Legal Fee, Lawyer s Effort and Optimal Legal Expenses Insurance

Legal Fee, Lawyer s Effort and Optimal Legal Expenses Insurance Yue Qiao Shandong University Zhewei Wang Shandong University Abstract In this paper a missing market, the lawyer s un-contractible effort, has been introduced to analyse legal expenses insurance under three fee arrangements: hourly fees, conditional fees and sliding fees. Three points are highlighted as our conclusions. First, to design an optimal insurancepayment system, demand-side cost-sharing is necessary. Second, supplyside cost-sharing is necessary only if the quantity and effort are substitutes and the payment contract involves hourly fees. Third, the optimal insurance-payment system could be achieved under conditional fees and sliding fees. Reputation incentives and side-contracts are also discussed in this paper. 1 Introduction The question of how to assure access to justice is a fundamental one in all jurisdictions. It involves making available suitable institutions and expertise to help access these, at an affordable price and in ways that help share the risk of what may be very uncertain negotiations. In addition, of course, the nature of the principal-agent relationship between lawyer and client means that there may be a need to provide incentives to the lawyer, and possibly to the client. Different countries achieve the latter in different ways. In the US, contingent fees seek to provide both risk sharing and incentives. In others, State-sponsored legal aid has been a means of encouraging access to justice but, as was suggested in the UK before legal aid was diluted in 2001, this may have involved lowpowered incentives for lawyers and clients (see Fenn, Gray & Rickman (1999)). In place of legal aid, the UK now has conditional fees, which combine an element of output-based pay, with insurance protection against the other side s costs. Other countries, especially those in Europe, also use legal expenses insurance (LEI) as a way to support litigants. LEI provides cover against the risk of making or defending a legal action, whether in court or not. It will pay for lawyers fees and other costs arising in legal actions, up to the limit of indemnity in the policy. It can be purchased either as before-the-event (BTE) insurance or as after-the-event (ATE) insurance. BTE insurance provides cover for legal claims that may happen and is very often sold in the same way and at the same time as other annual insurance contracts, for example when buying motor insurance or household insurance. It 1

is very commonly used as a way of bringing compensation claims arising from road accidents and from consumer contract disputes. ATE insurance provides cover for a specified legal claim after the disputed event or accident has happened. It basically protects the policyholder from the risk of having to pay his opponent s legal costs in that dispute. ATE is often sold, but not necessarily, through legal service providers who are using conditional fee agreements. This character of ATE insurance means it is likely to be available only where the chances of winning a case are high. The principal rationale of LEI is the same as other insurances: the risk averse individual shifts his risk to a risk neutral insurer who is better able to bear it by a certain premium. In England and Wales, however, even lawyers can work on a conditional fee basis, the risk of liability for opponents legal fees still encourages the development of legal expenses insurance. Also, Kirstein (2000) and others develop strategic grounds for demanding insurance: even risk neutral parties can gain by enhancing their bargaining position through lower costs. The introduction of LEI for the market of legal services results in a structural change of the market. Bowles & Rickman (1998) characterise the change as a three-way relationship between groups with an interest in the provision of goods and services. The client purchases insurance from the insurer and receives services from the lawyer. The lawyer supplies legal services to the client and receives reimbursement from the insurer. Asymmetric information exists on every side of this relationship. The insurer has imperfect information about the client s risk type and the lawyer s reimbursement. The client has imperfect information about the lawyer s effort. The lawyer has imperfect information about the client s position. The three-way relationship can be understood as the insurer (the principal) maximising profits by means of contracting with the client and the lawyer. The lawyer is an agent for both the insurer and the client. Figure 1 illustrates this three-way relationship. Figure 1: The lawyer-client-insurer nexus of contracts (Bowles & Rickman 1998) Several authors have examined LEI in the presence of risk neutral claimants, either as a strategic device (Kirstein (2000); Kirstein & Rickman (2004)) or 2

in terms of the effects on claim filing and settlement decisions (van Velthoven & van Wijck (2001)). Others have noted that insurance may be useful when clients are risk averse (Farmer & Pecorino (1998); Heyes, Rickman & Tzavara (2004)). None have yet considered the effects of insurance on lawyers efforts, clients preferred legal input, or on insurance companies themselves. In fact, the issues are often analogous to those appearing in insurance-based health care contexts. In both situations, a three-way relationship emerges between principal, agent and insurer. The agent (client/patient) buys insurance to protect against a risky loss (legal/health costs) and engages a principal (lawyer/physician) to supply expertise in terms of hours worked and effort. While the former can be observed, and monitored by the agent; the latter cannot be. Thus, there is an incentive problem between principal and agent, and also a monitoring problem for the ultimate payer, the insurer. Ma & McGuire (1997) consider appropriate contracting arrangements under these conditions for US health care and Bowles & Rickman (1998) note the potential for analogous work in the legal context. This paper represents a first attempt to begin exploring this issue. This paper is structured as follows. In Section 2, we describe our model. The section starts from a brief comparison between legal expenses insurance and health care insurance. After that, we present a model of a lawsuit that has reached trial (after failed settlement negotiations which are not modelled). In Section 3, in a structure similar to Ma and McGuire s, we examine the optimal insurance-payment systems. The analysis begins with subgame equilibrium effort of the lawyer under three forms of fee arrangement between the insurer and the lawyer. Based on the results of the subgame equilibrium, the existence of optimal insurance-payment systems is examined. Section 4 offers some discussions. In the section, we first look into the possibility of the plaintiff collusion with the lawyer. Then we examine the lawyer s reputation incentive and its effect on the insurance system. Section 5 briefly concludes. 2 The model 2.1 Health care insurance and legal expenses insurance In our introduction, we discussed the three-way relationship in the legal insurance system. A similar situation actually exists in many health care systems, where the doctor (the service provider), the patient (the client), and the insurer 1 interact in a framework of a three-way relationship. The similarities between legal insurance and health care are as follows. First, status-based insurance is missing in both systems. In health care markets, it is clear that if the insurance premium can be contingent on the individual s state of health, the insurance policy will be (first-best) efficient (Arrow 1963). Then, this policy can protect the individual from the risk of illness ex ante and retains incentives for the patient to utilise health care efficiently ex post. However, since health status is too costly to verify, this policy actually does not exist in the market. Similarly, in 1 More generally, in health care a third party payer may fund the treatment. For example, the NHS in UK and the social insurer in some European countries. 3

