Christian At
email: christian.at@univ-fcomte.fr.

Tim Friehe
email: tim.friehe@uni-merburg.de.

Yannick Gabuthy
email: yannick.gabuthy@univ-lorraine.fr.

On Lawyer Compensation When Appeals Are Possible

Keywords: Litigation, Appeals, Moral hazard, Optimal contract. JEL Codes: D82, K41

This paper describes how plaintiff should compensate lawyers, who choose unobservable effort, when litigation may proceed from the trial to the appeals court. We find that, when it is very likely that the defendant will appeal, transfers made to the lawyer only after an appeals court's ruling are key instruments in incentivizing both trial and appeal court effort. Indeed, the lawyer may not receive any transfer after the trial court's ruling. In contrast, when reaching the appeals stage is unlikely, a favorable trial court ruling triggers a positive transfer to the lawyer and first-best appeals effort. In our setup, the lawyer may receive a lower transfer after winning in both the trial and the appeals court as compared to the scenario in which the first-instance court ruled against the plaintiff and the appeals court reversed that ruling.

Introduction

1.1. Motivation and main results. The appeals process -whereby litigants dissatisfied with the first-instance authority's judgment may bring their case to a higher authority -is a widely observed feature of judicial systems. For example, from 2001 to 2005, about 15% of civil trials -including tort, contract, and property casesconcluded in a large set of U.S. counties were reviewed by an appellate court after the first-instance verdict [START_REF] Cohen | Appeals from General Civil Trials in 46 Large Counties, 2001-2005[END_REF]. 1 The frequency of appeals was even higher in product liability and medical malpractice trials, where litigants filed appeals in about 33% and 18% of these cases, respectively. In France, in the year 2015, approximately 22% (68%) of cases brought before the major courts (labor courts) were appealed [START_REF] Chambaz | Les chiffres-clés de la Justice 2017[END_REF]. Furthermore, appeals mechanisms are also ubiquitous outside of the judicial system since they are frequently used by administrative agencies, religious bodies, professional sports leagues, and many other organizations [START_REF] Shavell | The appeals process as a means of error correction[END_REF].

This paper explores how litigants should respond to the importance of appeals in their attempts to incentivize their lawyers. Specifically, we describe the contract that a plaintiff should offer to her lawyer, who chooses unobservable effort, when the case may proceed to the appeals court after the trial court's judgment. In our setup, the lawyer selects a level of effort specific to the stage of adjudication and each court either rules in favor or against the plaintiff regarding the fixed award at stake. By considering a dynamic moral hazard problem, the present paper complements the very important literature on how best to overcome the moral hazard problem when the case is finally decided in the trial court. This literature usually deals with the contingent-fee regime (e.g., [START_REF] Dana | Expertise and contingent fees: The role of asymmetric information in attorney compensation[END_REF][START_REF] Hay | Optimal contingent fees in a world of settlement[END_REF].

We find that the compensation contract offered by the plaintiff relies to a great extent on incentives that depend on the appeals court's judgment. We assume that payments after the appeal court's decision condition on the full history, that is, distinguish the scenario in which the plaintiff's lawyer won in trial court from the scenario in which she lost. This distinction creates incentives for trial court effort. In fact, we establish that, in some circumstances, transfers provided to the lawyer only after the appeals court's judgment fully substitute for positive transfers after the trial court's judgment (despite the presence of limited liability constraints regarding the lawyer and the possibility that the appeals stage will not be reached). In the appeals stage, the lawyer's optimization regarding effort depends on the trial court's ruling because the transfer after winning the case in appeals court is contingent on the trial court's ruling and since winning in appeals court is easier for the plaintiff's lawyer when the trial court decided in favor of the plaintiff. The marginal cost effect makes it possible that the lawyer receives a smaller transfer after winning in both courts as compared to winning only in appeals court.

There are relatively few contributions analyzing the appeals process and its implications theoretically. Most of the existing literature on appeals is concerned with their effect on judges' choices or focuses on the losing litigant's incentives to file appeal (see Section 1.2). In our paper, we abstract from both issues and instead focus on the compensation contract between the plaintiff and her lawyer. In fact, very much in the spirit of the litigation contest literature (e.g., [START_REF] Farmer | Discovery and disclosure with asymmetric information and endogenous expenditure at trial[END_REF], we consider the probability of winning to solely depend on effort. Moreover, for tractability, we assume exogenous probabilities for the filing of appeal.

1.2. Related literature. According to the seminal contribution by [START_REF] Shavell | The appeals process as a means of error correction[END_REF], the appeals mechanism may be viewed as a means to correct errors in judicial decision-making. Assuming both that litigants possess information about the occurrence of legal errors and that appellate courts identify them, only erroneously decided cases are appealed and the accuracy of judicial decisions is thereby improved. [START_REF] Daughety | Appealing judgments[END_REF] extend this framework by assuming that the imperfectly informed judge -interested in choosing the socially desirable outcomecan draw inferences from the decision to appeal. 2 Spitzer and Talley (2000) uses a similar incomplete-information framework, but considers that trial and appeal deal with the same issue, whereas [START_REF] Daughety | Appealing judgments[END_REF] distinguish between the trial tribunal's focus on facts and the appeals court's focus on issues of law. Following a different behavioral perspective, [START_REF] Levy | Careeist judges and the appeals process[END_REF] models opportunistic judges (with career concerns), and highlights that they can have incentives to provoke appeal in order to signal their talent and thus benefit from an enhanced reputation.

However, as emphasized by Shavell (2006) and [START_REF] Iossa | Information provision and monitoring of the decision-maker in the presence of an appeal process[END_REF], the threat of appeals may also serve as a disciplining device for such opportunistic adjudicators by preventing them from deviating too much from the socially desirable award, since their desire is obviously to avoid reversal by the higher tribunal. In a different perspective, [START_REF] Wohlschlegel | The appeals process and incentives to settle[END_REF] focuses on the litigants' behavior, and analyzes how the anticipation of appeals influences the likelihood of pre-trial and post-trial settlement between asymmetrically informed parties. [START_REF] Friehe | Rent seeking and bias in appeals systems[END_REF] consider the possibility of appeals in the context of litigation contests, focusing on justice, effort, and filing incentives.

The present paper is most closely related to [START_REF] Ohlendorf | Repeated moral hazard and contracts with memory: The case of risk-neutrality[END_REF] and [START_REF] At | Moral hazard and agency relationship in sequential litigation[END_REF] as both papers are also describing how compensation contracts can best address dynamic moral hazard. The first paper purposefully abstains from assuming a technological impact of a first-period success or failure on the second period, while we have to account for one since appeal court decisions build upon trial court decisions.3 Along similar lines, in [START_REF] Ohlendorf | Repeated moral hazard and contracts with memory: The case of risk-neutrality[END_REF], the moral hazard problem is repeated in the sense that there is a potential reward for the principal in the first and the second period. In our context, there is a payment for the plaintiff in the first period only if the case concludes after the trial court decision. Our paper thus also contributes to the literature by considering a scenario in which the second stage may not be reached. Whereas [START_REF] Ohlendorf | Repeated moral hazard and contracts with memory: The case of risk-neutrality[END_REF] provide a more general contribution to the literature on principalagent relationships, [START_REF] At | Moral hazard and agency relationship in sequential litigation[END_REF] are also concerned with plaintiff-lawyer relationships. However, they consider 'sequential trials', that is, trials in which liability is determined first and the level of damages next. Sequential trials are common in mass tort cases, such as asbestos claims [START_REF] White | Explaining the flood of asbestos litigation: Consolidation, bifurcation, and bouquet trials[END_REF].

1.3. Plan of the paper. The remainder of the paper is organized as follows.

Section 2 presents the model. Section 3 derives our main results concerning the plaintiff's compensation of the lawyer. Section 4 concludes and suggests some extensions. For ease of exposition, all proofs are relegated to the appendix.

The model

A risk-neutral plaintiff has suffered losses from an accident or a breach of contract and sues the defendant for damages. The dispute may concern, for example, whether the injurer caused the plaintiff's losses or whether the promisor is excused from the nonperformance of contractual duties due to impossibility. The case may reach the appeals court after the trial court's judgment. The plaintiff hires a risk-neutral lawyer for representation (as in [START_REF] Emons | Playing it safe with low conditional fees versus being insured by high contingent fees[END_REF].

