Question

A personal-injury lawyer works as an agent for his injured plaintiff. The expected award from the trial (taking into account the plaintiff's probability of prevailing and the damage award if she prevails) is $l$, where $l$ is the lawyer's effort. Effort costs the lawyer $l^2/2$ What is the lawyer's effort, his surplus, and the plaintiff's surplus in equilibrium when the lawyer obtains the customary $1/3$ contingency fee (i.e., the lawyer gets $1/3$ of the award if the plaintiff prevails)? b. Repeat part (a) for a general contingency fee of $c$ c. What is the optimal contingency fee from the plaintiff's perspective? Compute the associated surpluses for the lawyer and plaintiff. What would be the optimal contingency fee from the plaintiff's perspective if she could "scll" the case to her lawyer [i.c., if she could ask him for an up- front payment in return for a specified contingency fee, possibly higher than in part (c)]? Compute the up-front payment (assuming that the plaintiff makes the offer to the lawyer) and the associated surpluses for the lawyer and plaintiff. Do they do better in this part than in part (c)? Why do you think selling cases in this way is outlawed in many countries?

Short answer

Also, what is the optimal contingency fee from the plaintiff's perspective, and what are the outcomes if the plaintiff could sell the case to the lawyer? Answer: When the contingency fee is 1/3, the equilibrium values are as follows: the lawyer's effort is 1/3, the lawyer's surplus is 1/18, and the plaintiff's surplus is 2/9. The optimal contingency fee from the plaintiff's perspective is 1/2. If the plaintiff could sell the case to the lawyer, the upfront payment would be 1/8, and the optimal surpluses for the lawyer and plaintiff would be 1/8 and 3/8, respectively.

Step-by-step solution

Part a: Find the lawyer's effort, surplus and the plaintiff's surplus when the contingency fee is 1/3

First, let's find the lawyer's net payoff and the plaintiff's net payoff when the contingency fee is 1/3. The lawyer's net payoff will be expressed as a function of his effort (l), while the plaintiff's net payoff will be determined by the expected award from the trial minus the lawyer's fee. Lawyer's net payoff = (1/3) * l - (l^2 / 2) To find the surplus for the lawyer (i.e., the value that maximizes the net payoff), we need to find the first derivative of the net payoff with respect to l and set it equal to 0. d(Lawyer's net payoff) / dl = (1/3) - l = 0 l = 1/3 Now that we have the value of l, we can find the lawyer's and plaintiff's surpluses. Lawyer's surplus = (1/3) * (1/3) - ((1/3)^2 / 2) Lawyer's surplus = 1/9 - 1/18 Lawyer's surplus = 1/18 Plaintiff's surplus = (2/3) * (1/3) Plaintiff's surplus = 2/9

Part b: Repeat part (a) for a general contingency fee of c

Now let's redo the calculation for a general contingency fee (c) instead of specifically 1/3. Lawyer's net payoff = c * l - (l^2 / 2) We will again find the first derivative of the net payoff with respect to l and set it equal to 0. d(Lawyer's net payoff) / dl = c - l = 0 l = c Now, we can find the lawyer's and plaintiff's surpluses with the contingency fee, c, and effort, l. Lawyer's surplus = c * l - (l^2 / 2) = c^2 - (c^2 / 2) Lawyer's surplus = c^2 / 2 Plaintiff's surplus = (1 - c) * l Plaintiff's surplus = (1 - c) * c

Part c: Find the optimal contingency fee from the plaintiff's perspective

To find the optimal contingency fee from the plaintiff's perspective, we need to maximize the plaintiff's surplus found in part (b). Plaintiff's surplus = (1 - c) * c To maximize the surplus, we take the first derivative with respect to c and set it equal to 0. d(Plaintiff's surplus) / dc = (1 - c) - c = 0 2c = 1 c = 1/2 Now, we find the surpluses for the lawyer and plaintiff at the optimal contingency fee, c = 1/2. Lawyer's surplus = (1/2)^2 / 2 = 1/8 Plaintiff's surplus = (1 - 1/2) * (1/2) = 1/4

Part d: Calculate the optimal fee with the opportunity for the plaintiff to sell the case

