بهینگی هزینه های محتمل در مشکل سازمانی دادخواهی

Linear contracts are of particular interest to economists. They have a simple structure, yet they are very popular in practice. In this regard, plaintiff–lawyer contractual relationships are of particular interest. Lawyers’ fees are mostly paid by a sharing rule and they are typically a fixed proportion of the recovery across all lawsuits of the same type and this fixed proportion typically stays constant for many years. Such a simple and stable form of contract is puzzling to contract theorists. This paper presents a simple agency model with a risk-averse principal and a risk-neutral agent. We show that the observed puzzling features of contracts in litigation are in fact optimal behaviors, if a lawyer's effort has a constant marginal cost.

Linear contracts are of particular interest to economists. They have a very simple structure and they are very popular in practice. In theory, however, economists are only able to show the optimality of a linear contract if both the principal and agent are risk neutral (double risk neutrality).1 This may not be consistent with most real-world cases involving linear contracts, for which double risk neutrality does not seem to be an appropriate assumption. One such case is litigation in courts, where we expect the plaintiff to be risk averse, and yet linear contracts are a very popular form of contracts between lawyers and plaintiffs. More specifically, a fixed sharing rule is the most popular method of calculating lawyers’ fees in the United States, by which the lawyer receives a pre-determined share of the recovery if the case is won, and nothing if the case is lost. As pointed out by Kakalik and Pace (1986), Fisher (1988) and Dana and Spier (1993), 96% of individual plaintiffs involved in tort litigation in the United States pay their lawyers on a sharing-rule basis. Furthermore, lawyers’ fees are typically fixed share of the recoveries across all lawsuits of the same type, although there is a large variation across different types of lawsuits. As indicated by Hay (1996, Table 6), the lawyer's fee is typically 29% for auto accidents, 32% for medical malpractice, 34% for asbestos injury, and 17% for aviation accidents. An immediate question is why such a simple sharing rule is chosen by the overwhelming majority of plaintiffs and lawyers. Is the sharing rule optimal? Further, why must it be a fixed proportion of the recovery across many different cases, typically one-third? An answer to these questions may shed some light on our understanding of the popularity of linear contracts in many other cases and on contract theory in general. In theory, linear contracts are known to be optimal if and only if both the principal and the agent are risk neutral. But Kim and Wang (1998) show that even if there is a tiny bit of risk aversion, linear contracts are far from optimal. Thus, the existing contract theory cannot explain the popularity of linear contracts in practice. Because the sharing rule is so pervasive in practice, many published papers in the existing literature simply impose linearity on contracts, even though the linear contract is not optimal within the framework of their models. In particular, although there is a large literature on litigation, the particular form and the key features of the sharing rule adopted in litigation has never been shown to be optimal in theory. This paper presents a simple agency model with a risk-averse principal (the plaintiff) and a risk-neutral agent (the lawyer). Under the assumption of a constant marginal cost of a lawyer's effort, we present four results: (1) the optimal contract is a sharing rule; (2) the share of the sharing rule is fixed, in the sense that it is independent of the recovery; (3) the share is also independent of the plaintiff's preferences, that is, the lawyer may charge the same share for all clients involved in the same type of cases; and (4) such a sharing rule is efficient (the first best). Therefore, our simple agency model explains the key features of the sharing rule in litigation. This paper proceeds as follows. Section 2 presents our agency model of litigation. Section 3 derives the results. Section 4 discusses various issues. Finally, Section 5 concludes with a few concluding remarks. All the proofs are in the Appendix.