the legal expenses insurance market, the status of a client is also difficult to verify. Even in the after-the-event (ATE) insurance market 2, since the litigant may still have some private information (e.g. the merit of the case), the insurance premium can not entirely be based on the status of the client. Second, the lack of status-based policy further rules out any policy that commits certain amounts of service contingent on status. In the health care market, the quantity of treatment is not contractible even ex post. In the legal insurance market, insurance coverage of legal fees depends on the amount of lawyers s hours which can not be specified in an insurance policy. Third, in both insurance markets, verifying the quantity of the provider s service is costly and difficult to the insurer. Thus the payment contracts are based on the provider s reported quantities which may be different from the actual quantity. Distinguishing between a reported and actual quantity reveals an incentive problem. If the client (patient or plaintiff) bears some cost, he has an incentive to ask the provider (doctor or lawyer) to under report the quantity. Truthful reporting thus translates into restrictions on insurance-payment system in both insurance markets. Fourth, effectiveness of the service provider s input is non-contractible. In both health care and legal insurance markets, payment contracts based on effectiveness of quantity are missing. Therefore, when designing an optimal system, there must be certain incentives to mitigate this market failure. Of course, the legal expenses insurance market has some unique features. The differences between legal expenses insurance and health care are mainly due to three aspects. First, the legal system is more strategic than the health care system. The outcome of a medical treatment only depends on the patient s health status and the doctor s inputs, while the outcome of a lawsuit is affected by many factors. In addition to the lawyer s inputs, both the plaintiff s and the defendant s strategic decisions in the litigation process play important roles in the outcome of the case. This may imply that the model of legal expenses insurance has to contain some features of litigation strategies. Second, although legal insurance and health care insurance are analogous systems, fee regulations of a legal system distinguish the payment system of legal insurance from health care insurance. In principle, fixed fees, hourly fees and conditional fees are the only methods that lawyers can charge in England and Wales. These specific fee arrangements reflect the particularity of the legal insurance system. Moreover, the payment to the lawyer can be based on the outcome of the lawsuit. For example, under conditional fees, the lawyer is paid only if the case is won. This may provide certain incentives to the lawyer to input more into the case. However, in health care, since the health status is difficult to verify, the payment to the doctor can not be based on the outcome of the treatment. Third, the English cost rule complicates the legal expenses insurance market. According to the English cost rule, the loser has to pay the winner s legal costs. This makes the insurance contract liable to cover the client s own costs and also potential payments to the rival of the litigation. These risks are related with the outcome of the lawyer s inputs. However, health care insurance only needs 2 There are two kinds of legal expenses insurances in market: before-the-event insurance (BTE) and after-the-event insurance (ATE). The difference between them is the timing of purchasing the insurance policy. If purchasing before the accident, it is BTE, otherwise it is ATE. 4

to cover treatment costs of a potential sickness. There are no other risks to the insurer. These aspects illustrate the differences between legal expenses insurance and health care insurance. The unique features of legal expenses insurance distinguish modelling of optimal insurance-payment system from those current models of health insurance. Nevertheless, the studies of health care systems can still inspire modelling and analysis of legal expenses insurance. 2.2 The basic model Inspired by the work of Ma & McGuire (1997), our model describes the relationship between a client (the insured plaintiff), a service provider (the lawyer) and a third party payer (the insurer) in legal service markets. When a person is involved in a legal dispute, he may retain a lawyer to pursue a case. An insurer may share the financial risk if the plaintiff purchased legal expenses insurance 3. The process of civil litigation involves a number of stages. If an accident does happen, the potential plaintiff has to choose whether to start a lawsuit. Early economic studies suggest that, in a world without legal expenses insurance, the potential plaintiff will choose not to file the case if the legal cost is higher than his potential recovery of the accident loss. If the plaintiff is insured, since he may have less financial pressure, he becomes tougher in making litigation decisions, but he may still choose to drop the case if the insurance co-payment is too high. In principle, if the plaintiff decides to start a lawsuit, there are two situations. The case can be settled in the negotiation stage or, if the negotiation fails, the case goes to trial. To keep the analysis tractable, we omit some early stages of litigation and focus on cases that go to trial. 2.3 Quantity of the service and effort of the lawyer The plaintiff s probability of prevailing at trial can be modelled by a production process. This process requires two inputs, quantity of the legal service and effort of the lawyer. By quantity of the service, we mean the number of (billable) hours the lawyer worked for the plaintiff. The quantity of the service may be measured and verified ex post. By effort we mean the input contributed by the lawyer which increases the quality or intensity of the legal service but is difficult to measure and verify. Effort can be any costly activity of the lawyer that affects the plaintiff s valuation of the service he receives. For example, this may be the lawyer s tasks to communicate with the plaintiff regularly, to prepare the documents the plaintiff needed or wanted, to investigate the case and report her findings to the plaintiff, to explain expected outcomes and give suggestions, or even to learn about laws and regulations that are unfamiliar to her. The most concrete way to think about effort in our context is simply in terms of how many tasks the lawyer fulfilled per hour. The lawyer is paid by her work hours but not by tasks. Therefore, more tasks lead to higher quality of service to the plaintiff but more cost to the lawyer. The cost of the effort is defined as the lawyer s disutility. 3 As we mentioned in Introduction, there are two types of legal expenses insurance BTE and ATE. The main difference between them is the timing of purchasing the insurance policy. In this paper, we assume the insurance is BTE. 5