If the plaintiff can sell the case to the lawyer with an upfront payment (u), the plaintiff's net payoff function becomes: Plaintiff's net payoff = u + (1 - c) * l The lawyer's net payoff function becomes: Lawyer's net payoff = c * l - (l^2 / 2) - u Since the optimal effort level is still equal to the contingency fee (l = c), we can rewrite the lawyer's net payoff function as: Lawyer's net payoff = (c^2 / 2) - u Then, we can find the value of u that maximizes each net payoff function. Since the lawyer's net payoff is an increasing function of u and the plaintiff's net payoff is a decreasing function of u, we will set them equal to each other. Plaintiff's net payoff = Lawyer's net payoff u + (1 - c) * l = (c^2 / 2) - u Substitute l = c: u + (1 - c) * c = (c^2 / 2) - u The optimal upfront payment u = (c^2 / 4) Therefore, the optimal contingency fee remains to be c = 1/2 with an upfront payment of u = 1/8. The associated surpluses for the lawyer and the plaintiff are: Lawyer's surplus = 1/8 Plaintiff's surplus = 1/4 + 1/8 = 3/8 In this case, the lawyer and the plaintiff do better than in part (c). The reason why selling cases in this manner is outlawed in many countries is that it can create a conflict of interest between the lawyer and the plaintiff. By offering upfront payments for a higher contingency fee, the plaintiff is incentivizing the lawyer to take on cases that might not be in the best interest of the plaintiff or the justice system.

Key concepts

Contingency Fee

Contingency fees play a crucial role in the dynamic between a lawyer and their plaintiff in personal-injury cases. The standard model involves a lawyer accepting a portion of the plaintiff's award contingent upon winning the case. Typically, this fee is set at one-third or 33%. This arrangement allows clients, often with limited resources, to pursue a claim without the burden of upfront legal costs.

From an economic standpoint, the level of the contingency fee directly influences the lawyer's effort in handling the case. A higher fee offers greater motivation for the lawyer to exert effort, increasing the chances of winning the case and thus their own payoff. In contrast, if the fee is too high, the plaintiff's net gain diminishes, which might lead to less favorable terms for the client.

Therefore, the contingency fee must balance the interests of both parties, providing sufficient incentive for the lawyer while ensuring the plaintiff retains a satisfactory share of the winnings.

Plaintiff's Surplus

The plaintiff's surplus refers to the portion of the award they retain after paying the lawyer's contingency fee. It signifies the net benefit the plaintiff gains from the case outcome. The calculation of this surplus involves subtracting the lawyer's share (determined by the contingency fee) from the total award won at trial.

In practical terms, if a contingency fee is set at 33%, the plaintiff receives 67% of the award. However, the optimal contingency fee from the plaintiff's perspective maximizes this surplus. Through analysis, setting a fee at 50% results in the plaintiff and lawyer each receiving a more balanced share of the surplus.

This strategy also ensures that the plaintiff benefits adequately without being overcharged for legal services, maintaining a fair outcome from their perspective.

Lawyer's Surplus

The lawyer's surplus is the financial gain realized by the lawyer after subtracting the cost of their effort from their share of the case's award. It is crucial in determining whether the lawyer is incentivized to put forth the best effort.

Mathematically, the lawyer's surplus is calculated by considering the portion of the award based on the contingency fee and subtracting the effort cost, which is modeled as $l^2/2$. For example, with a general fee $c$, the surplus equals $(c^2/2)$.

An appropriately chosen fee can align the lawyer's interest with the plaintiff's, encouraging actions that maximize the collective benefit. A well-calculated fee ensures sustainable business practices for lawyers while offering clients quality representation.

Equilibrium Analysis

Equilibrium analysis helps in finding balance points where both the lawyer and plaintiff maximize their respective surpluses. It involves calculating effort and fee levels that optimize these surpluses under the conditions of a legal agreement.

In this context, equilibrium is reached when both parties have no incentive to deviate from the agreed terms because any change would result in a decreased surplus for one or both parties. For instance, a contingency fee of 50% results in equal surplus benefits for both parties, which can be considered an equilibrium.

This analysis is complex but essential for ensuring that both the lawyer's and the plaintiff's needs are adequately met, promoting fairness and efficiency in legal services under the current structure of contingency fee arrangements.