We consider the following timing of events (see Figure 1). 4 At date 0, the plaintiff makes a take-it-or-leave-it offer to the lawyer that concerns representation in both the trial court and the appeals court. We assume that the plaintiff can credibly commit not to renegotiate the contract after the trial court's decision. After accepting the contract, the lawyer chooses an unobservable effort level e 1 ∈ [0, 1] at date 1.

Litigation effort is a one-dimensional index of inputs such as attorney hours, pages of documentation, etc. The lawyer incurs a cost c 1 (e 1 ). At date 2, the trial court announces its verdict y 1 ∈ {0, 1}, where y 1 = 1 (0) denotes that the defendant is (not) found liable. The probability of winning is normalized to be equal to the effort level (i.e. P{y 1 = 1|e 1 } = e 1 ). After losing in trial court, the plaintiff always files for appeal. 5 The defendant files for appeal with probability q ∈ [0, 1], which is assumed to be exogenous and common knowledge. These are strong assumptions made to simplify the analysis. In other words, our game includes only the plaintiff and the lawyer as strategic players. After a trial court judgment in favor of the plaintiff (i.e., y 1 = 1), the game thus ends with probability 1 -q with a transfer of damages J > 0 to the plaintiff. Otherwise, at date 3, the lawyer chooses an unobservable effort level e 2 ∈ [0, 1] at a cost c 2y 1 (e 2 ) to influence the probability of winning the case in the appeals court. An appeals court judgment in favor of the (plaintiff) defendant is denoted y 2 = 1 (y 2 = 0), and implies a transfer of J (no transfer). We assume that the probability that the plaintiff succeeds in appeals court is given by P{y 2 = 1|e 2 } = e 2 and thus independent of the trial court's judgment. To reflect that appeals courts will to some extent be guided by the lower level court's selection and assessment of facts, for example, we let the cost of appeals court effort costs c 2y 1 (e 2 ) depend on the trial court's judgment y 1 ∈ {0, 1}, yielding c 20 (e 2 ) and c 21 (e 2 ), in a way made more precise below. Effectively, this is analogous to having a probability of winning depend on the previous judgment but not the cost of effort.

0 contract 1 y 1 = 1 e 1 y 1 = 0 1 -e 1 appeal q J 1 -q y 2 = 1, J e 2 y 2 = 0, 0 1 -e 2 appeal y 2 = 1, J e 2 y 2 = 0, 0 1 -e 2 2 3 4 Period 1: Trial stage Period 2: Appeals stage Figure 1:
The sequence of events (with y 1 /y 2 as the outcome in the trial/appeals court and J/0 as plaintiff gross payoff).

To be more precise regarding our assumptions about the cost of effort, we summarize as follows:

Assumption 1. (i) With regard to the cost of trial court effort, we assume: c 1 ≥ 0, c 1 ≥ 0, c 1 ≥ 0, c 1 (0) = 0, c 1 (0) = 0 and c 1 (e 1 ) > 0 for all e 1 > 0;

(ii) With regard to the cost of appeals court effort, we assume: c 2y 1 (0) = 0, As is standard, we assume that the lawyer incurs no cost if she makes no effort and that her marginal costs are convex. The cost of appeals court effort depends on the trial court' verdict. We postulate that the lawyer's effort at the appeals stage is absolutely and marginally less costly after winning in trial court. As explained above, this can also be interpreted as a different way of formalizing that legal arguments will be more effective in persuading the appeals court when they have the backing of the trial court's judgment.

c 2y 1 (0) = 0, c 2y 1 (e 2 ) ≥ 0, c 2y 1 (e 2 ) ≥ 0, c 2y

Optimal Lawyer Compensation

The plaintiff seeks to resolve a dynamic moral hazard problem. The lawyer's levels of effort in the trial court and the appeals court are unobservable and thus must be incentivized by outcome-contingent payments t 1 (y 1 ) (to be made at date 2) and t 2 (y 1 , y 2 ) (to be made at date 4 when it is reached). Due to the lawyer's limited liability, these transfers must be non-negative6 t 1 (y 1 ) ≥ 0

t 2 (y 1 , y 2 ) ≥ 0.
Since only the difference between the payment in case of success and the one after failure matters for the lawyer's incentives, these constraints reduce to

t 1 (1) ≥ 0 (LL1) t 2 (y 1 , 1) ≥ 0 (LL2) after setting t 1 (0) = t 2 (y 1 , 0) = 0.
The contract must also ensure the lawyer's participation. We normalize the lawyer's outside payoffs to zero, and obtain the individual rationality constraints concerning the lawyer's expected payoffs from the litigation in the trial court litigation and the appeals court:

L 1 = e 1 a(1) + (1 -e 1 )a(0) -c 1 (e 1 ) ≥ 0 (IR1) L 2 = e 2 t 2 (y 1 , 1) -c 2y 1 (e 2 ) ≥ 0 (IR2)
where a(y 1 ) is the lawyer's continuation payoff after a trial court verdict y 1 , and can be written as:

a(y 1 ) = t 1 (y 1 ) + (y 1 q + (1 -y 1 ))[e 2 (y 1 )t 2 (y 1 , 1) -c 2y 1 (e 2 (y 1 ))] (1)
When designing the contract, the plaintiff incorporates how payment schemes influence lawyer's incentives. The lawyer chooses the following levels of effort:

e 1 ∈ argmax e 1 ∈[0,1] L 1 = e 1 a(1) + (1 -e 1 )a(0) -c 1 (e 1 ) (2)
e 2 (y 1 ) ∈ argmax

e 2 ∈[0,1] L 2 = e 2 t 2 (y 1 , 1) -c 2y 1 (e 2 ) ( 3
)
that result from the first-order conditions

a(1) -a(0) = c 1 (e 1 ) (IC1) t 2 (y 1 , 1) = c 2y 1 (e 2 (y 1 )). ( IC2
)
The level of effort that is privately optimal for the lawyer is attained where the marginal benefit, which is the increase of the probability of obtaining a(1) instead of a(0) in (IC1) and the increase in the probability of obtaining t 2 (y 1 , 1) instead of t 2 (y 1 , 0) = 0 in (IC2), is equal to the marginal cost. In comparison, the first-best level of appeals court effort, e F B 2 (y 1 ), solves J = c 2y 1 (e 2 (y 1 )). The plaintiff maximizes her expected payoff using the contract terms C = t 1 (1), t 2 (y 1 , 1) subject to the lawyer's incentive, participation, and limited liability constraints 7 max C e 1 [(1 -q)J -t 1 (1) + qe 2 (1)(J -t 2 (1, 1))] + (1 -e 1 )e 2 (0)(J -t 2 (0, 1)) s.t. (LL1), (LL2), (IR1), (IR2), (IC1), (IC2)

The following intermediary result simplifies this problem.

Lemma 1. The constraints (LL2), (IR1) and (IR2) are satisfied.

Proof. See Appendix A

Given Lemma 1, the plaintiff's maximization problem may be rewritten using the lawyer's first-order conditions as: max e 1 ,e 2 (0),e 2 (1)

e 1 [(1 -q)J -t 1 (1) + qe 2 (1)(J -c 21 (e 2 (1)))] + (1 -e 1 )e 2 (0)(J -c 20 (e 2 (0))) s.t. t 1 (1) = -q[e 2 (1)c 21 (e 2 (1)) -c 21 (e 2 (1))] + a(0) + c 1 (e 1 ) ≥ 0 7 Remember that t 1 (0) = t 2 (y 1 , 0) = 0.
We can now characterize the contract that maximizes the plaintiff's expected payoffs subject to the constraints.