As in other optimal insurance literature, we assume that the insurance contract cannot specify the quantity of service ex ante. However, as we mentioned in the previous section, according to current laws and regulations, the insurer may contract with the lawyer regarding the outcome of the lawsuit, which is the case of conditional fees. Also, the payment contract is based on the quantity reported by the plaintiff and the lawyer: after the trial, the lawyer files a claim which must be agreed by the plaintiff. The lawyer s reimbursement and the plaintiff s copayment are based on the reported information. The lawyer s effort is assumed to be observable to the plaintiff but nonverifiable to the insurer. Thus, the lawyer can only be paid on the basis of the reported quantity of the service, not effort. 2.4 The legal expenses insurance game The extensive from of the legal expenses insurance game consists of five stages: 1. The insurer chooses the elements of the insurance and payment systems; 2. Nature decides whether the client is involved in a legal dispute with probability π. If not, the game ends; otherwise, the client (plaintiff) retains a lawyer to pursue a claim; 3. The lawyer chooses her effort, ε; 4. After observing the lawyer s choice of effort, the plaintiff chooses the quantity (hours) of the lawyer s work, τ; 5. The trial finishes. The lawyer and the plaintiff play a billing subgame. Subsequently, the insurer pays the resulting bill. Figure 2: Extensive form of the legal expenses game 6

Figure 2 illustrates this game in its extensive form. In Stage 3, the lawyer makes her effort decision before the plaintiff chooses how many units of quantity to purchase. Since effort is costly for the lawyer but cannot be rewarded by the payment contract directly, our structure encourages incentives through the plaintiff s reaction. Since effort will affect the probability of prevailing at trial, different levels of it will lead to different quantities chosen by the plaintiff. Therefore, the lawyer can induce different quantity demands by offering different effort. This is the typical environment of supplier-induced demand (Phelps 1986). Since the duration of litigation can be quite long 4, it is reasonable to assume that the plaintiff actually observes the lawyer s quality. In this paper, we are particularly interested in three fee arrangements. In addition to two most commonly used arrangements, hourly fees and conditional fees, we introduce sliding fees as the third fee arrangement. We now define them as follows: Hourly fees The lawyer s fee is charged based on the amount of hours the lawyer worked on the client s case. Neither the fee nor the rate is contingent on the result of the case. Conditional fees Conditional fees are a form of no win, no fee arrangement. If the case is lost, the lawyer will charge no fees. In the event of a win, the lawyer charges his normal fees plus a percentage uplift on the normal fees. Sliding fees Under sliding fees, the lawyer can charge hourly rates if the case is won. If the case is lost, the insurer still pays the lawyer, but the rate is decreasing with the lawyer s quantity input. For example, the insurer pays 200 per hour if the total billing hours are less than 10 hours, pays 160 per hour if the total billing hours exceed 10 hours, while pays 100 per hour if the total billing hours exceed 20 hours. The sliding fee is a natural intermediate arrangement between fully contingent (conditional fees) and fully non-contingent (hourly fees) payment 5. Moreover, we assume that the English cost rule is applied here; in all of the fee arrangements if the case is won the lawyer s fee is paid by the defendant. We now define the billing subgame of Stage 5. Since monitoring is costly to the insurer, the insurer has to use the information in the report filed by the lawyer to collect the plaintiff s co-payment and to reimburse the lawyer. Since payment to the lawyer is based on her reported quantity. The lawyer will suggest to the plaintiff a quantity τ R which is not necessarily equal to the actual quantity τ. If the plaintiff agrees, then τ R is reported. Since the plaintiff can observe the actual 4 For example, the average duration of medical negligence cases lasts 65 months, personal injury cases lasts 56 months and professional negligence cases lasts 41 months. (Annex 3, Access to Justice, Department for Constitutional Affairs, 1996.) 5 This setting is inspired by Polinsky & Rubinfeld (2003). Here, the authors propose a novel litigation system which is between the standard contingency fee and the standard hourly fee arrangement. The model we present assumes a continuous relationship between hourly rate and hours, rather than the piecewise linear example given above. 7

τ, if he does not agree the suggested quantity, the actual quantity τ is reported. In this subgame, either the plaintiff or the lawyer can reveal the quantity records to the insurer. This subgame captures the idea that a false report is possible if and only if it is in the self-interest of both the plaintiff and the lawyer and there is no money transfer between them. Another situation emerges when the false report is motivated by the joint interest of the lawyer-plaintiff coalition, so a side-contract can emerge. We will discuss this in Section 5. 2.5 Payoff functions of participants The plaintiff s utility depends on the outcome of the trial, his co-payment for the legal service and the insurance premium. The probability of the plaintiff prevailing at trial p(τ, ε) is increasing in quantity of hours τ, and the lawyer s effort ε (1 p 0, p ε > 0, p τ > 0, p εε < 0 and p ττ < 0). The variables τ and ε are bounded below at zero. The co-payment per unit of service is β which is paid to the insurer. 6 The co-payment is less than the lawyer s hourly rate: w > β 0. The insurance premium is α. The premium has to be paid before the event. If the plaintiff wins at trial, the award is a. The service quantity reported τ R does not have to be the true quantity τ. Hence, using the strictly concave function U( ) to represent the plaintiff s preference, his expected utility function is: EU = πu[p(τ, ε)a βτ R α] + (1 π)u( α) (1) The lawyer is risk neutral. She has a disutility of effort given by G(ε), G (ε) > 0 and G (ε) > 0. For convenience, it is assumed the marginal cost of the lawyer is c, which is borne by the lawyer. The lawyer s hourly rate is set to be w c. Therefore, her expected utility function is given by: V = π[pwτ R + (1 p)µwτ R G(ε) cτ]. (2) where µ is the payment function that allows us to switch between the payment mechanisms. In particular, if µ = 1, the lawyer is reimbursed by hourly fees; while if µ = 0, it represents conditional fees. And, if µ is a decreasing function of the quantity τ, where µ (τ) < 0, we have sliding fees. 7 For simplification, we assume the defendant s total legal cost is a fixed amount C d. Suppose the English cost rule applies here, whereby the winning party s legal fees and expenses are payable by the losing party. Hence, the insurer s profit function becomes: U I = α + π[βτ R (1 p)c d (1 p)µwτ R ]. (3) In a competitive insurance market, the premium paid by the plaintiff must be equal to the expected value of the insurer s payment to the lawyer and the 6 The purpose of this paper is to characterise the elements of optimal legal expenses system. Since co-payment plays an important role in solving problems of supplier induced demand, even in the practice of legal expenses insurance it is rare, we still choose it as an element of our model. 7 We assume that the insurer cannot decide under which fee arrangement to run the litigation. This includes selecting the precise features of the µ function. This, for example, was the situation under the system of regulated scale fees for conveyancing. 8

