# On Lawyer Compensation When Appeals Are Possible

Christian At, Tim Friehe and Yannick Gabuthy

# Abstract

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.

JEL Classification: D82; K41

# Appendix

## A Proof of Lemma 1

Using the t2(y1,1) implied by (9) in (4) yields:

L2(e2(y1))=e2(y1)ce2y1(e2(y1))c2y1(e2(y1))

Since L2(0)=0 and L2(e2(y1))=e2(y1)ce2′′(e2(y1),y1)0, we get L2(e2(y1))0 such that (4) is satisfied.

Restating (8) yields a(1)=a(0)+c1(e1), which allows to state (3) as follows:

L1(e1)=a(0)+e1c1(e1)c1(e1)0.

Since (4) is satisfied, we have a(0)0. By noting f(e1)=e1c1(e1)c1(e1), we have f(0) = 0 and f(e1)=e1c1′′(e1)0 since c1′′(e1)0 by assumption. As a result, (3) is satisfied.

(2) is induced by (9) since ce2(e2(y1),y1)0.

## B Proof of Proposition 1

Denote by x(y1)=qy1+1y1 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 t2(y1,1)=c2y1(e2(y1)), we consider:

P(e2(y1),y1)=e2(y1)(Jc2y1(e2(y1)))and A(e2(y1),y1)=e2(y1)c2y1(e2(y1))c2y1(e2(y1))

as the expected payoff of the plaintiff and the lawyer, respectively, from an appeal. The expected social surplus is thus given by:

S(e2(y1),y1)=e2(y1)Jc2y1(e2(y1)).

Hereafter, we omit y1 when the computations are true whatever the value of y1.

Using a restatement of (5), a=t1+xA(e2), the plaintiff’s program to determine the optimal continuation payoff π(a) may be written:

maxt1,e2xP(e2)t1s.t.t1=axA(e2)0

Replacing t1=axA(e2) in the plaintiff’s objective function, this becomes:

maxe2xS(e2)as.t.axA(e2)0

With λ0 as the Lagrange multiplier, the Lagrangian is:

L=xS(e2)a+λ(axA(e2))

The first-order conditions are then given by:

(10)S(e2)λA(e2)=0
(11)λ(axA(e2))=0

and allow us to reason about the plaintiff’s desired level of appeals court effort as a function of the outcome y0.

Step1. Proof of e20,e2FB

We first show that e2=0 cannot be a solution. We know from the proof of Lemma 1 that a(0)0. From (8) (i.e. a(1)a(0)=c1(e1)), and c1(e1)0, we deduce that a(1)a(0). Hence, since a0 and A(0) = 0, we get λ = 0 and Sʹ(0) = 0, which is impossible since Sʹ(0) = J.

Next, we show that any e2>e2FB cannot be a solution. For e2>e2FB, we have S(e2)<0, which violates (10) since λA(e2)0.

Step2. Proof of e2(0)0,e2FB(0) and e2(1)0,e2FB(1)

We know that t1(0)=0 is optimal. The plaintiff’s program with outcome y1=0 is thus:

maxe2(0)P(e2(0))

as x is a constant.

Since P(e2FB)=0 and P(e2FB)=e2FBc′′(e2FB)<0, we know that e2(0)<e2FB(0).

For the case with outcome y1=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:

1. e2(1)=e2FB(1) and t1(1)=a(1)qAe2FB(1)>0 if q<a(1)Ae2FB(1)=qˉ ;

2. e2(1)=A1a(1)q<e2FB(1) and t1(1)=0 if qqˉ.

## Proof 4

Consider the first-order conditions (10) and (11).

If λ = 0, then S(e2)=0 and, thus, e2=e2FB, implying π(a,e2FB)=qS(e2FB)a0 since aqA(e2FB)=qS(e2FB). Furthermore, we get t1(1)=a(1)qAe2FB(1)>0.

If λ > 0 or t1(1)=0, then the second-best effort e2 is the solution of a=qA(e2). We have π(a,e2)=qP(e2). Since P(e2FB)=0 and P(e2FB)=e2FBc′′(e2FB)<0, we get P(e2)>P(e2FB)=0 and, thus, π(a,e2)>π(a,e2FB).

