report the …t of the settlement in the pre-litigation phase in Figure 6, the degree of the …t is similar to that of lawsuit …ling and the dropping of cases. Figure 7 shows the …t of the model on the time it takes to reach a settlement and drop a case after …ling a lawsuit. The model captures the shape of the data, which increase over the …rst several periods and decrease gradually thereafter. The di¤erence between the predicted fractions of cases that will settle and the data from the …rst three periods and the last several periods may result from the linearity assumption regarding the rate of arrival. This assumption may have prevented the model from capturing some of the factors in the data. 6 Comparison to the First-Best Outcome In this section, I compare the estimated model with the …rst-best outcome for the side of the plainti¤. I de…ne the …rst-best outcome as the outcome under which the expected joint surplus for the side of the plainti¤ is maximized. The joint surplus for the side of the plainti¤ is the expected payment from the defendant minus the legal costs incurred by the s plainti¤’ lawyer. The …rst-best outcome is computed by simulating the model, setting parameter values to = 1 and = 1. This corresponds to a situation in which the side of the plainti¤ fully considers both the costs and the bene…ts as if the plainti¤ were a lawyer representing herself. In other words, the side of the plainti¤ internalizes all the legal costs incurred by the s plainti¤’ lawyer and 100% of the payment rather than fraction under a contingency-fee arrangement. Table 4 summarizes the comparison of the …tted model with the simulated …rst-best outcome. Under the …rst-best outcome, about 4% less cases are dropped before a lawsuit is 29 0.35 Time to dropping without a lawsuit 0.3 Time to dropping (fit) 0.25 Time to filing a lawsuit Time to filing (fit) fraction 0.2 0.15 0.1 0.05 0 1 2 3 4 5 6 7 8 9 10 11 12 quarters Figure 6: Histogram of Timing in the Pre-Litigation Phase 0.1 Time to settlement after filing (data) 0.08 Time to settlement (fit) Time to dropping after filing (data) fraction 0.06 Time to dropping (fit) 0.04 0.02 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 quarters Figure 7: Histogram of Timing after Filing a Lawsuit 30 Fitted Model First Best Fraction of Dropped Cases in Phase O 11.3% 7.4% Fraction of Cases with a Lawsuit 77.5% 80.8% Mean Time to Resolution (quarters) 7.77 8.10 Mean of Legal Cost (dollars) 29,885 31,557 Mean Payment (dollars) 1,180,982 1,448,523 Joint Surplus (dollars) 1,151,098 1,416,937 Table 4: First-best outcome …led, and 3% more lawsuits are …led. The mean time to resolution increases by 0.3 quarters. Re‡ecting a higher fraction of lawsuits, fewer dropped cases, and a longer time to resolution, the mean legal cost increases by 5.8%. However, the increase in legal costs is very small compared to the 22.7% increase in payments. Hence, the expected joint surplus under the …rst-best outcome is 23.1% larger. 7 Policy Experiments Finally, I conduct a counter-factual policy experiment to limit the contingency fee.25 Reg- ulation of contingency fees has been adopted by 15 states as of 200526 and is continuously under consideration at the federal level. In spite of its policy importance, little is known empirically about how regulation on contingency fees a¤ects the various outcomes of med- ical malpractice disputes, such as legal costs, settlement payments, and the probability of lawsuits, as well as how it a¤ects the agency problem of lawyers and their