defendant if the case is lost (actuarially fair). Thus, α = π[(1 p)c d + (1 p)µwτ R βτ R ]. (4) 2.6 Solution of the billing subgame Before proceeding to the next section, we solve the billing subgame. The lawyer first suggests to the plaintiff a quantity of hours. If the plaintiff agrees, the hours are reported in the bill. If he disagrees, then the real hours are reported. Under individual rationality, since the co-payment is β > 0, the plaintiff pays a positive amount for each of the reported quantity of hours. Because he can reveal the true quantity, he will reject the over-reported quantity, but accept the under-reported quantity 8. In the other hand, when the lawyer receives a positive payment per unit of reported hour (w > 0), she will never under-report the quantity. Nevertheless, she cannot over-report the quantity since the plaintiff will reject her suggestion. Hence, truthful billing must be the equilibrium in the subgame in Stage 5. If w < 0, which means the lawyer has to pay a positive amount to the insurer for every reported hour, since the plaintiff always prefers under-reporting, it is obvious both the lawyer and the plaintiff will under-report the quantity. Thus, under this situation, in equilibrium, the reported quantity will be zero. To summarise: Result 1: If and only if β 0 and w > 0, truthful reports will be the equilibrium in Stage 5. By having now determined the lawyer s and the plaintiff s behaviours in the billing subgame, we move backward to the earlier stages of our game to examine the plaintiff s quantity and the lawyer s effort choice. 3 Optimal insurance-payment system As we have shown before, when w > 0 and β 0 equilibrium truthful reporting occurs in Stage 5, therefore τ = τ R. We now proceed to Stage 4 to find the subgame equilibrium of this game. 3.1 The plaintiff s quantity choice Suppose in Stage 1 the elements of insurance and payment have been chosen (α, β, w). And in Stage 2, Nature has decided that the plaintiff has to go to court. Then, in Stage 3, the lawyer has chosen her effort, ε. In Stage 4, the plaintiff chooses service quantity to maximise his expected utility. The optimal quantity choice of the plaintiff is obtained by maximising U(pa βτ α) with respect to τ. The interior first order condition is given by: p τ (ε, τ)a = β. (5) 8 If β = 0, the plaintiff is indifferent between truthful and untruthful reporting. In this case, we assume that he tells the truth. 9

This description illustrates the plaintiff s demand behaviour in two ways: first, the insurance premium α does not affect his demand for legal service; second, he chooses τ to set the marginal benefit of dispute p τ (ε, τ)a equal to his copayment rate β the marginal cost of hours to the plaintiff 9. Since p ττ < 0 and p τ > 0, the plaintiff s quantity choice is increasing with the value of the case a and decreasing with his co-payment β. The importance of a co-payment in the insurance system is clear: if there is no co-payment, the plaintiff s quantity demand becomes infinite. The trade-off between risk sharing and moral hazard is also characterised here. The co-payment that the plaintiff must pay for each unit of service received exposes him to some risk ex ante, but also partly remedies his incentives to over-utilise the insurance ex post. The function (5) is also the plaintiff s reaction function against the lawyer s effort choice ε. Since the lawyer s payment relies on τ, the effort ε becomes her instrument to influence the plaintiff s quantity choice. Similarly, if the fee arrangements affect the lawyer s effort ε, they will affect the plaintiff s quantity choice indirectly. From (5), we also obtain: dτ dε = p τε p ττ. (6) Equation (6) is the Technical Rate of Substitution (TRS) of the lawyer s effort choice ε with respect to the plaintiff s quantity choice τ. It measures the change in the plaintiff s choice of quantity per unit increase of the lawyer s effort. Clearly, the sign of the TRS is the same as the sign of p τε because p ττ is defined as a negative value. If quantity and effort are substitutes, which means p τε < 0, then a lower effort ε can induce a higher quantity demand τ. Alternatively, if the quantity and effort are complements, which means p τε > 0, then a higher τ can only be induced by a higher ε. In general, the sign of the cross partial derivative of p(τ, ε) may change according to both τ and ε. 3.2 Effort choice and the subgame perfect equilibrium We now consider the lawyer s decision in Stage 3. Anticipating the plaintiff s quantity reaction in Stage 4, the lawyer chooses ε to maximise her utility EV = pwτ +(1 p)µwτ cτ G(ε) subject to (5). Thus, in any subgame perfect equilibrium, the lawyer s choice of ε and the plaintiff s subsequent choice of τ are given by the solution to the following program: Program A: For w > β 0, choose τ and ε to maximise subject to p τ (ε, τ)a = β. pwτ + (1 p)µwτ G(ε) cτ (7) 9 This result is actually similar to the one in Schwartz & Mitchell (1970). Their paper, where the insurance is absent, claims that the lawyer s gross marginal product should equal the wage rate. 10