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<qˉ, since e2FB(1)=supe2(1)0,e2FB(1)A(e2(1)).   □

Step3. Analysis of the optimal continuation payoff π(a).

Consider, first, the case qqˉ. We know from Step 2 that e20,e2FB and a=xA(e2). We also know that a(0,xA(e2FB)) since A(e2) is continuous and increasing in e2 and A(0) = 0. Furthermore, the derivative of the function π(a)=xP(e2)=xP(A1(a/x)) is π(a)=P(e2)A(e2). When axA(e2FB) (i.e. when e2e2FB), we have:

π(a)=P(e2)A(e2)P(e2FB)A(e2FB)=1<0

While, when a0 (i.e. when e20), we get:

π(a)P(0)A(0)+

Some algebra gives:

π′′(a)=P′′(e2)A(e2)P(e2)A′′(e2)A(e2)2=S′′(e2)A(e2)S(e2)A′′(e2)A(e2)2<0

implying that π(a) is concave. Therefore, the continuation payoff reaches a maximum in the interval (0,xA(e2FB)).

Second, if q<qˉ, then the continuation payoff is π(a(1))=qS(e2FB(1))a(1).

Step4. 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:

maxe1,a(0),a(1)e1[(1q)J+π(a(1))]+(1e1)π(a(0))

Replacing a(1) by a(0)+c1(e1) using (8), this problem becomes:

maxe1,a(0)e1[(1q)J+π(a(0)+c1(e1))]+(1e1)π(a(0))

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

(12)(1q)J+π(a(1))π(a(0))+π(a(1))e1c1′′(e1)=0
(13)e1π(a(1))+(1e1)π(a(0))=0

Consider that e1=0. From (8), we deduce that a(1) = a(0) since cʹ(0) = 0 by assumption. From (13), we get πʹ(a(0)) = 0 = πʹ(a(1)) and π(a(1)) = π(a(0)), which violates (12). Therefore, the lawyer exerts a positive effort at the trial stage (i.e. e1>0).

Note that qa(1)Ae2FB(1)>0 is satisfied since e1>0 (implying that a(1) > 0).

Step5. Assessing payments after the appeals court’s judgment.

Since e1>0, from (8) we deduce a(1) > a(0) or qA(e2(1),1)>A(e2(0),0)qA(e2(0),0). Remember A(e2,y1)=e2c2y1(e2)c2y1(e2). Our assumptions ensure that Ae(e,y1)=e2c2y1′′(e2)0 and Ae′′(e,y1)=c2y1′′(e2)+ec2y1′′′(e2)0. We have two cases to consider.

Case (a): Suppose that c21(e2)c20(e2)<c21(e2)c20(e2)e2<0e2(0,e2FB), implying A(e2(0),1)<A(e2(0),0). We must have A(e2(1),1)>A(e2(0),0) at the optimum, which can be satisfied only for e2(1)>e2(0). From (9), we know that t2(0,1)=c20(e2(0)) and t2(1,1)=c21(e2(1)). Furthermore, there exists eˆ<e2(1) such that c21(e2(1))=c20(eˆ) since c21(e2)<c20(e2), c2y1′′(e2)0 and e2(1)>e2(0). We deduce that t2(1,1)=c21(e2(1))<t2(0,1)=c20(e2(0)) if e2(0)>eˆ, and t2(1,1)>t2(0,1) otherwise.

### Figure 2:

e2(1)

must be greater than e2(1)_>e2(0) to ensure A(e2(1),1)>A(e2(0),0).

### Figure 3:

Case (b): If c21(e2)c20(e2)e2<c21(e2)c20(e2)<0, then A(e2,1)>A(e2,0)e(0,e2FB) and the signs of both e2(1)e2(0) and t2(1,1)t2(0,1) are ambiguous.

### Figure 4:

e2(0)

must be smaller than e2(0)>e2(1) to ensure A(e2(1),1)>A(e2(0),0); therefore, we can have e2(0)>e2(1) or e2(0)<e2(1) as a solution.

