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Chapter 18: Problem 1

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

Expert verified
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

01

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
02

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
03

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
04

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.

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Most popular questions from this chapter

A painting is auctioned to \(n\) bidders, each with a private value for the painting that is uniformly distributed between 0 and 1 a. Compute the equilibrium bidding strategy in a first-price sealed-bid auction. Compute the seller's expected revenue in this auction. Hint: Use the formula for the expected value of the \(k\) th-order statistic for uniform distributions in Equation 18.71 b. Compute the equilibrium bidding strategy in a second-price sealed-bid auction. Compute the seller's expected revenue in this auction using the hint from part (a). c. Do the two auction formats exhibit revenue equivalence? d. For each auction format, how do bidders' strategies and the seller's revenue change with an increase in the number of bidders?

Suppose that left-handed people are more prone to injury than right-handed people. Iefties have an 80 percent chance of suffering an injury leading to a \(\$ 1,000\) loss (in terms of medical expenses and the monetary equivalent of pain and suffering) but righties have only a 20 percent chance of suffering such an injury. The population contains equal numbers of lefties and rightics. Individuals all have logarithmic utility-of-wealth functions and initial wealth of \(\$ 10,000\). Insurance is provided by a monopoly company. a. Compute the first best for the monopoly insurer (i.e., supposing it can observe the individual's dominant hand). b. Take as given that, in the second best, the monopolist prefers not to serve rightics at all and targets only leftics. Knowing this, compute the second- best menu of policies for the monopoly insurer. c. Use a spreadsheet program (such as the one on the website associated with Example 18.5 ) or other mathematical software to solve numerically the constrained optimization problem for the second best. Make sure to add constraints bounding the insurance payments for righties: \(0 \leq x_{R} \leq 1,000\). Establish that the constraint \(0 \leq x_{R}\) is binding and so righties are not served in the second best.

Suppose there is a \(50-50\) chance that an individual with logarithmic utility from wealth and with a current wealth of \(\$ 20,000\) will suffer a loss of \(\$ 10,000\) from a car accident. Insurance is competitively provided at actuarially fair rates. a. Compute the outcome if the individual buys full insurance. b. Compute the outcome if the individual buys only partial insurance covering half the loss. Show that the outcome in part (a) is preferred. c. Now suppose that individuals who buy the partial rather than the full insurance policy take more carc when driving, reducing the damage from loss from \(\$ 10,000\) to \(\$ 7,000\). What would be the actuarially fair price of the partial policy? Does the individual now prefer the full or the partial policy?

Consider the principal-agent relationship between a patient and doctor. Suppose that the patient's utility function is given by \(U_{P}(m, x),\) where \(m\) denotes medical care (whose quantity is determined by the doctor) and \(x\) denotes other consumption goods. The patient faces budget constraint \(I_{c}=p_{m} m+x,\) where \(p_{m}\) is the relative price of medical care. The doctor's utility function is given by \(U_{d}\left(I_{d}\right)+U_{P}-\) that is, the doctor derives utility from income but, being altruistic, also derives utility from the patient's well-being. Moreover, the additive specification implies that the doctor is a perfect altruist in the sense that his or her utility increases one-for-one with the patient's. The doctor's income comes from the patient's medical expenditures: \(I_{d}=p_{m} m .\) Show that, in this situation, the doctor will generally choose a level of \(m\) that is higher than a fully informed patient would choose.

Suppose the agent can be one of three types rather than just two as in the chapter. a. Return to the monopolist's problem of computing the optimal nonlinear price. Represent the first best in a schematic diagram by modifying Figure \(18.4 .\) Do the same for the second best by modifying Figure 18.6 b. Return to the monopolist's problem of designing optimal insurance policies. Represent the first best in a schematic diagram by modifying Figure \(18.7 .\) Do the same for the second best by modifying Figure 18.8
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