Proposition 1. The contract optimal for the plaintiff entails:

(i) effort levels such that e 1 > 0, e 2 (0) ∈ 0, e F B 2 (0) , and e 2 (1) ∈ 0, e F B 2 (1) ;

(ii) transfer payments at the trial stage such that t 1 (0) = 0 and (a) t 1 (1) > 0 and e 2 (1) = e F B 2 (1) if q < q, where q > 0, (b) t 1 (1) = 0 and e 2 (1) < e F B 2 (1) if q ≥ q, (iii) transfer payments at the appeals stage such that t 2 (y 1 , 0) = 0 and for t 2 (y 1 , 1):

(a) If c 20 (e 2 ) [1 -ε 20 ce ] < c 21 (e 2 ) [1 -ε 21 ce ]
with ε 2y 1 ce as the elasticity of costs after an outcome y 1 , then e 2 (1) > e 2 (0); there exists ê < e 2 (1) defined by

c 21 (e 2 ) = c 20 (ê) such that * if e 2 (0) > ê then t 2 (1, 1) < t 2 (0, 1) * if e 2 (0) < ê then t 2 (1, 1) > t 2 (0, 1). (b) If c 20 (e 2 ) [1 -ε 20 ce ] > c 21 (e 2 ) [1 -ε 21 ce ] then the sign of t 2 (1, 1) -t 2 (0, 1) is ambiguous. Proof. See Appendix B
Naturally, the plaintiff induces some effort in the court of first-instance. We also find that the effort in the appeals court may reach the first-best level when the trial court judged in favor of the plaintiff, but not otherwise. The dependency of the appeals court on the trial court's judgment shows in effort levels. The plaintiff may induce lower effort after a defeat in trial court (see part (iii)).

With regard to the transfers after the trial court judgment, the plaintiff incentivizes the lawyer using solely the payment after a success in the appeals court when it is sufficiently likely that the defendant will appeal after a defeat in trial court (i.e., when q ≥ q). If that payment applies, there will be less than first-best lawyer effort in the appeals court, as would be expected in a moral hazard setup.

With regard to the transfers after the appeal court's decision, we find that it is possible that the payment received after a defeat in trial court decision may be higher or lower than the one in which the appeals court trial was due to the defendant's filing of an appeal (i.e., after the plaintiff winning the trial court case). In this context, it must be borne in mind that the absolute and the marginal cost of appeal court effort depend on the trial court's decision. Having t 2 (1, 1) > t 2 (0, 1) clearly creates additional incentives for trial court effort. However, the cost advantage that is due to y 1 = 0 contributes to this objective as well. When the latter effect is very strong, we may thus have that t 2 (0, 1) > t 2 (1, 1) turns out to be optimal, which contrasts with the result by [START_REF] Ohlendorf | Repeated moral hazard and contracts with memory: The case of risk-neutrality[END_REF].

Proposition 2. An increase in the likelihood of an appeal by the defendant (i.e. q) implies that:

(a) de 1 dq < 0 and dt 1 (1) dq has an ambiguous sign;

(b) de 2 (0) dq > 0, dt 2 (0,1) dq > 0, de 2 (1)
dq ≤ 0, and dt 2 (1,1) dq ≤ 0.

Proof. See Appendix C

Intuitively, the greater the likelihood that the defendant files for appeal, the lower the induced level of effort in trial court. Winning in trial court is less important because it concludes the case with a lower probability. The lower level of effort in trial court implies that the plaintiff-lawyer pair more often enters the appeals court with a record of y 1 = 0 such that it is relatively more important to provide incentives using t 2 (0, 1). The fact that this implies some substitution then explains the effects regarding t 2 (1, 1) and e 2 (1).

Conclusion

This paper analyzes how the plaintiff would like to compensate the lawyer when the adjudication process includes the possibility to file an appeal. The plaintiff's problem can be understood as a dynamic moral hazard problem, where the trial in the appeals court is linked to the first-instance trial as appeals courts are to some extent guided by the trial court's activity. We highlight that the plaintiff focuses incentives on the scenario in which the case is appealed and then decided in favor of the plaintiff. This is due to the fact that incentives provided regarding the appeals stage are also relevant for effort incentives in the trial stage. Interestingly, there are circumstances in which the payment received by the lawyer after the appeal court's judgment is higher after losing in trial court than after winning. This possibility arises because persuading the appeals court implies costs that depend on the trial court outcome.

In our framework, lawyers' compensation depends on the case's history, that is, the steps that litigation took. In the United States, it is common for plaintiffs to pay their lawyers on a contingency-fee basis, where the contingency fee may very well depend upon whether the case was settled, won in the court of first instance or after appeal (Rule 1.5(c) Model Rules of Professional Conduct; e.g., [START_REF] Baumann | Contingent fees with legal discovery[END_REF]. However, in other jurisdictions, tying compensation of the lawyer as closely to the outcome of litigation is not possible. The present paper presents some arguments that restrictions along these lines may come at a cost to plaintiffs and thus also bear on access justice, adding to those presented in the previous literature (e.g., [START_REF] Helland | Contingent-fee contracts in litigation: A survey and assessment[END_REF].

Our aim was to explore lawyer compensation in a simple setup. This paper may thus be considered as a first step, and several extensions should be made to improve our understanding of the various matters that realistically impinge on this issue. For example, as mentioned in the paper, we restrict our attention to a full commitment situation by ruling out the possibility of contract renegotiation after the trial stage and before the potential appeal period. Using, for example, the analyses by [START_REF] Wang | Renegotiation-proof dynamic contracts with private information[END_REF] and/or [START_REF] Zhao | Renegotiation-proof contract in repeated agency[END_REF], it would be obviously relevant to embed the present approach in a more general setup with renegotiation-proof contracts, in order to check the robustness of our results in such an alternative analytical framework.

Moreover, in this paper, we focus on the litigation in the trial and the appeals court, ignoring the possibility of settlement bargaining. Clearly, it is interesting to consider the lawyer compensation offered by the plaintiff when there is settlement bargaining before the two levels of court judgments are passed.

the optimal continuation payoff π(a) may be written: With λ ≥ 0 as the Lagrange multiplier, the Lagrangian is:

max t 1 ,e 2 xP (e 2 ) -t 1 s.t. t 1 = a -xA(e 2 ) ≥ 0 Replacing t 1 = a -xA(e 2 )
L = xS(e 2 ) -a + λ(a -xA(e 2 ))
The first-order conditions are then given by:

S (e 2 ) -λA (e 2 ) = 0 (B.1) λ(a -xA(e 2 )) = 0 (B.2)
and allow us to reason about the plaintiff's desired level of appeals court effort as a function of the outcome y 0 .

Step 1. Proof of e 2 ∈ 0, e F B

2

We first show that e 2 = 0 cannot be a solution. We know from the proof of Lemma 1 that a(0) ≥ 0. From (IC1) (i.e. a(1) -a(0) = c 1 (e 1 )), and c 1 (e 1 ) ≥ 0, we deduce that a(1) ≥ a(0). Hence, since a ≥ 0 and A(0) = 0, we get λ = 0 and S (0) = 0, which is impossible since S (0) = J.

Next, we show that any e 2 > e F B 2 cannot be a solution. For e 2 > e F B 2 , we have S (e 2 ) < 0, which violates (B.1) since λA (e 2 ) ≥ 0.

Step 2. Proof of e 2 (0) ∈ 0, e F B 2 (0) and e 2 (1) ∈ 0, e F B 2 (1)

We know that t 1 (0) = 0 is optimal. The plaintiff's program with outcome y 1 = 0 is thus: max e 2 (0)

P (e 2 (0)) as x is a constant.

Since P (e F B 2 ) = 0 and P (e F B 2 ) = -e F B 2 c (e F B 2 ) < 0, we know that e 2 (0) < e F B 2 (0).

For the case with outcome y 1 = 1 such that x(1) = q, we can state the following lemma.

Lemma 2. For a given lawyer's continuation payoff a(1), we have:

(i) e 2 (1) = e F B 2 (1) and t 1 (1) = a(1) -qA e F B 2 (1) > 0 if q < a(1)

A(e F B 2 (1)) = q ;

(ii) e 2 (1) = A -1 a(1) q < e F B 2 (1) and t 1 (1) = 0 if q ≥ q.

Proof. Consider the first-order conditions (B.1) and (B.2).