## C Proof of Proposition 2

Step1. Proof of de2(1)dq0 and dt2(1,1)dq0

Consider, first, the case where q>q. We know that t1(1)=0 and, thus, a(1)=qA(e2(1),1). For an optimal choice of a(1), we have de2(1)dq=a(1)q2A(e2(1),1)<0, since e2(1)=A1a(1)q. From (3), we deduce that dt2(1,1)dq<0.

Consider now the case where qq. We have e2(1)=e2FB(1), implying that de2(1)dq=dt2(1,1)dq=0.

Step2. Proof of de1dq<0, de2(0)dq>0 and dt2(0,1)dq>0

Consider the first-order conditions (12) and (13), and note for simplicity:

E(a(q),e1(q),q)=(1q)J+π(a(q)+c1(e1(q))π(a(q))+π(a(q)+c1(e1(q))e1(q)c1′′(e1(q))=0and A(a(q),e1(q),q)=e1(q)π(a(q)+c1(e1(q))+(1e(q))π(a(q))=0

By totally differentiating the system, we get:

Given that Aq=0, we get EeAa(EaAe)2>0, since the program is concave. We deduce that:

where Eq=J<0, Aa=e(q)π′′(a(q)+c1(e(q))+(1e(q))π′′(a(q))<0 and Ae=π(a(q)+c1(e(q))π(a(q))+π′′(a(q)+c1(e(q))e(q)c1′′(e(q))<0 (since, from (13), π(a(q)+c1(e(q)) and πʹ(a(q)) have opposite signs, and π(.) is concave). Therefore, we get de1dq<0.

We also have da(0)dq>0. We know that e2(0)=A1(a(0)) and, thus, de2(0)dq=1A(e2(0),0)da(0)dq>0, implying dt2(0,1)dq>0.

Step3. Determination of the effect of q on t1(1)

Only the case where q<q matters as t1(1)=0 otherwise. We know that t1(1)=a(1)qAe2FB(1) and obtain:

dt1(1)dq=da(0)dq+c1′′(e1)de1dqAe2FB(1)

which exhibits an ambiguous sign.

## D Endogenous Probability

Suppose that the plaintiff lost the case in trial court (i.e. we have y1=0) and now considers setting the probability of appeal, p ∈ [0,1]. The plaintiff would at this stage want to

maxe2(0),ppS(e2(0))a(0)s.t.a(0)pA(e2(0))0

where a(0) and A(e2(0)) were defined above. With λq0 as the Lagrange multiplier, the Lagrangian is (omitting y1=0 from the parentheses):

L=pS(e2)a+λ(apA(e2))

The first-order conditions are then given by:

(14)S(e2)λA(e2)=0
(15)S(e2)λA(e2)=0
(16)λ(apA(e2))=0

It is straightforward from (15) that λ = 0 cannot be a solution, hence using (16), we can replace a=pA(e2) in the program that simplifies to:

maxe2,ppS(e2)pA(e2)=pP(e2)

We deduce that the optimal choice of the probability of an appeal is p = 1. This is consistent with the result of Ohlendorf and Schmitz (2012) when the continuation costs are equal to 0.

# Acknowledgements

### References

At, C., and Y. Gabuthy. 2015. “Moral Hazard and Agency Relationship in Sequential Litigation.” International Review of Law and Economics 41: 86–90.10.1016/j.irle.2014.11.002Search in Google Scholar

Baumann, F., and T. Friehe. 2016. “Contingent Fees with Legal Discovery.” American Law and Economics Review 18: 155–75.10.1093/aler/ahv020Search in Google Scholar

Chambaz, C. 2017. Les chiffres-clés de la Justice 2017. Tech. rep., Ministère de la Justice.Search in Google Scholar

Cohen, T. H. 2006. Appeals from General Civil Trials in 46 Large Counties, 2001–2005. Tech. rep., Bureau of Justice Statistics.Search in Google Scholar

Dana, J. D., and K. E. Spier. 1993. “Expertise and Contingent Fees: The Role of Asymmetric Information in Attorney Compensation.” Journal of Law, Economics, & Organization 9 (2): 349–67.Search in Google Scholar

