price theory

Price Theory

Lecture 11: Information Economics

Topics for today’s lecture . . .

1. Market unravelling

2. Signalling

3. Contracting

Market unravelling

The market for health insurance

Health insurance compensates an individual for the costs incurred when receiving medical care.

The likelihood that an individual will require medical attention varies from person to person.

• Factors that influence the risk that a person will require medical care include age, medical history, family medical history, and lifestyle.

In many jurisdictions, insurance companies must offer health insurance at the same price to

all consumers.

• In these jurisdictions, the fair price of health insurance depends on the average risk of the insured population.

The (potential) consumers of health insurance

Consumer p

EV RP WTP

George 0.2

$74k $6.4k $22.4k

Tom 0.4

$58k $9.6k $41.6k

Theo 0.6

$42k $9.6k $57.6k

Abe 0.8

$26k $6.4k $70.4k

Don 1.0

$10k $0.0k $80.0k

Suppose that five consumers are in the

market for health insurance.

• Each consumer has an income of $90,000.

• The cost of medical care is $80,000.

• Each consumer’s utility function is U =

√ I .

Consumers differ in the probabilities that

they will require medical care.

Willingness to pay for health insurance

Consumer p EV RP WTP

George 0.2 $74k $6.4k $22.4k

Tom 0.4 $58k $9.6k $41.6k

Theo 0.6 $42k $9.6k $57.6k

Abe 0.8 $26k $6.4k $70.4k

Don 1.0 $10k $0.0k $80.0k

Using the methods we developed in

lecture 4, we can calculate for each

consumer,

• the expected value of the lottery he faces.

• his risk premium.

Using these values we can derive each

consumer’s willingness to pay for health

insurance.

Note: You should check the calculations

on this slide.

Exercise: Expected payouts

Consumer p WTP

EP

George 0.2 $22.4k

$16.0k

Tom 0.4 $41.6k

$32.0k

Theo 0.6 $57.6k

$48.0k

Abe 0.8 $70.4k

$64.0k

Don 1.0 $80.0k

$80.0k

Recall that the cost of medical care is

$80,000.

1. Calculate the expected payout for

each consumer.

2. How does each consumer’s willingness

to pay compare to his expected

payout?

Exercise solutions

Consumer p WTP EP

George 0.2 $22.4k $16.0k

Tom 0.4 $41.6k $32.0k

Theo 0.6 $57.6k $48.0k

Abe 0.8 $70.4k $64.0k

Don 1.0 $80.0k $80.0k

1. For each consumer the expected

payout is,

EP = 80,000 × p.

The corresponding values are listed in

the table.

2. For each consumer (with the

exception of Don who faces a risk-free

outcome) willingness to pay exceeds

the expected payout.

The fair price of health insurance

The fair price of a health insurance policy is the average of the expected payouts for the

consumers who purchase the policy.

If all five consumers buy the policy, the fair price is,

FP = 16,000 + 32,000 + 48,000 + 64,000 + 80,000

5 = $48,000.

The fair price is significant because it just covers the expected cost of the insurance policies

to the insurer.

Note: Recall that, in reality, the fair price is less than the insurer’s total cost of doing

business as it does not cover the insurer’s operating costs. We will ignore this for the

purposes of the present example.

Unravelling

Consumer p WTP EP

George 0.2 $22.4k $16.0k

Tom 0.4 $41.6k $32.0k

Theo 0.6 $57.6k $48.0k

Abe 0.8 $70.4k $64.0k

Don 1.0 $80.0k $80.0k

If the insurer offers policies at a price of

$48,000, George and Tom will not

purchase.

• The price exceeds their respective willingness to pay.

If George and Tom do not purchase, the

insurer requires a higher price to break

even.

• This is because George and Tom have the lowest expected payouts.

Further unravelling

Consumer p WTP EP

George 0.2 $22.4k $16.0k

Tom 0.4 $41.6k $32.0k

Theo 0.6 $57.6k $48.0k

Abe 0.8 $70.4k $64.0k

Don 1.0 $80.0k $80.0k

If Theo, Abe and Don purchase policies,

the fair price is $64,000.

• This price exceeds Theo’s willingness to pay.

If only Abe and Don purchase, the fair

price is $72,000.

• This price exceeds Abe’s willingness to pay.

If Don alone purchases a policy, the fair

price is $80,000; exactly his willingness to

pay.