Given a fee arrangement µ, in Program A, only β and w appear as parameters. To characterise the set of subgame perfect equilibrium (τ, ε) pairs, we define: Ω={(τ, ε): there exist (w, β), with w > β 0, for which (τ, ε) solves Program A, given (w, β) }. Ω is the implementable set. It contains all those (τ, ε) pairs which solve the subgame-perfect equilibrium given combinations of parameters w > β 0. We now discuss three different fee arrangements. 3.2.1 Hourly fees Under hourly fees, the payment function is a constant and equals one: µ = 1. Figure 3 illustrates the implementable set under substitutes. The plaintiff s reaction function (5) (p τ (ε, τ)a = 0 and p τ (ε, τ)a = β) are negative sloped. The slope of the lawyer s indifference curve is decided by the value of w. An equilibrium is a tangency of an indifference curve with a reaction function. As w and β vary over their ranges, Ω is established (the shaded area). The boundary of Ω reflects the constraint w > β 0. Any effort level out of this boundary is infeasible. Figure 3: The set of implementable effort and quantity From (7) and (6), the solution system of Program A becomes: dτ dε (w c) G (ε) = p τε p ττ (w c) G (ε) = 0 (8) p τ (ε, τ)a β = 0 (9) Given w c > 0 and p ττ < 0, if effort and quantity choices are complements, that is p τε > 0, it is clear that the above equations have interior solutions in the implementable set Ω, which depends on the values of w and β. However, if effort and quantity choices are substitutes, since G (ε) is non-negative, only a corner 11

solution exists, that is ε = 0. This means the lawyer will reduce her effort as much as she can. From (8), only when w < c, can the plaintiff s quantity choice motivate the lawyer s effort. This suggests that in the case of substitutes, to give effort, the lawyer must bear some costs (w c < 0). To summarise: Result 2: Consider a quantity-effort pair (τ, ε) belonging to the implementable set Ω. Under hourly fees, suppose that τ and ε are substitutes for a given (τ, ε) pair, that is p τε < 0, the lawyer s effort choice only stays at its minimum (ε = 0) when w > c. Alternatively, suppose τ and ε are complements, that is p τε > 0, then ε is above its minimum, ε > 0, when w > c. Result 2 implies a fundamental incentive problem in the fees arrangement: If quantity can induce effort, the lawyer does not need to share any cost of the plaintiff (w > c). If the relationship between quantity and effort is one of substitutes, cost-sharing (w < c) brings incentives to the lawyer to work harder. 3.2.2 Conditional fees Under conditional fees, the payment function is set to µ = 0. Then, the solution system of Program A becomes: dτ dε (p τ wτ + pw c) + p ε wτ G (ε) = 0 (10) p τ (ε, τ)a β = 0 (11) We define the quantity elasticity of prevailing as ɛ 1 = pτ τ p > 0. This reflects the nature of the lawsuit. For example, if the case is unfavourable to the plaintiff, the quantity elasticity of prevailing will be low, which means to achieve the same outcome the lawyer must supply more hours. The quantity elasticity of prevailing also reflects features of the prevailing probability function p(τ, ε). Equation (10) now becomes: dτ dε pw(ɛ 1 + 1) dτ dε c + p εwτ G (ε) = 0 Since p ε wτ > 0 and G (ε) > 0, whether effort and quantity choices are complements or substitutes, the above equations always have interior solutions. The size of the implementable set Ω depends on both the parameters (w, β) and the nature of lawsuit itself (ɛ 1 ). For example, for an inelastic ɛ 1, to gain the same implementable set, w should increase. In other words, the lawyer will only produce the same (τ, ε) combinations in a relatively difficult case if w is high enough to compensate for the disutility created. 3.2.3 Sliding fees Now consider sliding fees, where the payment function is a decreasing function of quantity, µ τ < 0. The solution system of Program A is: dτ dε [pw(ɛ 1 + 1)(1 µ) + (1 p)w(ɛ 2 + 1)µ c] + p ε wτ(1 µ) G (ε) = 0 (12) 12

p τ (ε, τ)a β = 0 (13) where ɛ 2 is defined as the elasticity of the payment function: ɛ 2 = µτ τ µ < 0, i.e. how sharp are the incentives for economising on hours. By the same method used in the analysis of conditional fees, we find that under the sliding fees arrangement, interior solutions are affected by the parameters (w, β), the nature of lawsuit (ɛ 1 ) and the payment function µ (via ɛ 2 ). To compare conditional fees and sliding fees, we state: Result 3: Under both conditional fees and sliding fees, for w > c and β > 0, there would exist ε > 0 whatever τ and ε are substitutes or complements. Compared to conditional fees, suppose at (τ, ε), τ and ε are substitutes, when ɛ 2 < 1, sliding fees arrangement could have a smaller implementable set Ω. PROOF: Since in both cases the existence of interior solutions is determined by β and w, through changing the values of them, an interior solution occurs. When τ and ε are substitutes, since the plaintiff s reaction functions are the same, the steeper the slope of the indifference curve (lawyer s utility), the smaller the size of the implementable set. Under the conditional fees arrangement, dτ the slope of the lawyer s utility function is given by: dε = G (ε) p εwτ pw(ɛ ; and 1+1) c under the sliding fees arrangement, the slope of the lawyer s utility function is: dτ dε = G (ε) p εwτ(1 µ) pw(ɛ. When ɛ 1+1)(1 µ)+(1 p)w(ɛ 2+1)µ c 2 < 1, the slope of sliding fees is steeper than the slope of conditional fees. 3.2.4 Discussion The key difference between the above fee arrangements is whether the lawyer shares the litigation risk. Under hourly fees, the lawyer s income is independent from the case s outcome. When w > c, her income is increasing with the plaintiff s quantity requirements (hours). If τ and ε are substitutes, the lawyer will reduce effort to maximise her income. Therefore, in subgame perfect equilibria, we would not observe hourly fees. Accordingly, we conclude that, under hourly fees, supply-side cost sharing (w < c) is necessary when effort and quantity are substitutes. Since the existence of the implementable set Ω is a necessary condition for an optimal insurance-payment system, this also implies that the optimal system does not exist under hourly fees if quantity and effort are substitutes. Under the conditional fee, the lawyer has to share the litigation risk since it is a no win no fee arrangement. Her income is based on the outcome of the case. Even if τ and ε are substitutes, for winning the case, she has to provide a positive effort (ε > 0). Under this situation, supply-side cost sharing is not necessary. To summarise, the above results imply that, to induce a positive effort when effort and quantity are substitutes, if the lawyer does not share the litigation risk, she has to share some litigation costs. Under sliding fees, the lawyer also shares the litigation risk since her income is affected by the outcome of the case. Unlike conditional fees, here the payment function acts as an implementation to the lawyer s effort provision. The purpose of supply-side cost sharing is actually achieved by setting the payment function. 13