We first present the solution for problem (3) and later present two extensions. All the proofs and derivations are in the Appendix. 3.1. The optimality of contingent fees There is a unique solution for problem (3), which is stated in the following proposition. Proposition 1. UnderAssumption 1, if the recovery R is a publicly known and fixed constant, there is a unique optimal contract and it is in the form of contingent fees. Specifically, 1. The optimal contract is View the MathML sources∗(R˜)=cR˜, where View the MathML sourceR˜≡(R,p;   0,1−p). 2. The optimal wining chance is p* = 1. 3. This solution is the first best. The linear contract in Proposition 1 states that the plaintiff pays the lawyer nothing if the case is lost and the plaintiff pays the lawyer a fixed proportion of the recovery if the case is won; this fixed proportion equals the marginal cost of effort. This solution matches very well with contingent fees in practice. It is a sharing contract with a fixed proportion. This proportion is fixed regardless of the plaintiff's preferences and the potential size of the recovery. Only in the case of different types of lawsuits, marginal costs may differ and thus the shares of recovery for the lawyer may differ. One phenomenon in reality on litigation that is particularly puzzling to economists is that, although the sharing rule is fixed across different states and provinces within one country, it generally varies across different countries under similar law systems. This leads to one simple explanation to the fixed sharing rule in the literature: the fixed sharing rule is determined by traditional and culture. However, this phenomenon can be well explained by our solution. Our solution suggests that the fixed sharing rule is determined by the marginal cost of a lawyer's effort. The marginal cost of lawyer is typically fixed within one country due to labor mobility, while it varies across countries due to labor immobility. Hence, the seemingly puzzling phenomenon is actually a direct implication of a simple agency problem. It is well known that if there is no risk aversion, an optimal linear contract exists in the standard agency model. Proposition 1 is the first time to show the existence of an optimal linear contract with the presence of risk aversion. Our linear contract is the unique optimal contract, while in the standard agency model under double risk neutrality, the linear contract is one of infinitely many optimal contracts. This may be due to two crucial differences between our model and the standard agency model. First, in our model, the agent is risk neutral and the principal is risk averse, while, in the standard model, the agent is risk averse and the principal is risk neutral. Second, although the constancy of the marginal cost does not play a role in the standard agency model, it may be crucial for the linearity of our optimal contract. Having a zero constant term in our linear contract is interesting, since it matches with reality well. Optimal linear contracts are found in certain agency models, but this is perhaps the first time that a pure sharing rule (with a zero constant term) is found to be optimal in a principal-agent model. No constant term means that there is no initial payment for the service and hence the lawyer's income is completely contingent on outcome. Having a fixed share that is independent of preferences and output is also interesting. Linear contracts are not only popular in reality; they are well known to tend to have a stable structure within the same class of contractual relationships, that is, the share term of linear contracts in practice tends to be the same within the same class of contractual relationships. See, for example, the discussion of the uniformity of contract terms in Bhattacharyya–Lafontaine (1995). This phenomenon puzzles economists to this day. Our solution represents the first time that this possibility is shown in theory. This result may shed some light on our understanding of similar observations in many other business relationships. 3.2. A Random recovery By the time when a lawyer is hired and the case is brought to the court, the plaintiff and the lawyer generally do not know the precise size of the recovery, although they do have some rough idea of the size. In technical terms, it means that, by the time when a contract is initiated, the recovery is a random variable and the two parties know the density function of the recovery. It turns out that, with a random recovery, the optimal contract is still in the form of contingent fees and this arrangement is efficient. Proposition 2. UnderAssumption 1, if R is random with a publicly known density function f(R), the unique optimal contract is still in the form of contingent fees. Specifically, 1. The optimal contract is View the MathML sources∗(R˜)=cR˜ for all R ≥ 0, where View the MathML sourceR˜≡(R,p; 0,1−p). 2. The optimal winning chance is p*(R) = 1 for all R ≥ 0. 3. This solution is the first best. Note that allowing the winning probability to be a function of recovery in Proposition 2 is a general case. We can alternatively restrict the winning probability to be a fixed number (independent of R). Our result will not be affected by this restriction, since in both circumstances the optimal winning probability is p* = 1. 3.3. An effort-dependent random recovery The recovery may include punitive damages, which may be affected by the lawyer's effort in convincing the court of the defendant's malicious nature. This means that R may be dependent on p. More generally, when R is random, its density function may be dependent on p and the density function can be written as f(R; p). Assume that more effort from the lawyer can generally improve the chance of getting a high recovery. This is the well-known assumption of first-order stochastic dominance (FOSD), which is stated in Assumption 2. Assumption 2 FOSD. The conditional density function f(R; p) of R satisfies the FOSD property. Proposition 3. Under Assumption 1 and Assumption 2, if R is random with a publicly known conditional density function f(R; p), the unique optimal contract is still in the form of contingent fees. Specifically, 1. The optimal contract is View the MathML sources∗(R˜)=cR˜ for all R ≥ 0, where View the MathML sourceR˜≡(R,p;   0,1−p). 2. The optimal winning chance is p* = 1. 3. This solution is the first best.