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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.
a. 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.
d. What would be the optimal contingency fee from the plaintiff 's perspective if she could ''sell'' the case to her lawyer [i.e., 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?
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a) If the contingency fee is 1/3 rd , the surplus to the lawyer is given by
The first order condition for the above function is
Hence, the surplus maximizing effort for the lawyer is
The lawyer's surplus then is
The plaintiff's surplus then is
b) If the contingency fee is c , the surplus to the lawyer is given by
The first order condition for the above function is
Hence, the surplus maximizing effort for the lawyer is
The lawyer's surplus then is
The plaintiff's surplus then is
c) The plaintiff's surplus is
The first order condition for the above function is
Hence, the surplus maximizing contingency fee for the plaintiff is
The plaintiff's surplus then is
If contingency fees is
, surplus maximizing effort for the lawyer is
The lawyer's surplus then is
d) If the plaintiff can sell the case to her lawyer she can extract the whole surplus of the lawyer and increase her surplus. This can be done by setting a 100% contingency fee. The lawyer will then choose l = 1 and earn a surplus ½. The plaintiff can then sell the case for ½ and extract his surplus.
As this scheme does not leave the lawyer with any surplus it is outlawed in many countries.
Solve for the optimal linear price per ounce of coffee that the coffee shop would charge in Example 18.4. How does theshop's profit compare to when it uses nonlinear prices? Hint: Your first step should be to compute each type's demand at alinear price p.
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If instead of two tariffs, the coffee shop charges price p per ounce of coffee, then the
the utility function of a consumer is given by
The first order condition for the above yields
So, the inverse demand function of consumer is given by
As
, we get
Putting this in the inverse demand function, we get
So,
The expected profit of the coffee shop is given by
Substituting the demand function for the two types of demanders, we get
The first order condition for this is
Re arranging the above we get,
Solving for p, we get,
The expected profit of the shop is given by
If instead the firm sets non-linear price, the expected profit is found to be 50. Hence, non-linear price is a better strategy.
Return to the nonlinear pricing problem facing the monopoly coffee shop in Example 18.4, but now suppose the proportion ofhigh demanders increases to 2/3 and the proportion of low demanders decreases to 1/3. What is the optimal menu in thesecond-best situation? How does the menu compare to the one in Example 18.4?
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a) We know that to maximize the monopolists' expected profit
As
, we get
Substituting the values, we get
The tariff for the low type is given by
The optimal quantity and tariff for high demander remains unchanged at 16 and 160 respectively.
Only the quantity and tariff of the low type is affected, that of the high type remains unaffected. This shows that there is "no distortion at the top".
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 care 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?
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Suppose that left-handed people are more prone to injury than right-handed people. Lefties 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 righties. 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 righties at all and targets only lefties. 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:
. Establish that the constraint
is binding and so righties are not served in the second best.
. Establish that the constraint
is binding and so righties are not served in the second best. Essay
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Consider the same setup as in Problem 18.5, but assume that insurance is offered by competitive insurers.
a. Ignore the issue of whether consumers' insurance decisions are rational for now and simply assume that the equal numbers of lefties and righties both purchase full insurance whatever the price. If insurance companies cannot distinguish between consumer types and thus offer a single full-insurance contract, what would the actuarially fair premium for this contract be?
b. Which types will buy insurance at the premium calculated in (a)?
c. Given your results from part (b), will the insurance premiums be correctly computed?
Reference: Problem 18.5
Suppose that left-handed people are more prone to injury than right-handed people. Lefties 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 righties. 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 righties at all and targets only lefties. 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:
. Establish that the constraint
is binding and so righties are not served in the second best.
. Establish that the constraint
is binding and so righties are not served in the second best. Essay
Answer:
Suppose 100 cars will be offered on the used-car market. Let 50 of them be good cars, each worth $10,000 to a buyer, and let 50 be lemons, each worth only $2,000.
a. Compute a buyer's maximum willingness to pay for a car if he or she cannot observe the car's quality.
b. Suppose that there are enough buyers relative to sellers that competition among them leads cars to be sold at their maximum willingness to pay. What would the market equilibrium be if sellers value good cars at $8,000? At $6,000?
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Consider the following simple model of a common values auction. Two buyers each obtain a private signal about the value of an object. The signal can be either high (H) or low (L) with equal probability. If both obtain signal H, the object is worth 1; otherwise, it is worth 0.
a. What is the expected value of the object to a buyer who sees signal L? To a buyer who sees signal H?
b. Suppose buyers bid their expected value computed in part (a). Show that they earn negative profit conditional on observing signal H-an example of the winner's curse.
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Doctor-patient relationship
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
is the relative price of medical care. The doctor's utility function is given by U d (I d ) + 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.
is the relative price of medical care. The doctor's utility function is given by U d (I d ) + 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. Essay
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Diagrams with three types
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.
Reference:
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Increasing competition in an auction
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 kth-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?
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Team effort
Increasing the size of a team that creates a joint product may dull incentives, as this problem will illustrate. Suppose
n partners together produce a revenue of
to exert.
a. Compute the equilibrium effort and surplus (revenue minus effort cost) if each partner receives an equal share of the revenue.
b. Compute the equilibrium effort and average surplus if only one partner gets a share. Is it better to concentrate the share or to disperse it?
c. Return to part (a) and take the derivative of surplus per partner with respect to n. Is surplus per partner increasing or decreasing in n ? What is the limit as n increases?
d. Some commentators say that ESOPs (employee stock ownership plans, whereby part of the firm's shares are distributed among all its workers) are beneficial because they provide incentives for employees to work hard. What does your answer to part (c) say about the incentive properties of ESOPs for modern corporations, which may have thousands of workers?
to exert.
a. Compute the equilibrium effort and surplus (revenue minus effort cost) if each partner receives an equal share of the revenue.
b. Compute the equilibrium effort and average surplus if only one partner gets a share. Is it better to concentrate the share or to disperse it?
c. Return to part (a) and take the derivative of surplus per partner with respect to n. Is surplus per partner increasing or decreasing in n ? What is the limit as n increases?
d. Some commentators say that ESOPs (employee stock ownership plans, whereby part of the firm's shares are distributed among all its workers) are beneficial because they provide incentives for employees to work hard. What does your answer to part (c) say about the incentive properties of ESOPs for modern corporations, which may have thousands of workers? Essay
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