Problem Set 4: Adverse selection, moral hazard, and principal-agent

problems

1. High productivity workers produce £2,000,000 worth of goods over their lifetime (in present value

terms), whereas low productivity workers only produce £500,000 worth in their lifetime. 20% of all

workers are high productivity, but an employer is unable to tell to which group a given worker belongs

(even after they have worked for them for a lifetime). It costs high productivity workers £100,000 in

disutility to successfully complete a year in higher education, while it costs low productivity workers

£300,000 to do the same thing (as it takes them longer to do the problem sets). Assume that people

can stay in higher education for as long as they wish, and that people who leave education later both

retire later and die later, so the number of years they spend in education has no impact on how many

years they work in total (or how many years of retirement they get).

a) Will there be a separating and/or pooling equilibrium in this market? How many years of higher

education will high and low types complete in the equilibria that you find?

b) A possible implication of this model is that education is a socially wasteful activity. Explain why. Do

you agree?

2. There are two types of people wishing to buy flowers: business people and young couples. Suppose

there are equal numbers of people in both groups. Business people value flowers at £5 + 𝑡£4 where

𝑡 = 1 if they are able to pick up the flowers in the train station on the way home from work and 𝑡 = 0

if they have to buy them online. Young couples value flowers at £4 + 𝑡£2. Suppose that there is an

infinite supply of flowers both at the train station and online, and that they can be produced at zero

cost to the flower seller (a monopolist). Given the flower seller cannot tell whether someone is a

business person when they buy their flowers, what price should they set for flowers at the train station

and flowers on the internet?

3. [Very tricky—too hard for an exam, don’t worry!] There are two possible states of the world, good

and bad. In the good state of the world, individuals get an income of 𝑦𝐺 , and in the bad state they get

an income of 𝑦𝐵 , where 𝑦𝐵 < 𝑦𝐺 . The probability of the bad state of the world is max{0,1 − 𝑧𝑒}

where 𝑒 is the amount of effort agents expend in trying to avoid it (and 𝑧 is some constant). Individuals

seek to insure themselves against the bad state of the world by purchasing insurance at the actuarially

fair premium (i.e. 𝜋 = max{0,1 − 𝑧𝑒}) from the insurer. Individuals are risk averse with vNM utility

function 𝑢(𝑐, 𝑒) = −(𝑐 − 𝑏)2 − 𝑒, where 𝑏 is some constant. The insurer is wise to the problems

caused by moral hazard however, and prevents anyone from taking out more than 𝑘(𝑦𝐺 − 𝑦𝐵) units

of insurance, where 𝑘 is some constant between 0 and 1. Show that in an internal solution, individuals

will purchase 𝑘(𝑦𝐺 − 𝑦𝐵) units of insurance. Thus solve for their effort levels in an internal solution.

Now suppose that the insurer seeks to choose 𝑘 to maximise the amount of insurance purchased

(perhaps the insurer can sell a secondary good to people purchasing insurance from them). What level

of 𝑘 will they choose, when 𝑦𝐺 = 2, 𝑦𝐵 = 1, 𝑏 = 2 and 𝑧 = 2? [Hint: find the highest level that is

consistent with your solution for 𝑒 being valid.]

4. An owner wishes to incentivise his risk neutral manager. The manager attends for a full working

week, but the hours of effort he puts in are unobservable to the owner. The cost of each hour’s effort

for the manager is equivalent to £20. Profits are maximised where output is £1000, which is achieved

when the manager puts in 25 hours of effort. The manager has an outside option that enables him to

earn £200 and put in zero hours of effort.

a) What is the best compensation scheme from the owner’s point of view?

b) Why is the optimal incentive scheme that you have identified relatively uncommon in the real

world?

5. [Slightly tricky] An agent has mean-variance utility given by 𝑢 = 𝔼𝑤 − 𝜌𝔼(𝑤 − 𝔼𝑤)2 −𝑐

2𝑥2, where

𝑤 is their risky income, 𝑥 is their unobserved effort level, 𝜌 measures their risk aversion, and 𝑐

measures their cost of effort. They have an outside option which delivers a utility of 𝑢 with certainty.

A risk-neutral principal wishes to maximise 𝜋(𝑥) − 𝑤 , where 𝜋(𝑥) is their risky profits, which is

normally distributed with mean 𝑑𝑥 and variance 𝑣𝑑𝑥. The principal is constrained to only offer linear

contracts to the agent, of the form 𝑤 = 𝑎 + 𝑏𝜋(𝑥). Prove that the optimal choice of 𝑏 is only a

function of 𝜌 and 𝑣, not a function of 𝑐 or 𝑑. Show that when 𝜌𝑣 = 0, 𝑏 = 1. Verify that when 𝜌𝑣 =

1, 𝑏 ≈ 0.3660254040, by substitution.

6. a) To what extent does the NHS solve the inefficiencies caused by asymmetric information in private

health care markets?

b) Provide an economist’s perspective on the incentive problems that a system like the NHS might

face instead.