If λ = 0, then S (e 2 ) = 0 and, thus,

e 2 = e F B 2 , implying π(a, e F B 2 ) = qS(e F B 2 ) -a ≤ 0 since a ≥ qA(e F B
2 ) = qS(e F B 2 ). Furthermore, we get t 1 (1) = a(1) -qA e F B 2 (1) > 0.

If λ > 0 or t 1 (1) = 0, then the second-best effort e 2 is the solution of a = qA(e 2 ).

We have π(a, e 2 ) = qP (e 2 ). Since P (e F B 2 ) = 0 and P (e

F B 2 ) = -e F B 2 c (e F B
2 ) < 0, we get P (e 2 ) > P (e F B

2 ) = 0 and, thus, π(a, e * 2 ) > π(a, e F B 2 ).

Consequently, it is always optimal to make the constraint binding. However, the constraint cannot be bind if the probability that the defendant will appeal is too small, q < q, since e F B 2 (1) = sup e 2 (1)∈(0,e F B 2 (1)] A(e 2 (1)).

Step 3. Analysis of the optimal continuation payoff π(a).

Consider, first, the case q ≥ q. We know from Step 2 that e 2 ∈ 0, e F B 2 and a =

xA(e 2 ). We also know that a ∈ (0, xA(e F B 2 )) since A(e 2 ) is continuous and increasing in e 2 and A(0) = 0. Furthermore, the derivative of the function π(a) = xP (e 2 ) =

xP (A -1 (a/x)) is π (a) = P (e 2 )
A (e 2 ) . When a → xA(e F B 2 ) (i.e. when e 2 → e F B 2 ), we have:

π (a) = P (e 2 ) A (e 2 ) → P (e F B 2 ) A (e F B 2 ) = -1 < 0
While, when a → 0 (i.e. when e 2 → 0), we get:

π (a) → P (0) A (0) → +∞
Some algebra gives: π (a) = P (e 2 )A (e 2 ) -P (e 2 )A (e 2 ) A (e 2 )2 = S (e 2 )A (e 2 ) -S (e 2 )A (e 2 ) A (e 2 ) 2 < 0 implying that π(a) is concave. Therefore, the continuation payoff reaches a maximum in the interval (0, xA(e F B 2 )).

Second, if q < q, then the continuation payoff is π(a(1)) = qS(e F B 2 (1)) -a(1).

Step 4. Determination of the second-best effort level at the trial stage.

The plaintiff can choose any pair of non-negative a(1) and a(0) to solve the following maximization problem: max e 1 ,a(0),a(1)

e 1 [(1 -q)J + π(a( 1))] + (1 -e 1 )π(a(0))

Replacing a(1) by a(0) + c 1 (e 1 ) using (IC1), this problem becomes:

max e 1 ,a (0)
e 1 [(1 -q)J + π(a(0) + c 1 (e 1 ))] + (1 -e 1 )π(a( 0))

The first-order conditions, which are sufficient since the objective function is concave, are given by:

(

1 -q)J + π(a(1)) -π(a(0)) + π (a(1))e 1 c 1 (e 1 ) = 0 (B.3) e 1 π (a(1)) + (1 -e 1 )π (a(0)) = 0 (B.4)
Consider that e 1 = 0. From (IC1), we deduce that a(1) = a(0) since c (0) = 0 by assumption. From (B.4), we get π (a(0)) = 0 = π (a(1)) and π(a(1)) = π(a(0)), which violates (B.3). Therefore, the lawyer exerts a positive effort at the trial stage (i.e. e 1 > 0).

Note that q ≡ a(1)

A(e F B qA(e 2 (0), 0). Remember A(e 2 , y 1 ) = e 2 c 2y 1 (e 2 ) -c 2y 1 (e 2 ). Our assumptions ensure that A e (e, y 1 ) = e 2 c 2y 1 (e 2 ) ≥ 0 and A e (e, y 1 ) = c 2y 1 (e 2 ) + ec 2y 1 (e 2 ) ≥ 0. We have two cases to consider.

Case (a): Suppose that c 21 (e 2 )-c 20 (e 2 ) < c 21 (e 2 )-c 20 (e 2 ) e 2 < 0 ∀e 2 ∈ (0, e F B 2 ), implying A(e 2 (0), 1) < A(e 2 (0), 0). We must have A(e 2 (1), 1) > A(e 2 (0), 0) at the optimum, which can be satisfied only for e 2 (1) > e 2 (0). From (IC2), we know that t 2 (0, 1) = c 20 (e 2 (0)) and t 2 (1, 1) = c 21 (e 2 (1)). Furthermore, there exists ê < e 2 (1) such that c 21 (e 2 (1)) = c 20 (ê) since c 21 (e 2 ) < c 20 (e 2 ), c 2y 1 (e 2 ) ≥ 0 and e 2 (1) > e 2 (0). We deduce that t 2 (1, 1) = c 21 (e 2 (1)) < t 2 (0, 1) = c 20 (e 2 (0)) if e 2 (0) > ê, and t 2 (1, 1) > t 2 (0, 1) otherwise.

Case (a)

e A(e 2 , y 1 ) 0 e 2 (0) e 2 (1)

A(e 2 , 0)

A(e 2 , 1)

A(e 2 (0), 0)

Figure 2: e 2 (1) must be greater than e 2 (1) > e 2 (0) to ensure A(e 2 (1), 1) > A(e 2 (0), 0) 2 ) and the signs of both e 2 (1) -e 2 (0) and t 2 (1, 1) -t 2 (0, 1) are ambiguous.

t 2 (1, 1) < t 2 (0, 1) t 2 (1, 1) > t 2 (0, 1)

Case (b)

e A(e, y 1 )

0 e 2 (1)
e 2 (0)

A(e, 1)

A(e, 0)

A(e 2 (1), 1)

Figure 4: e 2 (0) must be smaller than e 2 (0) > e 2 (1) to ensure A(e 2 (1), 1) > A(e 2 (0), 0); therefore, we can have e 2 (0) > e 2 (1) or e 2 (0) < e 2 (1) as a solution.

Appendix C. Proof of proposition 2

Step 1. Proof of de 2 (1) dq ≤ 0 and dt 2 (1,1)

dq ≤ 0
Consider, first, the case where q > q. We know that t 1 (1) = 0 and, thus, a(1) =

qA(e 2 (1), 1). For an optimal choice of a(1), we have de 2 (1) dq = -a(1) q 2 A (e 2 (1),1) < 0, since e 2 (1) = A -1 a(1) q . From (3), we deduce that dt 2 (1,1) dq < 0. Consider now the case where q ≤ q. We have e 2 (1) = e F B 2 (1), implying that de 2 (1) dq = dt 2 (1,1) dq = 0.

Step 2. Proof of de 1 dq < 0, de 2 (0) dq > 0 and dt 2 (0,1) dq > 0

Consider the first-order conditions (B.3) and (B.4), and note for simplicity:

E(a(q), e 1 (q), q) = (1 -q)J + π(a(q) + c 1 (e 1 (q)) -π(a(q)) + π (a(q) + c 1 (e 1 (q))e 1 (q)c 1 (e 1 (q)) = 0 and A(a(q), e 1 (q), q) = e 1 (q)π (a(q) + c 1 (e 1 (q)) + (1 -e(q))π (a(q)) = 0

By totally differentiating the system, we get:

E a da dq + E e de dq + E q = 0 A a da dq + A e de dq + A q = 0 Given that A q = 0, we get E e A a -(E a A e ) 2 > 0, since the program is concave. We deduce that: sign de dq = sign -E q A a sign da dq = sign E q A e where E q = -J < 0, A a = e(q)π (a(q) + c 1 (e(q)) + (1 -e(q))π (a(q)) < 0 and A e = π (a(q) + c 1 (e(q)) -π (a(q)) + π (a(q) + c 1 (e(q))e(q)c 1 (e(q)) < 0 (since, from (B.4), π (a(q) + c 1 (e(q)) and π (a(q)) have opposite signs, and π(.) is concave).

Therefore, we get de 1 dq < 0. We also have da(0) dq > 0. We know that e 2 (0) = A -1 (a(0)) and, thus, de 2 (0) dq = 1 A (e 2 (0),0) da(0) dq > 0, implying dt 2 (0,1) dq > 0.