clients. For the counterfactual experiment, I set the limit on contingency fees at 20% ( = 0:2), and simulate the distribution of equilibrium outcomes using the estimated parameters. With s a 20% limit, the fraction of the payment to the plainti¤’ lawyer decreases from 33% to 20%, while the per-period legal costs incurred by the lawyer remain the same. This implies s that the plainti¤’ lawyer has less incentive to maximize the joint-surplus because she can receive an even smaller fraction of the payment. Table 5 summarizes the comparison of the …tted model with the outcome of the counter- factual experiment. The policy of limiting the contingency fee would increase the number of cases to be dropped in the pre-litigation phase from 11.3% to 12.7% and reduce the …ling 25 Note that the results of the counterfacutal policy experiments are limited in the sense that the e¤ects of policy changes only a¤ect the behavior through the model. Given that the estimated parameter is a s reduced form of the various factors that a¤ect the incentive of plainti¤’ lawyers, the results is a benchmark for what would happen if policy change would a¤ect factors only in the way the model considers. For a more complete experiments, the model need to be expanded so that factors such as reputaiton and capacity constraint are included. Also, more detailed data on lawyers is crucial to estimate such models. 26 See the database on state tort law reform constructed by Avraham (2006). This regulation typically imposes a limit on the fraction lawyers can receive as a contingency fee (e.g., 40% or 33%). 31 Fitted Model Experiment Fraction of Dropped Cases in Phase O 11.3% 12.7% Fraction of Cases with a Lawsuit 77.5% 76.3% Mean Time to Resolution (quarters) 7.77 7.66 Mean Legal Cost (dollars) 29,885 29,020 Mean Payment (dollars) 1,180,982 943,123 Joint Surplus (dollars) 1,151,098 914,103 Table 5: Policy Experiment on Limiting Contingency Fees at 25% of lawsuits from 77.5% to 76.3%. The mean time to resolution also decreases from 7.77 quarters to 7.66 quarters. These changes are consistent with the change in the lawyer’s incentive due to the limit on the contingency fee. Re‡ecting a shorter time to resolution, more frequent dropping of cases, and less frequent …ling of lawsuits, legal costs also decrease from $29,885 to $29,020. However, the 20.1% decrease in the mean payment is much larger than the savings on legal costs. This results in a 20.6% decrease in the joint surplus. 8 Conclusion In this paper, I investigated the degree to which expert agents work in the best interest of their clients in the case of medical malpractice lawyer in Florida. Instead of estimating a model of agency problem, I constructed a bargaining model, which nests the convex s combination of two extreme cases – one in which the objective of the plainti¤’ lawyer is to maximize her shot-run pro…t and one in which her objective is to maximize her client’s payment. The results show that lawyers are much more likely to maximize their own short- run pro…ts than those of their clients. Finally, I conducted a counterfactual experiment to simulate the …rst-best outcome as well as the policy experiment of limiting the contingency fee to 20%. One of the issues the paper could not address is which of the mitigating factors, such as reputational concern and altruism, are working. In other words the degree of the experts’ agency problem can only be modelled as a reduced form parameter in this paper. Dis- entangling the relative importance of these factors would be possible if more information on lawyers were available as data. Another topic the paper could not address is the welfare issue with respect to the medical liability system. This is because the model is about the bargaining that occurs after alleged medical malpractice incidents occur and not about how the liability system a¤ects the probability that such incidents will occur. 