More precisely, if the lawyer only offers her minimum effort, her income will fall below her cost. In addition, since the size of implementable set Ω is affected by the hourly rate w, the co-payment β, the payment function µ and the difficulty of the case (via ɛ 1 ), given a fee arrangement, can the insurer change the implementable set through changing the payment contract (w) or the insurance contract (β). 3.3 Optimal insurance-payment system We now characterise the optimal insurance-payment system. That is, we analyse the choice of the insurance-payment parameters that maximise the plaintiff s expected utility, given that the insurance premium is actuarially fair, that the lawyer s expected utility is at least zero, and that the choices of ε and τ are given by an equilibrium of the subgame in Stage 3. In the previous section, we found that subgame perfect equilibria exist in each fee arrangement given appropriate parameter settings. Therefore the optimal insurance system may exist in each fee arrangement. In an optimal insurance-payment system, the lawyer is only paid her reservation utility level. Without loss of generality, we set the reservation utility level to zero. This gives: V = π[pwτ + (1 p)µwτ G cτ] = 0 Combining the lawyer s binding reservation utility constraint and the insurer s breakeven constraint (4), we obtain: α + πβτ + πpwτ = π[g(ε) + cτ + (1 p)c d ]. (14) Here, πpwτ is the cost transfer under the English cost rule. Therefore, the right-hand side of (14) is the total expected costs that the insurer has to bear; the left-hand side is the total expected gains. The above equation replaces the lawyer s and insurer s participation constraints. Formally, the equilibrium insurance-payment, as well as quantity-effort pair, will be given by the solution to following program: Program B: Choose α, β, w, τ and ε to maximise subject to (14) and (ε, τ) Ω. πu(pa βτ α) + (1 π)u( α) (15) Immediately, inspection of Program B in the light of Result 2 tells us that there cannot be an optimal insurance system involving hourly fees when ε and τ are substitutes: in this case, Ω =. For the remaining fee arrangements, including hourly fees when ε and τ are compliments, we need to examine the solution to Program B. 14

Potentially, Program B is complex to solve since it is high-dimensional and highly constrained. One approach to overcoming this is to solve a relaxed version of the problem, then to check whether the remaining constraints are satisfied at this (relaxed) solution (see Ma & McGuire (1997); also Laffont & Tirole (1993)). Thus, assume that ε is verifiable and contractible so that we can ignore the lawyer s first-order condition. The new program is: Program C: Choose α, β, w and τ to maximise subject to (14) and (5). πu(pa βτ α) + (1 π)u( α) (16) Since ε is contractible, a payment contract can directly specify a payment to the lawyer contingent on her performance of a particular effort level. Program C is a second best regime in the sense that hours are not contractible ex ante. In Program B, in addition to hours, the lawyer s effort level is non-contractible. In this sense, Program B is a third best regime. Let α C, β C, w C, τ C and ε C denote the solution to Program C, and let EU C be the plaintiff s expected utility at this solution. Accordingly, let EU B be the plaintiff s expected utility of Program B. Because EU C is from a more relaxed program (second best), EU C EU B. Moreover, because (5) (Program C) is less constricted than Ω (Program B), Ω (ε C, τ C ). Therefore, if the (ε B, τ B ) = (ε C, τ C ), the second best is achieved in the third best regime. If (ε B, τ B ) (ε C, τ C ), the plaintiff s expected utility is inferior to the second best. We now go to fee arrangements. Truthful claim condition of Stage 5 requires w > 0. From Program B, we know that if τ and ε are complements, all three fee arrangements could produce optimal systems, and if τ and ε are substitutes, only conditional fees and sliding fees could induce optimal systems. Of course, to compare these optimal systems we at least need to know the specific functions of p and µ, but we still can conclude that if there is a system can solve both Program B and C, in this system the plaintiff s expected utility is superior to other systems which can only solve Program B. This implies that if the insurer can change fee arrangements (µ) to align the lawyer s effort incentive with that in the second best, then the non-contractible effort is inconsequential. By choosing a proper fee arrangement, the second best utility can be achieved. 4 Discussions In this section, we discuss two issues of our model. First, we look into an alternative form of billing subgame for Stage 5. Second, we consider the lawyer s reputation incentive and its effect on the insurance system. 15