Step 3. Determination of the effect of q on t 1 (1)

Only the case where q < q matters as t 1 (1) = 0 otherwise. We know that t 1 (1) = a(1) -qA e F B 2 (1) and obtain:

dt 1 (1) dq = da(0) dq + c 1 (e 1 ) de 1 dq -A e F B 2 (1) which exhibits an ambiguous sign.

where a(0) and A(e 2 (0)) were defined above. With λ ≥ 0 as the Lagrange multiplier, the Lagrangian is (omitting y 1 = 0 from the parentheses):

L = pS(e 2 ) -a + λ(a -pA(e 2 ))

The first-order conditions are then given by: We deduce that the optimal choice of the probability of an appeal is p = 1. This is consistent with the result of [START_REF] Ohlendorf | Repeated moral hazard and contracts with memory: The case of risk-neutrality[END_REF] when the continuation costs are equal to 0.

  1 (e 2 ) ≥ 0 and c 2y 1 (e 2 ) > 0 for all e 2 > 0; c 21 (.) < c 20 (.), c 21 (e 2 ) < c 20 (e 2 ) and c 21 (1) ≥ J.

  in the plaintiff's objective function, this becomes: max e 2 xS(e 2 ) -a s.t. a -xA(e 2 ) ≥ 0

Figure 3 :

 3 Figure 3:

S

  (e 2 ) -λA (e 2 from (C.2) that λ = 0 cannot be a solution, hence using (C.3), we can replace a = pA(e 2 ) in the program that simplifies to max e 2 ,p pS(e 2 ) -pA(e 2 ) = pP (e 2 )

[START_REF] Eisenberg | Plaintiphobia in state courts redux? an empirical study of state court trials on appeal[END_REF], for example, report a ratio of 11.6% over the period

2005-2009.

In a broader context, the adjudicator may also gather information from other sources. For instance,[START_REF] Oytana | The judicial expert in a two-tier hierarchy[END_REF] develops a model where the appellate court makes decisions on the basis of information provided by -potentially biased -experts.

The literature on dynamic moral hazard problems includes several papers assuming that the outcome of the first stage bears on the second stage in terms of effectiveness of the effort or costs. See, for example,[START_REF] Schmitz | Allocating control in agency problems with limited liability and sequential hidden actions[END_REF] andSchmitz (2013), and the literature cited therein.

We neglect discounting and assume that the plaintiff incurs no fixed cost in filing the lawsuit. These assumptions are made to alleviate notations and do not alter the gist of our arguments.

We introduce this as an assumption in our main analysis. In our Appendix D, we outline the scenario in which the plaintiff chooses the probability of appeal after having lost the trial court case.

This restriction is consistent with the champerty doctrine in the U.S. and the forbidden pactum de cuota litis in continental Europe.

(1)) > 0 is satisfied since e 1 > 0 (implying that a(1) > 0).Step 5. Assessing payments after the appeals court's judgment. Since e 1 > 0, from (IC1) we deduce a(1) > a(0) or qA(e 2 (1), 1) > A(e 2 (0), 0) ≥

Acknowledgments

We are grateful for the helpful comments received from two anonymous reviewers.

Appendix A. Proof of Lemma 1

Using the t 2 (y 1 , 1) implied by (IC2) in (IR2) yields:

L 2 (e 2 (y 1 )) = e 2 (y 1 )c e 2 y 1 (e 2 (y 1 )) -c 2y 1 (e 2 (y 1 ))

Since L 2 (0) = 0 and L 2 (e 2 (y 1 )) = e 2 (y 1 )c e 2 (e 2 (y 1 ), y 1 ) ≥ 0, we get L 2 (e 2 (y 1 )) ≥ 0 such that (IR2) is satisfied.

Restating (IC1) yields a(1) = a(0) + c 1 (e 1 ), which allows to state (IR1) as follows:

L 1 (e 1 ) = a(0) + e 1 c 1 (e 1 ) -c 1 (e 1 ) ≥ 0 Since (IR2) is satisfied, we have a(0) ≥ 0. By noting f (e 1 ) = e 1 c 1 (e 1 ) -c 1 (e 1 ), we have f (0) = 0 and f (e 1 ) = e 1 c 1 (e 1 ) ≥ 0 since c 1 (e 1 ) ≥ 0 by assumption. As a result, (IR1) is satisfied.

(LL2) is induced by (IC2) since c e 2 (e 2 (y 1 ), y 1 ) ≥ 0.

Appendix B. Proof of Proposition 1

Denote by x(y 1 ) = qy 1 + 1 -y 1 the probability that there will be an appeal as a function of the trial court's judgment. Moreover, define π(a) as the plaintiff's maximum continuation payoff when she implements a lawyer continuation payoff a.

Using t 2 (y 1 , 1) = c 2y 1 (e 2 (y 1 )), we consider:

P (e 2 (y 1 ), y 1 ) = e 2 (y 1 )(J -c 2y 1 (e 2 (y 1 )))

and A(e 2 (y 1 ), y 1 ) = e 2 (y 1 )c 2y 1 (e 2 (y 1 )) -c 2y 1 (e 2 (y 1 )) as the expected payoff of the plaintiff and the lawyer, respectively, from an appeal.

The expected social surplus is thus given by: S(e 2 (y 1 ), y 1 ) = e 2 (y 1 )J -c 2y 1 (e 2 (y 1 )).

Hereafter, we omit y 1 when the computations are true whatever the value of y 1 .

Using a restatement of (1), a = t 1 + xA(e 2 ), the plaintiff's program to determine

Appendix D. Endogenous probability

Suppose that the plaintiff lost the case in trial court (i.e., we have y 1 = 0) and now considers setting the probability of appeal, p ∈ [0, 1]. The plaintiff would at this stage want to max e 2 (0),p pS(e 2 (0)) -a(0) s.t. a(0) -pA(e 2 (0)) ≥ 0


Christian At
email: christian.at@univ-fcomte.fr.

Yannick Gabuthy
email: yannick.gabuthy@univ-fcomte.fr.

Moral Hazard and Agency Relationship in Sequential Litigation

Keywords: Sequential trials, Moral hazard. JEL codes: D82, K41

The aim of this paper is to determine the optimal fee contract between a plaintiff and his lawyer in a trial process where liability and damages are treated sequentially: the court determines liability at the first stage and chooses damages at the second one. The plaintiff-lawyer relationship is embedded in a dynamic principal-agent setup where the lawyer's effort is unobservable by her client. It is shown that such a sequential process induces an interesting spillover effect for the plaintiff: the prospect of payment at the second period acts as carrot for the first period by inciting the lawyer to make effort in the liability stage in order to enjoy the reward in case of winning when damages are litigated. This effect implies a particular contractual design where the lawyer gets no transfer from the plaintiff at the first period.

Introduction

The moral hazard problem is common in litigation since the plaintiff does not have the ability or the expertise to directly observe how much time his lawyer is spending on the case and how hard she is working during those hours. 1 Absent concerns for reputation or long-run relationship, paying the lawyer by the hour would clearly induce shirking. In this context, there is a large literature which shows that contingent fees may be a response to this problem by tying the attorney's pay on the outcome of litigation. Typically, the contingent fee mechanism implies that the lawyer gets a share of the judgement if her client wins and nothing if he loses. This mechanism is widely used in the U.S., where the typical contract involves a percentage of 33%, while several European countries are considering to allow such payment schemes (which are currently forbidden). 2 Overall, contingent fees are shown to be better than fixed fees because the individual with more information becomes residual claimant (see, e.g., [START_REF] Halpern | Legal fees contracts and alternative cost rules: An economic analysis[END_REF][START_REF] Danzon | Contingent fees for personal injury litigation[END_REF][START_REF] Hay | Contingency fees and agency costs[END_REF][START_REF] Wang | The optimality of contingent fees in the agency problem of litigation[END_REF].