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[46] Wolinsky, Asher (1993) “Competition in a Market for Informed Experts’ Services,” Rand Journal of Economics, 24(3), pp. 380-398. [47] Wittman, Donald (1989) “Dispute Resolution, Bargaining, and the Selection of Cases for Trial: A Study of the Generation of Biased and Unbiased Data,” Journal of Legal Studies, 17, pp. 313-352. [48] Yildiz, Muhamet (2003) “Bargaining Without a Common Prior - An Immediate Agree- ment Theorem,” Econometrica, 71, pp. 793-811 [49] Yildiz, Muhamet (2004) “Waiting to Persuade,”Quarterly Journal of Economics, 119, pp. 223-248 9 Appendix Appendix A Proof of Proposition 1 Consider two separate cases depending on whether the players will settle or continue. p p p d d d First, consider the case of Et [Vt+1 tL (kt+1 ) CL ] + A( )Et [Vt+1 tL (kt+1 ) CL ] > 0; in which players do not settle. I will show that they will not settle under this condition using proof by contradiction. Suppose that the players settle in period t with monetary p p p transfer xt . The plainti¤ agrees only if A( )xt Et Vt+1 tL (kt+1 ) CL , while the d d d 1 defendant agrees only if xt Et Vt+1 tL (kt+1 ) CL . This requires 0 = (xt 1 p p p d d d xt ) A( ) Et Vt+1 tL (kt+1 ) CL + Et Vt+1 tL (kt+1 ) CL . This contradicts with p p p d d d Et [Vt+1 tL (kt+1 ) CL ] + A( )Et [Vt+1 tL (kt+1 ) CL ] > 0, which proves that the players will not settle at t. Hence, the continuation value of each player before the nature chooses i proposer at date t denoted as V t tL (kt ) can be written as p p p p Vt tL (kt ) = Et Vt+1 tL (kt+1 ) CL ; h i d d d d Vt tL (kt ) = Et Vt+1 tL (kt+1 ) CL : 36 p p p d d d Second, consider the case of Et [Vt+1 tL (kt+1 ) CL ] A( )Et [Vt+1 tL (kt+1 ) CL ]; in which players settle. Both players accept an o¤er if it gives them at least their continuation value. If the plainti¤ is recognized as a proposer, she chooses to o¤er xt = d Et d Vt+1 tL (kt+1 ) d CL ; the defendant’ continuation value, which is the lowest s s s o¤er the defendant accepts. In such case, the plainti¤’ lawyer’ payo¤ is A( )xt , which is larger than her continuation value, i.e., A( )xt = A( ) d Et d Vt+1 tL (kt+1 ) d CL p p p Et [Vt+1 tL (kt+1 ) CL ]. In the case that the defendant is a proposer, he chooses to o¤er 1 p p p xt = A( ) Et Vt+1 tL (kt+1 ) s CL , which is the lowest o¤er the plainti¤’ lawyer accepts. s In such case, the defendant’ payo¤ is xt , which is larger than his continuation value, i.e. 1 p p p d d d A( ) Et Vt+1 tL (kt+1 ) CL Et Vt+1 tL (kt+1 ) CL . Thus, in equilibrium, the proposer o¤ers the continuation value of the opponent, and the opponent accepts. The continuation value of each players before the nature chooses proposer at date t will be h i p d d d p p p Vt tL (kt ) = ( A( )Et Vt+1 tL (kt+1 ) CL ) + (1 ) Et Vt+1 tL (kt+1 ) CL ; h i 1 d d d d p p p Vt tL (kt ) = Et Vt+1 tL (kt+1 ) CL + (1 ) ( Et Vt+1 tL (kt+1 ) CL ): A( ) Combining both cases, the continuation value is written as n h i o p d d d p p p Vt tL (kt ) = max A( )( Et Vt+1 tL (kt+1 ) CL ); Et Vt+1 tL (kt+1 ) CL p p p +(1 ) (Et Vt+1 tL (kt+1 ) CL ; h i d d d d Vt tL (kt ) = Et