Definition: Adverse selection

A phenomenon whereby an individual’s behaviour in a market depends on their private

information, to the detriment of uninformed market participants.

In insurance markets, one consequence of adverse selection is that an increase in the

insurance premium increases the overall riskiness (average expected payout) of the pool of

individuals who purchase an insurance policy.

Universal health care

In many nations, governments pursue the objective of universal health care.

• With universal health care, every citizen is insured against the cost of medical treatment.

• The average expected payout is held down by the inclusion of all citizens in the pool of insured individuals.

Broadly speaking, there are two methods for achieving universal health care:

Government provision: The government provides health insurance (sometimes called health

coverage) to all citizens, financing the costs through taxation.

Government mandate: The government requires all citizens to purchase health insurance

(regardless of willingness to pay), providing subsidies to citizens who would otherwise be

unable to afford the premiums.

Quiz 1

Consumer p WTP EP

Theo 0.6 $57.6k $48.0k

Abe 0.8 $70.4k $64.0k

Theo and Abe are in the market for

health insurance. If an insurer offers

policies with a premium of $57k then,

(a) neither consumer will purchase a policy.

(b) only Abe will purchase a policy, and the premium is greater than the fair price.

(c) both consumers will purchase a policy, and the premium is lower than the fair price.

(d) both consumers will purchase a policy, and the premium is greater than the fair price.

Quiz 2

Which of the following statements is false?

(a) Market unravelling is one possible consequence of adverse selection.

(b) A government mandate requiring consumers to purchase an insurance policy raises the

willingness to pay of individuals with the lowest expected payouts.

(c) A risk averse consumer’s willingness to pay for health insurance increases as her/his risk

of illness or injury increases.

(d) Market unravelling does not occur when health insurance is provided by the

Government.

The market for lemons

Adverse selection, and market unravelling, can also arise when sellers have private information.

• Sellers typically know more about the quality of the products they sell than buyers, and have an incentive to overstate the quality.

• An example can be found in the homework problem for this topic.

There are a number of method by which governments may address the problem of market

unravelling.

Standards & information disclosure: The government can mandate that products are

produced to minimum standards, and require sellers to reveal significant information to buyers.

Guarantees & consumer protections: The government can require sellers to guarantee the

quality and performance of their products.

Signalling

Adverse selection in the labour market

A worker’s productivity depends, at least in part, on her/his level of ability (or talent).

• When a worker first seeks employment, her/his level of ability is private information; not directly observable by prospective employers.

Employers are willing to pay higher wages to workers with high ability (and therefore a high

marginal product of labour).

• Thus a worker has an incentive to overstate her/his ability.

Knowing this, an employer will only offer higher wages to workers who provide a credible

signal of having a high level of ability.

Definition: Signalling

An action taken by an individual with private information, which credibly conveys the

information to uninformed parties.

For a signal to be credible, an individual must not be willing to send the signal if it is more

favourable than her/his private information.

A labour market with two types of worker

Suppose that there are two types of worker in the labour market, high ability and low ability.

• If an employer cannot determine a worker’s level of ability, she/he receives a wage of $40,000.

• A worker who is identified as high ability receives wage of $90,000.

The utility that a worker derives from a wage w is U = √

w − c .

• In this equation c is the disutility a worker incurs if she/he sends a signal of high ability.

A university degree may serve as a credible signal of high ability, as workers with a high level

of ability tend to find studying easier (less effort) than people with a low level of ability.

Educational attainment as a signal of ability

degree √ $90,000 − 50 = 250

high

ability

worker no degree √ $40,000 = 200

degree √ $90,000 − 150 = 150

low

ability

worker no degree √ $40,000 = 200

A worker always has the option of not

sending a signal.

• That is, not study for a degree.

Suppose that the disutility of studying is,

• 50 for a high ability worker.

• 150 for a low ability worker.

In this instance a university degree is a

credible signal of high ability, as only a

high ability worker would choose to study.

The signalling theory of education

The signalling theory of education states that education can be valuable to a student, even if

the content has no application in the workplace.

• Attaining a degree is a means by which a student can send a credible signal of her/his ability.

One implication of the signalling theory of education is that a degree becomes more valuable

as the courses become more difficult.

• Students of low ability are less likely to attempt courses that requires a lot of effort.