Though this literature has generated insightful results, it is also limited by considering litigation only as a unitary/one-shot process while legal systems often feature a sequence of decisions before a final judgment is reached. In criminal procedures, convinction and sentencing are indeed frequently treated sequentially: the jury first decides whether the defendant committed the crime while the judge sentences him. This division finds an analogue in the tort law where civil proceedings may imply a separation between liability and damages. For example, when Polaroid successfully sued Kodak for infringing its patents on instant photography, the questions of liability and damages were litigated in different trials [START_REF] Landes | Sequential versus unitary trials: An economic analysis[END_REF]. In the same way, the New York law encourages judges to separate liability and damages issues into sequential proceedings in any action for personal injury. More precisely, it presumptively assigns the determination of liability to the first stage and determination of damages to the second stage, unless the court orders otherwise [START_REF] De Mot | Are sequential trials really better than unitary trials?[END_REF]. 3 Overall, sequential trials are common in antitrust 1 We use the pronon 'she' to refer to the lawyer and 'he' to refer to the client. 2 Pactum cuota litis is not allowed by the ethical code of the European association of lawyers. However, as explained by [START_REF] Emons | US-style contingent fees and UK-style conditional fees: Agency problems and the supply of legal services[END_REF], market pressure has led some countries (e.g. the U.K., Belgium and the Netherlands) to allow conditional fees in which the lawyer receives an upscale premium -unrelated to the amount adjudicated -in case of winning.

3 In practice, a so-called 'sequential litigation' is not limited to separating liability and damages, and refers to any situation where the issues of disagreement are adjudicated sequentially by the court. In contrast, in 'unitary litigation', the distinct issues of a dispute are bundled together in a single proceeding.

litigation, employment litigation, and complex litigation such as mass tort cases where a very important sphere of application is related to asbestos claims. 4 In this context, several legal systems specify the courts' option of conducting separate trials. For example, in U.S. federal courts, Rule 42(b) of the Federal Rules of Civil Procedure allows the court to separate liability and damages "in furtherance of convenience or to avoid prejudice".

Likewise, in the U.K., Rule 3.1(2) of the Civil Procedure Rules states that the court may "direct a separate trial of any issue". Finally, in Canada, procedure rules are determined independently by each jurisdiction and most jurisdictions allow sequential litigation.

The aim of our paper is then to determine the optimal fee contract arising between the plaintiff and his lawyer in such a sequential litigation, considering that the lawyer's effort is unobservable by her client. Following this objective, we broaden the discussion of attorneys' fee arrangements in mainly two ways.

First, as mentioned above, the law and economics literature examined the moral hazard problem exclusively in the situation of unitary trials while many cases of sequential litigation are observed in practice. Our approach considers such cases by modelling the plaintiff-lawyer relationship as a dynamic principal-agent setup under moral hazard where the agent is protected by limited liability. This relationship may be then considered as a particular problem of repeated moral hazard (see, e.g., [START_REF] Bierbaum | Repeated moral hazard under limited liability[END_REF][START_REF] Ohlendorf | Repeated moral hazard and contracts with memory: The case of risk-neutrality[END_REF].

Second, we consider that the lawyer and her client are unrestricted in the contracts they can make, and we then try to highlight what types of fee arrangements arise in equilibrium.

This approach contrasts sharply with most of the existing literature which restricts attention a priori to percentage fees, without worrying about the additional and important issue of what type of payment mechanism might be more efficient. 5 The argument behind such a restriction is based on the fact that some of the proposed alternatives would not be allowed by existing laws, following the champerty doctrine in the U.S. and the forbidden pactum cuota litis in continental Europe. However, this does not mean that these alternatives will never be allowed in any jurisdictions at any time. Furthermore, we argue that a normative perspective may be useful to provide insightful recommendations/guidelines to alter some inefficiencies of the existing legal systems.

Our paper complements also the previous literature on sequential litigation which ignores the issue of lawyer-client relationship by focusing the analysis on the parties' strategic interactions. [START_REF] Landes | Sequential versus unitary trials: An economic analysis[END_REF] presents the first formal study of the incentives to file, settle, and spend in sequential versus unitary trials, while there is a related informal discussion in the pioneering paper of [START_REF] Schwartz | Severance -A means of minimizing the role of burden and expense in determining the outcome of litigation[END_REF]. It is shown that separating liability and damages tends to lower the expected cost of litigating a given dispute because the cost of litigating damages will be saved if the plaintiff loses liability. This expected cost reduction increases his incentive to sue and decreases his willingness to settle. Therefore, even though the litigation costs per case will fall, the number of cases will rise so that the overall effect on the total litigation costs is uncertain. 6 Chen et al. (1997) extend the analysis by considering asymmetric information: the defendant is privately informed about both the probability that he will found liable and the damages, conditional upon a finding for liability. The Landes' results are challenged due to the information revealing effect of sequential litigation: the information produced in the first stage of a sequential dispute may increase the likelihood of settlement between the litigants, relative to a unitary trial.

Following this argument, [START_REF] Gensler | Bifurcation unbound[END_REF] encourages federal courts to favor sequential proceedings. Finally, interesting results are obtained when the costs of litigation are assumed to be a choice variable for the conflicting parties. De [START_REF] De Mot | Are sequential trials really better than unitary trials?[END_REF] analyze endogenous litigation expenditures, by using a rent-seeking model, and show that the optimal trial process (i.e. unitary or sequential) depends on the strength of the plaintiff's case. In fact, separating issues incentivizes the defendant to front-load his expenditures in order to avoid the costs associated with the second step of the trial. This front-loading will promote expenditures by the plaintiff if his case is sufficiently strong (viz. if the returns on his investment is sufficiently high). The opposite effect is obtained when the plaintiff has a weak case. This result provides a theoretical foundation to procedural rules that give courts discretion on whether to join or separate issues.

Considering the plaintiff-lawyer relationship, our analysis shows that the optimal -secondbest -contract exhibits a particular design. Indeed, the separation between liability and damages in sequential litigation implies an interesting spillover effect for the plaintiff which enables him not to to reward his lawyer at the liability stage. In fact, the transfer provided at the second period acts as carrot for the first period by inciting the lawyer to make effort in the liability stage in order to get this transfer in case of victory when damages are litigated.

The remainder of the paper is organized as follows. Section 2 lays down the model, and derives our main results concerning the optimal fee contracts. Section 3 concludes and suggests some possible extensions. Proofs are relegated to the appendix.

The Model

2.1. Framework. A plaintiff has been a victim of an accident or a breach of contract, and sues the defendant to be paid damages following a sequential litigation process. In this context, the lawyer and plaintiff enter into a contractual relationship for the lawsuit, which implies the following timing of events.7 At date 0, the plaintiff makes a take-it-or-leave-it contract offer to the lawyer. Having accepted the offer, at date 1 the lawyer chooses an unobservable first-period effort level h 1 ∈ [0, 1], incurring disutility c 1 (h 1 ). At date 2, the verifiable result y ∈ {0, 1} is realized, where y = 0 denotes failure (viz. the defendant is not found liable) and induces that the litigation process is aborted, while y = 1 denotes success (viz. the defendant is found liable) and implies that the game goes on. The probability of success is normalized to equal the effort level (i.e. P r{y = 1|h 1 } = h 1 ).

In case of winning, the relationship continues to date 3 where the lawyer chooses an unobservable second-period effort level h 2 ∈ [0, 1], incurring disutility c 2 (h 2 ). Finally, at date 4, the plaintiff gets the verifiable damages J ∈ {0, J H }, where

P r{J = J H |h 2 } = h 2 . 8
In other words, the attorney's efforts affect the winning probability of the plaintiff at the first period and the level of compensation awarded by the court at the second one. The effort cost functions satisfy c i > 0, c i > 0, c i ≥ 0, c i (0) = 0, c i (0) = 0 and c i (1) ≥ J H

(with i = 1, 2). We also assume that c i is high enough to ensure the concavity of the objective functions. The sequence of events is illustrated in Figure 1.