Vt+1 tL (kt+1 ) CL 1 h i +(1 ) max Ep V p tL (kt+1 ) p d d CL ; Et Vt+1 tL (kt+1 ) d CL : A( ) t t+1 s At the beginning of the period t, the plainti¤’ lawyer chooses whether to drop the case or not. She drops the case if her continuation payo¤ is lower than 0 because she can always have outside option of 0 by dropping the case Hence, the continuation values at the beginning of period t are written as p Vtp tL (kt ) = max 0; V t tL (kt ) s The continuation value for the defendant depends on the decision of the plainti¤’ lawyer. If she drops the case, the defendant has no obligation to pay, while his continuation value is d Vt tL (kt ) s s if plainti¤’ lawyer does not drop the case. Thus, the defendant’ continuation value is written as ( p 0 if 0 > V t tL (kt ) Vtd tL (kt ) = d : V t tL (kt ) otherwise. 37 Appendix B Proof of Proposition 2 I consider two separate cases depending on whether the players will settle or continue the case. First, consider the case of Ytp (kt ) > A( )Ytd (kt ), in which players do not settle. Sup- pose that the players settle in period t with monetary transfer x. s The plainti¤’ lawyer agrees only if A( )x Ytp (kt ), while the defendant agrees only if x Ytd (kt ). This p requires 0 = x x 1 d A( ) Yt (kt ) + Yt (kt ). This contradicts with Ytp (kt ) > A( )Ytd (kt ), which proves that the players will not settle at t. Hence, the continuation value of each player before settlement decision at date t will be i W t (kt ) = Yti (kt ): Now, consider the case of Ytp (kt ) A( )Ytd (kt ); in which players settle. Each player accepts an o¤er if and only if it gives him/her at least his/her continuation value, and a party recognized by nature to propose chooses to o¤er this continuation value because it s is least costly o¤er to be accepted by the other party. If the plainti¤’ side is recognized as the proposer, she o¤ers Ytd (kt ) to the defendant, which is defendant’ continuation value s and defendant accepts. In this case, the plainti¤’ lawyer receives s A( )Ytd (kt ): If the 1 p defendant is recognized as the proposer, he o¤ers x = A( ) Yt (kt ) with which the plainti¤’s lawyer receives her continuation value of Ytp (kt ). Thus, I can write the continuation value of each player before the settlement decision at date t as h i p W t (kt ) = A( )Ytd (kt ) + (1 )Ytp (kt ); d 1 W t (kt ) = Ytd (kt ) + (1 ) Y p (kt ) : A( ) t Combining the above two cases, I can rewrite the continuation value as n o p W t (kt ) = max A( )Ytd (kt ); Ytp (kt ) + (1 )Ytp (kt ); d 1 W t (kt ) = Ytd (kt ) + (1 ) max Ytp (kt ); Ytd (kt ) : A( ) s At the beginning of the period t, the plainti¤’ lawyer chooses whether to drop the case or not. Same as in Phase O; she drops the case if her continuation payo¤ is lower than 0 be- cause she can always have outside option of 0 by dropping the case Hence, the continuation 38 s value for the plainti¤’ lawyer at the beginning of period t is written as p Wtp (kt ) = max 0; W t (kt ) s The continuation value for the defendant depends on the decision of the plainti¤’ lawyer. If she drops the case, the defendant has no obligation to pay, while his continuation value is d s s W t (kt ) if plainti¤’ lawyer do not drop the case. Thus, the defendant’ continuation value is written as ( p 0 if 0 > W t (kt ) Wtd (kt ) = d W t (kt ) otherwise. Appendix C Computation of Conditional Choice Probabilities In Chapter 2, I characterized the equilibrium outcome on the time, mode, cost, and terms of settlement, as well as the time of …ling a lawsuit (if a lawsuit is …led). However, the outcomes are contingent on the realization of the state of information kt = (nt ; mt ) 2 f0; :::; tg f0; :::; tg which is not observable to the econometrician. Thus, from the perspec- tive of the econometrician, the litigation timing tL , the resolution timing tS , the compen- sation payment x, and the total legal cost are all random variables because they depend on the (unobservable) realization of kt . Hence, I …rst compute the conditional probability of observing the realization of such random variables in order to compute the likelihood contribution of each observation. Let dDO (nt ; mt ), dSO (nt ; mt ), dDL (nt ; mt ; tL ), dSL (nt ; mt ; tL ), and dF I (nt ; mt ) denote t t t t t the indicator function respectively for dropping in Phase O, settlement in Phase O, dropping in Phase L, settlement in Phase L, and …ling of a lawsuit at period t. Using the results of Propositions 1 and 2, these indicator functions are written as p dDO t (nt ; mt ) = I W t (nt ; mt ) < 0 ; n o dSO t (nt ; mt ) = I Ytp (nt ; mt ) + A( )Ytd (nt ; mt ) < 0 ; p dF I t (nt ; mt ) = I Et V1p (nt+1 ; mt+1 ) > Et Wt+1 (nt+1 ; mt+1 ) ; p dDL t (nt ; mt ; tL ) = I V t tL (nt+1 ; mt+1 ) < 0 ; n o p d p dSL t (nt ; mt ; tL ) = I Et V t+1 tL (nt+1 ; mt+1 ) + A( )Et V t+1 tL (nt+1 ; mt+1 ) CL d A( )CL < 0 ; where dDO (nt ; mt ) = 1, t dSO (nt ; mt ) = 1, dDL t (nt ; mt ; tL ) = 1, dSL (nt ; mt ; tL ) = 1, t t dF I (nt ; mt ) = 1 indicate dropping in Phase O, settlement in Phase O, dropping in Phase t L, settlement in Phase L, and litigation at period t given (nt ; mt ) and (nt ; mt ; tL ), while 0 indicate otherwise. In computing equilibrium, we assume the rates of information arrival 39 as = + O Ot O0 O1 t and Lt = L0 + L1 t: Let qt (nt ; mt ) denote the probability of being in state kt = (nt ; mt ) 2 f0; :::; tg L f0; :::; tg in Phase O, and let qt (nt ; mt ; tL ) denote the probability of being in state kt in Phase L given the litigation date of tL < t. I can compute O qt recursively as follows: O q0 (0; 0) = 1; O O qt (n; m) = Ot qt 1 (n 1; m 1) 1 dSO (n t 1; m 1) dDO (n t 1; m 1) 1 dF I (n t 1; m 1) O + Ot (1 )qt 1 (n 1; m) 1 dSO (n t 1; m) dDO (n t 1; m) 1 dF I (n t 1; m) O +(1 Ot )qt 1 (n; m) 1 dSO (n; m)1 t dDO (n; m) 1 t dF I (n; m) : t L L The initial condition for qt is qtL ; which can be written as tL X tL X L qtL (n; m; tL ) = O qtL (ntL ; mtL )dF I (ntL ; mtL ): t ntL =0 mtL =0 L I can compute qt recursively for t > tL as follows: L L qt (n; m; tL ) = Lt qt 1 (n 1; m 1; tL ) 1 dSL (n t 1; m 1; tL ) dDL (n t 1; m 1; tL ) L + Lt (1 )qt 1 (n 1; m; tL ) 1 dSL (n t 1; m; tL ) dDL (n t 1; m; tL ) L +(1 Lt )qt 1 (n; m; tL ) 1 dSL (n; m; tL ) t dDL (n; m; tL ) : t Let l = fF ILE; N OF ILEg denote if a case is litigated (l = F ILE) or not (l = N OF ILE), and s 2 fDROP; SET T LE; JU DGEg denote if the case is dropped (s = DROP ), settled (s = SET T LE) or reaches judgment by the court (s = JU DGE). A case can be resolved in one of the …ve way; i) dropped without …ling a lawsuit ((l; s) = (N OF ILE; DROP )), ii) settlement without a lawsuit ((l; s) = ((N OF ILE; SET T LE)), iii) dropped after a lawsuit ((l; s) = (F ILE; DROP )), iv) settlement after lawsuit ((l; s) = (F ILE; SET T LE)), and judgement of the court ((l; s) = (F ILE; JU DGE)) on equilibrium path. With this notation, the probability that the econometrician observes litigation at date tL can be written as tL X tL X Pr(F ILE) Pr(tL jF ILE) = O qtL (ntL ; mtL )dF I (ntL ; mtL ); t ntL =0 mtL =0 O where qtL (ntL ; mtL ) is the