Note: In reality, a university education both provides you with knowledge and skills that are

valuable in the workplace, and sends a signal of your ability.

Quiz 3

Grade c (high) c (low)

High distinction 80 160

Distinction 60 120

Credit 40 80

Pass 20 40

Fail 0 0

A worker’s utility is given by the function

U = √

w − c .

The wage for high skilled workers is

$90,000, and the wage for a worker

without a signal is $40,000.

What is the lowest grade that provides a

credible signal of high ability?

(a) High distinction

(b) Distinction

(c) Credit

(d) Pass

Quiz 4

Grade c (high) c (low)

High distinction 60 120

Distinction 45 90

Credit 30 60

Pass 15 30

Fail 0 0

A worker’s utility is given by the function

U = √

w − c .

The wage for high skilled workers is

$90,000, and the wage for a worker

without a signal is $40,000.

If the difficulty of the course is lowered,

as set out in the table, what is the lowest

grade that provides a credible signal of

high ability?

(a) High distinction

(b) Distinction

(c) Credit

(d) Pass

Discussion: Grade inflation

Grade inflation is the phenomenon of individual lecturers, or universities as a whole, making it

easier for students to earn high grades.

• When grade inflation occurs, a higher proportion of students receive high grades.

How do you think grade inflation affects a student’s employment prospects?

How do you think that the grading practices at one university affect the employability of

students at another university?

Contracting

Employing a manager

A firm employs Don to manage a store.

The store’s profit is uncertain:

• If business is good, the store will make $1.2M in sales.

• If business is bad, the store will make $0.5M in sales.

The probability that business is good depends on the level of effort exerted by Don.

• With high effort, business is good with probability pH = 0.5.

• With low effort, business is good with probability pL = 0.2.

Don’s welfare

Don’s utility function is U = √

w − e.

• In this equation w is Don’s wage, and e is the disutility Don incurs by exerting effort.

• If Don exerts low effort then e = 0, while if Don exerts high effort e = 100.

Don’s next best alternative to managing the store is a job that would deliver him 300 units of

utility.

• It follows that Don will only accept a contract from the firm if the (expected) utility is at least 300.

The optimal contract with observable effort

Suppose that the firm can dictate Don’s level of effort. The wage the firms pays to Don

depends on the level of effort it wants Don to exert.

• If the firm wants low effort then the wage must satisfy √

w = 300 or w = $90,000.

• If the firm wants high effort then the wage must satisfy √

w − 100 = 300 or w = $160,000.

The firm’s expected profit with low effort is therefore,

E Π = 0.2 × 1,200,000 + 0.8 × 500,000 − 90,000 = $550,000.

This is lower than the firm’s expected profit with high effort,

E Π = 0.5 × 1,200,000 + 0.5 × 500,000 − 160,000 = $690,000.

Exercise: An author’s contract with observable effort

A publisher commissions Stephen to author a book. If the book sells well the publisher will

make $6M, while if sales are bad the publisher will earn $2M. The probability of good sales is

pH = 0.2 if Stephen exerts high effort, and pL = 0.1 if he exerts low effort.

Stephen’s utility is described by the function U = √

w − e, where e = 0 if Stephen exerts low effort, and e = 200 if he exerts high effort. If Stephen does not write the book, he will

receive 500 units of utility from other endeavours.

Suppose that the effort exerted by Stephen is observable:

1. How much must the publisher pay Stephen to exert low effort?

2. How much must the publisher pay Stephen to exert high effort?

3. Which level of effort delivers the publisher the highest profit (earnings minus the author’s

wage)?

Exercise solutions

1. For low effort e = 0. Therefore, Stephen’s wage must satisfy √

w = 500. Squaring both

sides gives us w = $250,000.

2. For high effort e = 200. Therefore, Stephen’s wage must satisfy √

w − 200 = 500. Adding 200 to both sides squaring gives us w = $490,000.

3. With low effort, the publisher’s expected profit is,

E Π = 0.1 × 6,000,000 + 0.9 × 2,000,000 − 250,000 = $2,150,000.

With high effort, the publisher’s expected profit is,

E Π = 0.2 × 6,000,000 + 0.8 × 2,000,000 − 490,000 = $2,310,000.

Therefore, the publisher would prefer Stephen to exert high effort.

Definition: Moral hazard

A phenomenon whereby an individual’s actions cannot be observed and the individual has an

incentive to shirk, exerting less effort than her/his employer would prefer.