Before analyzing the main question of interest, several comments on the framework and its main assumptions are in order. First, this paper deals with the lawsuit only, ignoring the possibility of pre-trial settlement or considering that this attempt has failed. This assumption may seem somewhat puzzling given that, in practice, a large number of disputes do not rise the level of a lawsuit and end in settlement. However, as mentioned by [START_REF] Hay | Contingency fees and agency costs[END_REF], in most situations the settlement amount reflects the expected value the claim would have if it went to trial. Therefore, the optimal lawyer-client contract in a world 0 contract 1 h 1 chosen y = 0 is realized, the game stops in which cases settle should not be very different from the optimal contract in which all cases go to trial. Second, we consider a full commitment situation, where the principal can commit not to renegotiate the contract that is written at date 0. This feature has been deliberately chosen to develop our main insights in a tractable manner, and could be relaxed in an extended version of the model. In the same way, we assume that the agent cannot be replaced at the end of the liability stage. While this assumption is restrictive, it seems reasonable to consider that the first-period relationship induces learning and specific training, which make replacement unprofitable for the principal and imply that the parties are 'locked-in' as a bilateral monopoly. Finally, while litigation generates external costs and benefits (including administrative costs and deterrence of harmful activities), we abstract from the complex problem of the social optimality by analyzing the contractual arrangement that maximizes the client's -private -welfare in litigation.9

y = 1 is realized 2 y = 1 is realized 3 h 2 chosen 4 J ∈ {0, J H } is realized
2.2. Optimal Contracting. As a benchmark, we briefly consider the first-best situation where the effort levels maximize the expected total surplus:

S(h 1 , h 2 ) = h 1 (h 2 J H -c 2 (h 2 )) -c 1 h 1 . (1)
where h 2 J H -c 2 (h 2 ) is the expected total surplus of the damages' game, which occurs with the probability h 1 of winning the liability's game. The first-best effort levels h F B 1 and h F B 2 are then the unique solutions to the following first-order conditions:

h F B 2 J H -c 2 (h F B 2 ) = c 1 (h F B 1 ), (2)
J H = c 2 (h F B 2 ). (3)
The left-hand sides of these expressions define the marginal aggregate benefits of the firstand second-periods' efforts. Efficiency requires that such marginal returns be equated to the corresponding marginal costs (which are characterized in the right-hand sides of ( 2) and ( 3)). Furthermore, since c i (0) = 0, c i (0) = 0 and c i (1) ≥ J H , we get interior solutions (i.e. h F B 1 ∈ (0, 1) and h F B 2 ∈ (0, 1)).

Let us now turn to the asymmetric information situation where the lawyer's effort choices are not observable by her client. A contract specifies a recommanded effort level h i and a transfer payment t i from the plaintiff to his lawyer at each period i. The lawyer is assumed to be protected by limited liability which implies that the transfers must always be non-negative:

t 1 ≥ 0, (4)
t 2 ≥ 0.

(5)

We thus rule out contracts with payments from the attorney to the client. This assumption restricts the class of feasible contracts, however it is commonly used in law and economics literature (see, e.g., [START_REF] Emons | US-style contingent fees and UK-style conditional fees: Agency problems and the supply of legal services[END_REF][START_REF] Emons | Playing it safe with low conditional fees versus being insured by High contingent fees[END_REF][START_REF] Emons | Conditional versus contingent fees[END_REF], and it is consistent with the champerty doctrine in the U.S. and the forbidden pactum cuota litis in continental Europe: the lawyer cannot buy the lawsuit from the client or incur penalties if the case is lost. 10 Furthermore, the condition stated in (5) implies that the lawyer cannot be forced to refund payments that he received in the past. In fact, this assumption is made without loss of generality and we could easily relax the constraint by considering only t 2 ≥ -t 1 .

Moreover, the recommendations formulated by the plaintiff must satisfy suitable incentive compatibility constraints. Following the standard principal-agent approach, we consider that the lawyers' payments are made contingent on the results of the trial (viz. the lawyer will never be rewarded for a failure). Moving backward, conditional on a success arising at the first period, the lawyer chooses her effort level h 2 at the second period such that:

h 2 = arg max h 2 ∈[0,1] U 2 = h 2 t 2 -c 2 (h 2 ). ( 6
)
10 Furthermore, without such limited-liability conditions, the -risk-neutral -agent could become the residual claimant and the moral-hazard problem would disappear. In other words, the plaintiff could costlessly structure the lawyer's payments so that the first-best effort levels would be implemented:

the contract would imply negative (positive) transfers from the plaintiff to his lawyer in case of failure (success) in both periods. See, e.g., [START_REF] Laffont | The theory of incentives. The principalagent model[END_REF] who analyze this issue in a more simple and general setup.

The first-order condition of the second-period lawyer's optimization problem is then given by:

t 2 = c 2 (h 2 ). ( 7
)
This condition implies that the marginal benefit from an increase in the second-period effort (i.e. t 2 ) is equated with its marginal cost (i.e. c 2 (h 2 )).

Anticipating her future expected payoffs, the lawyer chooses h 1 at the first period such that:

h 1 = arg max h 1 ∈[0,1] U 1 = h 1 (t 1 + U 2 ) -c 1 (h 1 ). ( 8
)
The first-order condition of the first-period lawyer's optimization problem is then given by:

t 1 + h 2 t 2 -c 2 (h 2 ) = c 1 (h 1 ). ( 9
)
This expression can be interpreted like the condition in ( 7), except that an increase in the first-period effort will be not only rewarded by the immediate transfer t 1 , but also by the possibility to continue the litigation process and, then, the prospect to get the secondperiod transfer in case of winning (which implies a net -ex-ante -expected payment of

U 2 ).
Finally, the contract must ensure the lawyer's participation. Assuming a competitive market for attorneys so that they expect zero profit or normalizing to zero their outside opportunity wages, the individual rationality conditions are:

U 1 ≥ 0, (10)
U 2 ≥ 0. ( 11
)
The maximization problem of the plaintiff may then be written as:

max h i ,t i Π = h 1 (-t 1 + h 2 (J H -t 2 )) s.t. t 1 ≥ 0, t 2 ≥ 0, U 1 ≥ 0, U 2 ≥ 0 and t 1 + h 2 t 2 -c 2 (h 2 ) -c 1 (h 1 ) = 0, t 2 -c 2 (h 2 ) = 0.
The next result follows.

Proposition 1 When the lawyer's efforts are unobservable, the second-best contract {h SB i , t SB i } entails:

0 < h SB i < h F B i (∀i = 1, 2), t SB 1 = 0 and t SB 2 = c 2 (h SB 2 ).

Proof. See Appendix

The intuition behind these results is the following. First, the prospect of a second-period payment (i.e. t SB 2 > 0) following a first-period success motivates the lawyer to exert effort in the first period, which enables the client not to reward it (i.e. t SB 1 = 0). In other words, the separation between liability and damages in sequential trial creates a desirable spillover effect for the principal: the incentives provided in the second period act as carrot for the first period since the agent works hard in the liability stage in order to enjoy the reward in case of winning when damages are litigated. 11

Furthermore, more classically, the second-best contract exhibits a conflict between the limited liability and the incentive constraints, implying a downward distorsion of the firstperiod effort. Punishment being infeasible, the client is restricted to use only rewards to induce effort from the lawyer. This restriction of the principal's incentive instrument implies that he must give up some ex-ante rent to the agent. Following ( 6), ( 8) and ( 9), this expected limited liability rent is given by:

U SB 1 = h SB 1 (t SB 1 + h SB 2 t SB 2 -c 2 (h SB 2 )) -c 1 (h SB 1 ) = h SB 1 c 1 (h SB 1 ) -c 1 (h SB 1 ) ≥ 0. (12)
Hence, the client distorts the first-period effort level below to its first-best value (i.e.

h SB 1 < h F B 1
) in order to reduce the cost of this rent. 12 In the same way, the plaintiff implements h SB 2 < h F B 2 because the second-period transfer is increasing in the lawyer's effort level (i.e. t SB 2 = c 2 (h SB 2 ) > 0). Overall, the conjunction of moral hazard and limited liability induces a trade-off between rent extraction and efficiency which prevents the principal to implement the first-best effort levels.