probability of reaching state ktL = (ntL ; mtL ) in period tL and dF I (ntL ; mtL ) is the indicator of …ling a lawsuit at tL given the state ktL . Dropping and t 40 settlement in Phase L occurs only after litigation in Phase O. Thus, the probability of dropping and settlement at period tS in Phase L conditional on litigation at Phase L is written as tS X tS X Pr(tS ; DROP jtL ; F ILE) = L qtS (ntS ; mtS ; tL )dDL (ntS ; mtS ; tL ); t ntS =0 mtS =0 tS X tS X L Pr(tS ; SET T LEjtL ; F ILE) = qtS (ntS ; mtS ; tL ) 1 dDL (ntS ; mtS ; tL ) dSL (ntS ; mtS ; tL ); t t ntS =0 mtS =0 L where qtS (ntS ; mtS ; tL ) is the probability of reaching state ktL = (ntL ; mtL ) in period tS given the case is litigated at tL and dDL (ntS ; mtS ; tL ) and dSL (ntS ; mtS ; tL ) are the indicators of t t dropping and settlement in period tS at state (ntS ; mtS ) given the case is litigated at tL : The probability of reaching judgment can be calculated similarly. If players do not settle for the whole period in Phase L between tL + 1 and tL + T , the case reaches judgment (s = JU DGE) by the court. Hence, I can write the probability of reaching judgement as T T X X Pr(JU DGEjtL ; F ILE) = L qT (nT ; mT ; tL )[1 dDL (nT ; mT ; tL ) dSL (nT ; mT ; tL )]: t t nT =0 mT =0 In some cases, players may drop or settle in Phase O and there will be no litigation. Thus, the probability of observing dropping and settlement at date tS without litigation (l = N OF ILE) are tS X tS X Pr(tS ; DROP j;; N OF ILE) = O qtS (ntS ; mtS )dDO (ntS ; mtS ); t ntS =0 mtS =0 tS X tS X O Pr(tS ; SET T LEj;; N OF ILE) = qtS (ntS ; mtS ) 1 dDO (ntS ; mtS ) dSO (ntS ; mtS ); t t ntS =0 mtS =0 O where qtS (ntS ; mtS ) is the probability of reaching state ktS = (ntS ; mtS ) in period tS , and dDO (ntS ; mtS ) and dSO (ntS ; mtS ) are the indicators of dropping and settlement at tS given t t the state ktS . Regarding payment and cost, I compute conditional probabilities for payment and cost to fall into bins xk and ck since supports of payments and costs are continuous. A piece of information that a¤ects payment but that is unobservable to the econometrician is the identity of the proposer. The identity of the proposer at the time of the settlement a¤ects the payment as shown by both Propositions 1 and 2. Thus, we will consider this in 41 computing the probability. Let dp and dd be the indicator function such that h i dp (nt ; mt ; xk ; tS ; tL ; F ILE) = If Et Vtd S tL +1 (ntS +1 ; mtS +1 ) d CL = x 2 xk g; dd (nt ; mt ; xk ; tS ; tL ; F ILE) = If Et Vtp S tL +1 (ntS +1 ; mtS +1 ) p CL = x 2 xk g; dp (nt ; mt ; xk ; tS ; ;; N OF ILE) = If Ytd (ntS ; mtS ) = x 2 xk g; dd (nt ; mt ; xk ; tS ; ;; N OF ILE) = IfYtp (ntS ; mtS ) = x 2 xk g: Because the nature chooses the plainti¤ as a proposer with probability and the defendant with probability 1 , the probability of observing payment x falling into bin xk given the settlement date tS and the litigation date tL ; is Pr(x 2 xk jtS ; SET T LE; tL ; F ILE) tS X X tS = L qtS (ntS ; mtS ; tL )dp (ntS ; mtS ; xk ; tS ; tL ; F ILE) ntS =0 mtS =0 tS X tS X + (1 L )qtS (ntS ; mtS ; tL )dd (ntS ; mtS ; xk ; tS ; tL ; F ILE): ntS =0 mtS =0 Similarly, I can express the probability of payment to be in bin xk with settlements without lawsuit in Phase O as Pr(x 2 xk jtS ; SET T LE; ;; N OF ILE) tS X X tS = O qtS (ntS ; mtS )dp (ntS ; mtS ; xk ; tS ; ;; N OF ILE) ntS =0 mtS =0 tS X tS X + (1 O )qtS (ntS ; mtS )dd (ntS ; mtS ; xk ; tS ; ;; N OF ILE): ntS =0 mtS =0 In the case of dropping, there is no monetary