4.1 Collusion In previous sections, we ruled out the possibility of a side-contract. Now we relax this constraint to enable the plaintiff and the lawyer to make decisions in their joint interest. Since the payment to the lawyer is based on reported quantities, the asymmetry of information on the actual quantities between the lawyer and the plaintiff on the one hand and the insurer on the other may motivate false reporting. If the lawyer and the plaintiff as a group can exploit the insurer by misrepresenting the quantity information, explicit incentives must be introduced to mitigate this effect. If the quantity information between the lawyer and the plaintiff on one side and the insurer on the other side is asymmetric and monitoring is costly to the insurer, the lawyer and the plaintiff may play as a group to over-claim on τ. This subgame is described as follows. First, the plaintiff and the lawyer jointly decide on a report of quantity and a transfer from the lawyer to the plaintiff (the side contract). Second, the quantity τ R = τ + τ o is reported in the claim. Here, τ is the true quantity and τ o is the extent of over-reporting. Accordingly, the co-payment β is paid and the legal fee is received by the lawyer-plaintiff coalition. The case where τ o < 0 involves under reporting. For each unit of quantity the coalition over-reports, it pays co-payment β but receives the legal fee. However, the probability of winning is not affected by the over-reporting quantity τ o. In the cases of hourly fees and conditional fees, the joint interest of the lawyer and the plaintiff is given by: p(ε, τ)w(τ + τ o ) + [1 p(ε, τ)]µw(τ + τ o ) β(τ + τ o ) G(ε) cτ (17) Here, since τ does not affect the coalition s over-reporting decision, we separate its effects from the effects of over-reporting 10. Rearranging (17) gives µwτ o + p(wτ o µwτ o ) βτ o + pwτ + (1 p)µwτ βτ G(ε) cτ (18) As a group, the plaintiff and the lawyer choose τ o to maximize their joint interest. The First Order Condition is µw + pw(1 µ) β = 0. (19) We first assume µ is a constant, either µ = 0 for conditional fees or µ = 1 for hourly fees. We will relax this assumption when we analyse sliding fees. First, under hourly fees, µ = 1. (19) reduces to w = β. Effectively, the coalition can trade w for β. Hence, when w > β, the coalition will over report the quantity. Accordingly, when w < β, the coalition will under report the quantity. Truthful reporting obtains if and only if w = β. This means only if the 10 The purpose of this section is examining if side-contracting puts more constraints on the optimal insurance systems of Section 4. Since the true quantity is already defined by the optimal systems, under side-contracting, the lawyer-plaintiff coalition seeks extra benefit by misrepresenting the quantity information. Therefore we are particularly interested in the difference between the true quantity and the reported quantity. 16

co-payment β and the hourly rate w set to be identical is truthful reporting ensured in a side-contracting regime. Comparing with the constraint w > β of Program A, we find that under hourly fees, side-contracting imposes an inconsistent constraint on the insurance-payment parameters. Substituting w = β into (4), we have α = π[(1 p)c d pw(τ + τ o )] = π[(1 p)c d pwτ R ]. (20) Here, π(1 p)c d is the litigation risk and πpwτ R is the cost transfer exacted from the defendant by the lawyer-plaintiff coalition. They are both caused by the English cost rule. Thus, (20) tells us that, in a side-contracting regime under hourly fees, the optimal insurance premium covers only the net litigation risk but not the lawyer s payment. Second, under conditional fees, where µ = 0, (19) becomes pw = β. When pw > β, the coalition will over report the quantity; when pw < β, the coalition will under report the quantity. Truthful reporting obtains if and only if pw = β: under conditional fees, side-contracting imposes a more stringent constraint on the insurance-payment parameters. However, since p < 1, the new constraint does not rule out the constraint w > β. Finally, under sliding fees, if µ is a function of reported quantity τ + τ o, the First Order Condition of (18) becomes µw + pw(1 µ) + (1 p)µ wτ + (1 p)µ wτ o β = 0. Substituting the elasticity of the payment function ɛ 2 to the above equation yields pw + (1 p)(ɛ 2 + 1)µw β (p 1)µ = τ o. (21) w Now, the condition for truthful reporting is pw + (1 p)(ɛ 2 + 1)µw = β. Under sliding fees, side-contracting imposes a more stringent constraint on the insurance-payment parameters. Similar as conditional fees, since ɛ 2 < 0 and µ < 1, the new constraint does not rule out the constraint w > β. We now summarise: Result 4: In a side-contracting regime, the hourly fee imposes an inconsistent constraint on the insurance-payment parameters, while the conditional fee and the sliding fee do not. The rationale for the above result is not difficult to understand. To mitigate the harm of side-contracting, the insurer may have to set some constraints on the insurance-payment parameters. Under hourly fees, since the payment to the lawyer is independent from the outcome of the case, only when the insurer s net payment to the lawyer-plaintiff coalition (wτ βτ) equals zero, the coalition can not benefit from false reporting. However, this condition is inconsistent with the Program A constraint w > β. Under conditional fees and sliding fees, the 17