11 Notice that our model treats the outcome of the two stages of the trial as independent, while it would be relevant to consider that the probability to win at the damages stage increases when the effort chosen at the liability stage is higher (viz. P r{J = J H |h 2 , h 1 } is increasing in h 1 ). This restriction enables us to develop our main results in a focused and tractable manner. Furthermore, we conjecture that such a modification would not alter the gist of our arguments. Indeed, assuming that P r{J = J H |h 2 , h 1 } is increasing in h 1 would create only an expected additional benefit for the lawyer to exert effort h 1 in the first period. Therefore, the main insight of our analysis should not be impacted: the plaintiff should pay nothing for success in the liability stage, and should reward the lawyer in the damages stage in case of victory.

12 It is straightforward to show that there is no liability rent in the first-best situation:

t F B i = c i (h F B i )/h F B i ⇒ U F B 1 = 0.

Conclusion

This paper studies the plaintiff-lawyer contractual relationship when the litigation process implies a separation between liability and damages. This relationship is then considered as a particular problem of repeated moral hazard which induces a specific fee arrangement:

following the spillover effect highlighted in the analysis, the prospect of payment at the damages stage induces the lawyer to provide effort at the liability stage, which implies no transfer at the first period. As noted in Introduction, this is, to our knowledge, the first paper that analyzes the moral hazard problem between a lawyer and her client within the specific context of sequential litigation, whereas this context is common in practice.

However, our analysis has only just scratched the surface of the various matters that impinge on this issue, and some extensions and generalizations suggest themselves. First, as mentioned in the paper, we consider a full commitment regime by ruling out the possibility of renegotiation after the liability stage and before the damages one. In other words, we assume that the plaintiff can ex-ante commit not to renegotiate the contract after the first period is over. A further step toward realism would be to embed the present framework in a more general setup implying renegotiation-proof contracts. 13

In the same way, the analysis has been deliberately restricted to the case where the damages may take only two values (i.e. J ∈ {0, J H }), which enables us to consider t 2 as a fixed payment. Following some analytical developments in the literature on contract theory (see, e.g., [START_REF] Innes | Limited liability and incentive contracting with ex-ante action choices[END_REF][START_REF] Poblete | The form of incentive contracts: Agency with moral hazard, risk neutrality, and limited liability[END_REF], it would be interesting to extend our framework by considering that both effort and output (i.e. damages) are continuous. Such an analysis would be significantly more intricate, but would provide a more complete and robust approach of the moral hazard problem in sequential litigation.

It would certainly allow to develop new insights on the optimal payment scheme arising in this context.

Furthermore, we restrict our attention to the case of separation between liability and damages, while the sequentiality of the litigation process may refer to any situation where the issues of disagreement are treated successively by the court. In such a context, the parties litigate 'issue 1' after which the court makes a decision about 'issue 1'. Then, the parties litigate 'issue 2' after which the court makes a decision about 'issue 2', and so forth. As mentioned by De Mot (2012), courts have opportunities to separate issues at trial in many countries, and it would be then interesting to extend our framework in order

13 See, e.g., [START_REF] Wang | Renegotiation-proof dynamic contracts with private information[END_REF] and [START_REF] Zhao | Renegotiation-proof contract in repeated agency[END_REF] who analyze renegotiation problems in more general frameworks.

to embed such -more general -situations. For example, in the U.K., according to Rule 3.1(2)e of the Civil Procedure Rules, the court may order that part of any proceedings (such as counterclaim) be dealt with separate proceedings. Similar possibilities may be found in the U.S. Federal Rules of Civil Procedure.

Appendix. Proof of Proposition 1

The maximization problem of the plaintiff may be written as: max {h 1 ,h 2 ,t 1 ,t 2 } Π = h 1 (-t 1 + h 2 (J H -t 2 )) s.t.

t 1 ≥ 0 (LL 1 ) t 2 ≥ 0 (LL 2 ) U 1 = h 1 (t 1 + U 2 ) -c 1 (h 1 ) ≥ 0 (IR 1 ) U 2 = h 2 t 2 -c 2 (h 2 ) ≥ 0 (IR 2 ) t 1 + h 2 t 2 -c 2 (h 2 ) -c 1 (h 1 ) = 0 (IC 1 ) t 2 -c 2 (h 2 ) = 0 (IC 2 )
Furthermore, recall that the effort cost functions satisfy c i ≥ 0, c i > 0, c i ≥ 0, c i (0) = 0, c i (0) = 0 and c i (1) ≥ J H (with i = 1, 2). Moreover, we assume that c i is high enough to ensure the concavity of the objective function.

Let us show that (IR 1 ), (IR 2 ) and (LL 2 ) are satisfied:

• From (IC 1 ) we get t 1 = -h 2 t 2 + c 2 (h 2 ) + c 1 (h 1 ). Replacing t 1 and U 2 by their values in U 1 gives U 1 (h 1 ) = h 1 c 1 (h 1 ) -c 1 (h 1 ). Since U 1 (0) = 0 and U 1 (h 1 ) = h 1 c 1 (h 1 ) ≥ 0, we have U 1 (h 1 ) ≥ 0, which implies that (IR 1 ) is satisfied.

• Replacing t 2 (obtained from (IC 2 )) in U 2 gives U 2 (h 2 ) = h 2 c 2 (h 2 ) -c 2 (h 2 ). Since U 2 (0) = 0 and U 2 (h 2 ) = h 2 c 2 (h 2 ) ≥ 0, we get U 2 (h 2 ) ≥ 0 and (IR 2 ) is satisfied.

• (LL 2 ) is induced by (IC 2 ) since c 2 (0) = 0 and c 2 > 0.

After substituting for the values of t 1 and t 2 (obtained from (IC 1 ) and (IC 2 )) into the plaintiff's objective function, we get the following reduced program: max

h 1 ,h 2 Π = h 1 (-c 1 (h 1 ) -c 2 (h 2 ) + h 2 J H ) s.t. t 1 = c 1 (h 1 ) + c 2 (h 2 ) -h 2 c 2 (h 2 ) ≥ 0 (LL 1 )
The Lagrange function is given by: L = h 1 (-c 1 (h 1 ) -c 2 (h 2 ) + h 2 J H ) + λ(c 1 (h 1 ) + c 2 (h 2 ) -h 2 c 2 (h 2 )),

Figure 1 :

 1 Figure 1: The sequence of events.

See White (2002) for some quantitative data.

Some notable exceptions are[START_REF] Rubinfeld | Contingent fees for attorneys: An economic analysis[END_REF], and[START_REF] Polinsky | Aligning the interests of lawyers and clients[END_REF].

See also De Mot (2012) who considers the effects of sequential trials in conjunction with the English fee-shifting rule (inducing that the losing party bears all litigation costs).

To simplify the exposition, we neglect discounting and assume that the plaintiff incurs no fixed cost in filing the suit. In real-world cases, such an amount includes the cost of preparing a complaint and making the fact of the lawsuit known to the defendant.

The assumption that J L = 0 is made to alleviate notations and could be relaxed without altering the gist of our arguments.

See Kaplow (1994) and[START_REF] Shavell | The fundamental divergence between the private and the social motive to use the legal system[END_REF] for discussions on the divergence between the private and the social value of litigation.

where λ is the Lagrange multiplier.

The first-order conditions are then:

-h 1 c 1 (h 1 ) -c 1 (h 1 ) -c 2 (h 2 ) + h 2 J H + λc 1 (h 1 ) = 0, (A.1)

First, consider that λ = 0. In this case, we get

The constraint (LL 1 ) may be rewritten

Hence, λ = 0 cannot be a solution.

Now, assuming that λ > 0, the constraint (LL 1 ) is binded (i.e. t SB 1 = 0). Considering

2) is satisfied if and only if h 2 = 0 (since c 2 > 0). However, in this case, the condition (A.1) cannot be satisfied since λc 1 > 0. Hence, h 1 = 0 cannot be a solution. Considering h 2 = 0, the condition (A.2) becomes h 1 J H = 0 and it cannot be satisfied since h 1 > 0.

We conclude that h

3), we know that c 1 (h 1 ) = U 2 (h 2 ) in both first-and second-best situations,

where

Finally, the optimal value of t 2 is induced by the constraint (IC 2 ), such that t SB 2 = c 2 (h SB 2 ).