transfer between the side of plainti¤ and defendant. Hence, the payment is 0 with probability one given that the case is dropped, i.e. Pr(x = 0jtS ; DROP; ;; N OF ILE) = 0 Pr(x = 0jtS ; DROP; ;; F ILE) = 0 If a case reaches judgement, the identity of the proposer does not matter because there is no bargaining taking place. The defendant pays the amount the jury awards if he loses, and makes no payments otherwise. Hence, I can express the density of the payment in 42 judgment as ( 1 if x = 0 Pr(x 2 xk jtS ; JU DGE; tL ; F ILE) = t Pr( V = x 2 xk ) if x > 0: Probabilities that the total defense cost C will fall into a bin Ck can be computed similarly. For cases settled after …ling lawsuit, the probability of total defense cost falling into a bin Ck is calculated as Pr(C 2 Ck jtS ; SET T LE; tL ; F ILE) ( d 1 if C = tL CO + (tS tL )CL 2 Ck d = d d = 0 if C = tL CO + (tS tL )CL 2 Ck ; while those for the cases settled without …ling can be written as Pr(C 2 Ck jtS ; SET T LE; ;; N OF ILE) ( d 1 if C = tS CO 2 Ck = d = 0 if C = tS CO 2 Ck : The total defense cost for dropped cases are written in the same way as settled case as Pr(C 2 Ck jtS ; DROP; tL ; F ILE) ( d 1 if C = tL CO + (tS tL )CL 2 Ckd = d d = 0 if C = tL CO + (tS tL )CL 2 Ck ; and Pr(C 2 Ck jtS ; DROP; ;; N OF ILE) ( d 1 if C = tS CO 2 Ck = d = 0 if C = tS CO 2 Ck : Regarding the cases concluded by a court judgement, we can calculate it similarly as Pr(C 2 Ck jtS ; JU DGE; tL ; F ILE) ( d 1 if C = tL CO + (tS tL )CL 2 Ckd = d d = 0 if C = tL CO + (tS tL )CL 2 Ck ; This completes the computation of conditional probabilities. Appendix D Likelihood Function 43 The likelihood function is written as Z Z Z Z Z L( jx; tS ; s; t; i) = Pr(x; tS ; s; tL ; ljZ; )dFC dF dFT dFT dFV ; where Pr(x; tS ; s; tL ; ljZ; ) is computed using the conditional probabilities computed above as follows. As I cannot compute the integration exactly, I use Monte Carlo integration by generating 10,000 draws from the distribution of unobserved heterogeneity. For each draw, I compute Pr(x; tS ; s; tL ; ljZ; ) as follows, and sum the logarithm of the probability over all the elements in the sample to obtain the lo-likelihood. (i) For the cases dropped without …ling a lawsuit, Pr(x; tS ; s; tL ; ljZ; ) is Pr(x; tS ; s; tL ; ljZ; ) = Pr(N OF ILEjZ; )) Pr(tS jDROP; ;; N OF ILE; Z; ) Pr(x 2 xk jtS ; DROP; ;; N OF ILE; Z; ) Pr(C 2 Ck jtS ; DROP; ;; N OF ILE; Z; ): (ii) For the cases settled without …ling a lawsuit, Pr(x; tS ; s; tL ; ljZ; ) is Pr(x; tS ; s; tL ; ljZ; ) = Pr(N OF ILEjZ; )) Pr(tS jSET T LE; ;; N OF ILE; Z; ) Pr(x 2 xk jtS ; SET T LE; ;; N OF ILE; Z; ) Pr(C 2 Ck jtS ; SET T LE; ;; N OF ILE; Z; ): (iii) For the cases dropped after …ling a lawsuit, Pr(x; tS ; s; tL ; ljZ; ; i) is computed by Pr(x; tS ; s; tL ; ljZ; ) = Pr(F ILEjZ; )) Pr(tL jF ILE; Z; ) Pr(DROP jtL ; F ILE; Z; ) Pr(tS jDROP; tL ; F ILE; Z; ) Pr(x 2 xk jtS ; DROP; tL ; F ILE; Z; ) Pr(C 2 Ck jtS ; DROP; tL ; F ILE; Z; ): 44 (iv) For the cases settled after …ling a lawsuit, Pr(x; tS ; s; tL ; ljZ; ; i) is computed by Pr(x; tS ; s; tL ; ljZ; ) = Pr(F ILEjZ; )) Pr(tL jF ILE; Z; ) Pr(SET T LEjtL ; F ILE; Z; ) Pr(tS jSET T LE; tL ; F ILE; Z; ) Pr(x 2 xk jtS ; SET T LE; tL ; F ILE; Z; ) Pr(C 2 Ck jtS ; SET T LE; tL ; F ILE; Z; ): while (v) for the cases resolved in court judgment, I have Pr(x; tS ; s; tL ; ljZ; ) = Pr(F ILEjZ; ) Pr(tL jF ILE; Z; ) Pr(JU DGEjtL ; F ILE; Z; ) Pr(tS jJU DGE; tL ; F ILE; Z; ) Pr(x 2 xk jtS ; JU DGE; tL ; F ILE; Z; ) Pr(C 2 Ck jtS ; JU DGE; tL ; F ILE; Z; ): 45