payment to the lawyer is contingent on the outcome of the case. Therefore, the net payment to the lawyer-plaintiff coalition is also contingent on the outcome of the case. In this situation, even new constraints are more stringent, they do not conflict with w > β. 4.2 Reputation incentive In previous sections, we assumed that the lawyer is solely stimulated by her economic interest and derived optimal legal insurance systems accordingly. We now turn to the effect of reputation incentive on the legal insurance system. Reputation is regarded as an important factor which will affect lawyers service provision. Early studies explain this in terms of lawyers long term returns. For example, Galanter & Palay (1991) suggest that reputation may stimulate lawyers to work hard since it will affect their future business. Of course, the lawyer can improve her reputation by various ways. In this section, we incorporate the lawyer s concern for reputation in a simple way by assuming that she tries to guarantee the probability of prevailing above a certain level. Formally, the reputation incentive becomes a new constraint to the system, which is p(τ, ε) P (22) where P is a constant. Following backward induction, we can find that, in Stage 5 and Stage 4, this new constraint does not affect the choices of quantity and reported quantity. In Stage 3, the lawyer must choose an effort level, which makes the probability of prevailing higher than P when combined with the plaintiff s choice of quantity. It is an interesting question whether the plaintiff will benefit from the reputation constraint. In Stage 4, the plaintiff s optimal quantity decision is given by (5), p τ (τ, ε)a = β. We can find that, if β is sufficiently low, the plaintiff may already receive a probability of prevailing more than P in the equilibrium in Stage 3. Therefore, the reputation constraint is not always necessarily binding. In this section, we only look into the equilibria in which the reputation constraint (22) binds. Returning to the subgame perfect equilibrium of Stage 3, without the reputation constraint, the lawyer s choice of effort ε and the plaintiff s choice of quantity τ are given by the solution of Program A. When the reputation constraint does bind, then ε and τ will be determined by (5) together with (22). We use α, β, w, ε, τ and EU to denote the elements of the optimal insurance-payment system of Section 3 where the reputation constraint is absent. Given the equilibrium, suppose we put the reputation constraint at p p(τ, ε). We are going to investigate how the lawyer s effort choice ε, which satisfies p p(τ, ε) affects the equilibrium. As shown in Figure 4, the original equilibrium (without the reputation constraint) is the tangency point between the reaction function p τ and the lawyer s indifferent curve EV. Since the reaction function does not change, suppose the 18

Figure 4: The set of implementable effort and quantity hourly rate w could be relaxed, then setting w < w would result EV < EV. This would implement a point (τ, ε ) outside Ω and therefore increase the plaintiff s expected utility. Since the line of probability of prevailing p is steeper than the reaction function line p τ 11, we find p(τ, ε ) > p(τ, ε). Now consider the reputation constraint p(τ, ε) p(τ, ε ) > p(τ, ε). For the policy β = β, τ = τ, ε = ε and α satisfying (14), the plaintiff s expected utility becomes EU = EU. Thus, under the reputation constraint, the plaintiff s expected utility is at least EU. Considering p(τ, ε) p(τ, ε), the plaintiff s utility is EU EU. We now look into the effect of the reputation constraint on the co-payment. If (τ, ε) satisfy the reputation constraint p p and belong to Ω, the plaintiff s expected utility can not excess EU. Thus, to achieve an expected utility above EU, the equilibrium (τ, ε) (with the reputation constraint) must be outside the Ω. Moreover, the reputation constraint binds, therefore (τ, ε) must be on p and outside Ω. When p > p τ, (τ, ε) must lie on a reaction function which has a higher co-payment β. Thus, we state: Result 5: Under the reputation constraint, the plaintiff may be better off and cannot be worse off; and the insurance co-payment may increase. Comparing to the equilibrium of Section 3, the insurer can enlist the reputation constraint by raising the co-payment β (therefore lowering τ), forcing the lawyer to raise her effort. However, the plaintiff s equilibrium expected utility will not fall. Result 5 actually points out that the plaintiff will benefit from the reputation constraint. 11 Since p ε > 0, the probability of prevailing is non-increase if the lawyer reduces her effort. 19

5 Summary and conclusions In this paper a missing market, the lawyer s un-contractible effort, has been introduced to analyse legal expenses insurance under three fee arrangements: hourly fees, conditional fees and sliding fees. Unlike other early studies in litigation, we add the insurer as one player and look into the three-way relationship that often appears in litigation rather then the classic two-way relationship that is typically studied. We assume that insurance-payment contracts are based on reported information, and discover that truthful reporting imposes constraints on the contract parameters. In our model, the plaintiff chooses the amount of an input (quantity) after observing the lawyer s input (effort). Three points are highlighted as our conclusions. First, to design an optimal insurance-payment system, demand-side cost-sharing is necessary. The plaintiff is required to share the cost of the service received to mitigate his incentive to over utilise the insurance. A prospective payment (fixed excess fee) fails to achieve this function. Second, supply-side cost-sharing is necessary only if the quantity and effort are substitutes and the payment contract involves hourly fees. Under conditional fees and sliding fees, since the payment to the lawyer is contingent on the outcome of the lawsuit, the lawyer always has incentives to input effort to win the case. In addition, the lawyer s expected income also relies on the character of the lawsuit. This is consistent with Plott (1987). In his argument, the legal production function (the prevailing function in our model) is an important factor affecting legal costs. Third, the optimal insurance-payment system could be achieved under conditional fees and sliding fees. Under hourly fees, if the quantity and effort are complements, the optimal system could be achieved, while if they are substitutes pure hourly fees fail to induce an optimal system. Reputation incentives and side-contracts are also discussed in this paper. Unsurprisingly, we find that the plaintiff will benefit from the lawyer s concern for her reputation. In the case of the lawyer and the plaintiff colluding against the insurer, we find that under hourly fees, to prevent the insurer losing from side-contracting, the constraint on the insurance contract is inconsistent with the one in the non-side-contracting regime. One of the assumptions we made in this paper is that the plaintiff can observe the lawyer s effort. Under other circumstances, arguments on the observability of effort may arise. If the duration of a case is comparatively short, repeated interactions between the plaintiff and his lawyer may not occur. In this situation, we can interpret our model as a more general model with many clients. Suppose that the plaintiff may get some information from his friends and other clients about the effort of the lawyer. Given the information on the lawyer s past effort, the plaintiff may believe the lawyer will provide the same effort since the lawyer needs to maintain her reputation. Even non-observability of effort can be accommodated in the model. If the lawyer s effort is unobservable, Stage 3 and Stage 4 of our model become one stage. In this stage, the plaintiff and the lawyer make their choices simultaneously. The other stages of the model need no change. However, if the lawyer s 20