Daughety, A. F., and J. F. Reinganum. 2000. “Appealing Judgments.” RAND Journal of Economics 31 (3): 502–25.10.2307/2600998Search in Google Scholar

Eisenberg, T., and M. Heise. 2015. “Plaintiphobia in State Courts Redux? An Empirical Study of State Court Trials on Appeal.” Journal of Empirical Legal Studies 12 (1): 100–27.10.1111/jels.12066Search in Google Scholar

Emons, W. 2006. “Playing it Safe with Low Conditional Fees versus Being Insured by High Contingent Fees.” American Law and Economics Review 8 (1): 20–32.10.1093/aler/ahj002Search in Google Scholar

Farmer, A., and P. Pecorino. 2013. “Discovery and Disclosure with Asymmetric Information and Endogenous Expenditure at Trial.” Journal of Legal Studies 42 (1): 223–47.10.1086/667932Search in Google Scholar

Friehe, T., and A. Wohlschlegel. 2017. “Rent Seeking and Bias in Appeals Systems,” Working Papers in Economics & Finance 2017-01, University of Portsmouth.Search in Google Scholar

Hay, B. L. 1997. “Optimal Contingent Fees in a World of Settlement.” Journal of Legal Studies 26 (1): 259–78.10.1086/467995Search in Google Scholar

Helland, E., and S. A. Seabury 2013. “Contingent-fee Contracts in Litigation: A Survey and Assessment.” In Research Handbook on the Economics of Tort, Edited by J. Arlen. Cheltenham, UK: Edward Elgar.Search in Google Scholar

Iossa, E., and G. Palumbo. 2007. “Information Provision and Monitoring of the Decision-Maker in the Presence of an Appeal Process.” Journal of Institutional and Theoretical Economics 163 (4): 657–82.Search in Google Scholar

Levy, G. 2005. “Careeist Judges and the Appeals Process.” RAND Journal of Economics 36 (2): 275–97.Search in Google Scholar

Ohlendorf, S., and P. W. Schmitz. 2012. “Repeated Moral Hazard and Contracts with Memory: The Case of Risk-Neutrality.” International Economic Review 53 (2): 433–52.10.1111/j.1468-2354.2012.00687.xSearch in Google Scholar

Oytana, Y. 2014. “The Judicial Expert in a Two-Tier Hierarchy.” Journal of Institutional and Theoretical Economics 170 (3): 537–70.Search in Google Scholar

Schmitz, P. W. 2005. “Allocating Control in Agency Problems with Limited Liability and Sequential Hidden Actions.” Rand Journal of Economics 36 (2): 318–36.Search in Google Scholar

Schmitz, P. W. 2013. “Job Design with Conflicting Tasks Reconsidered.” European Economic Review 57 (1): 108–17.10.1016/j.euroecorev.2012.11.001Search in Google Scholar

Shavell, S. 1995. “The Appeals Process as a Means of Error Correction.” Journal of Legal Studies 24 (2): 379–426.10.1086/467963Search in Google Scholar

Shavell, S. 2006. “The Appeals Process and Adjudicator Incentives.” Journal of Legal Studies 35 (1): 1–29.10.1086/500095Search in Google Scholar

Spitzer, M., and E. Talley. 2000. “Judicial Auditing.” Journal of Legal Studies 29 (2): 649–83.10.1086/468088Search in Google Scholar

Wang, C. 2000. “Renegotiation-Proof Dynamic Contracts with Private Information.” Review of Economic Dynamics 3 (3): 396–422.10.1006/redy.2000.0099Search in Google Scholar

White, M. J. 2002. “Explaining the Flood of Asbestos Litigation: Consolidation, Bifurcation, and Bouquet Trials,” NBER Working Paper Series, Working Paper No. 9362.Search in Google Scholar

Wohlschlegel, A. 2014. “The Appeals Process and Incentives to Settle,” MPRA Paper No. 59424.Search in Google Scholar

Zhao, R. R. 2006. “Renegotiation-Proof Contract in Repeated Agency.” Journal of Economic Theory 131 (1): 263–81.10.1016/j.jet.2005.05.003Search in Google Scholar

Published Online: 2019-03-23

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