In the presence of moral hazard, an employer can create an incentive for an individual to exert

effort by linking the individual’s wage to her/his observed performance.

The participation constraint

Suppose now that the firm cannot observe the level of effort that Don exerts, but can still

observe the sales made by the store.

An incentive contract pays Don a wage wG if business is good, and wB if business is bad.

For Don to accept this contract, the expected utility of the contract when he exerts high

effort must be equal to (or greater than) his second best job offer,

0.5 ( √

wG − 100) + 0.5 ( √

wB − 100) = 300 or √

wG + √

wB = 800.

This is know as Don’s participation constraint because it describes the condition under which

Don will accept the contract.

The incentive compatibility constraint

The expected utility of the contract when Don exerts high effort, must be equal to (or

greater than) the expected utility of the contract when Don exerts low effort,

0.5 ( √

wG − 100) + 0.5 ( √

wB − 100) = 0.2 √

wG + 0.8 √

wB,

or, √

wG = √

wB + 1000

3 .

This is known as the incentive compatibility constraint because it describes the condition

under which Don cannot improve his welfare by shirking (exerting low effort).

Note: If Don shirks, he does not incur the disutility of high effort, but also lowers the

probability of earning wG.

The optimal contract with hidden effort

The optimal values of wG and wB can be found by solving the two constraints simultaneously.

Substituting for √

wG, from the incentive compatibility constraint, into the participation

constraint, ( √

wB + 1000

3

) + √

wB = 800,

or,

wB =

( 700

3

)2 ≈ $54,444.

Substituting for √

wB into the incentive compatibility constraint,

√ wG =

( 700

3

) +

1000

3 or wG =

( 1700

3

)2 ≈ $321,111.

The cost of moral hazard

By utilising an incentive contract the firm is able to motivate Don to exert a high level of

effort, even though it cannot directly observe Don’s actions.

Don’s expected wage under the incentive contract is,

0.5 × 54,444 + 0.5 × 321,111 = $187,778,

higher than $160,000 the firm would pay the manager if it could observe Don’s effort.

The higher cost of the incentive contract is a result of the need to compensate the

risk-averse Don for carrying some of the risk associated with the store’s sales.

Exercise: An author’s contract with hidden effort

A publisher commissions Stephen to author a book. If the book sells well the publisher will

make $6M, while if sales are bad the publisher will earn $2M. The probability of good sales is

pH = 0.2 if Stephen exerts high effort, and pL = 0.1 if he exerts low effort.

Stephens utility is described by the function U = √

w − e, where e = 0 if Stephen exerts low effort, and e = 200 if he exerts high effort. If Stephen does not write the book, he will

receive 500 units of utility from other endeavours.

Suppose that the effort exerted by Stephen is hidden:

1. Derive an expression for Stephen’s participation constraint?

2. Derive an expression for Stephen’s incentive compatibility constraint?

3. What is the optimal incentive contract? Does this contract deliver a higher expected

profit than paying Stephen a fixed wage of $250,000, and having Stephen exert low effort?

Exercise solutions

1. The incentive contract must deliver Stephen the same expected utility as his next best

alternative,

0.2 ( √

wG − 200) + 0.8 ( √

wB − 200) = 500 or 0.2 √

wG + 0.8 √

wB = 700.

2. The incentive contract must not give Stephen an incentive to shirk,

0.2 ( √

wG − 200) + 0.8 ( √

wB − 200) = 0.1 √

wG + 0.9 √

wB,

or, √

wG = √

wB + 2000.

Exercise solutions

3. Substituting for √

wG from the incentive compatibility constraint, into the participation

constraint,

0.2 ( √

wB + 2000) + 0.8 √

wB = 700 or √

wB = 300.

Squaring both sides of this equation wB = $90,000. Substituting for √

wB into the

incentive compatibility constraint,

√ wG = 300 + 2000 or wG = $5,290,000.

The publisher’s expected profit is,

E Π = 0.2(6,000,000 − 5,290,000) + 0.8(2,000,000 − 90,000) = $1,670,000.

This is less than the expected profit of $2,150,000 that the publisher earns if it pays

Stephen the fixed wage $250,000, and Stephen applies low effort.

Questions?

Key concepts from today’s lecture

You can use these concepts (as search terms) to conduct further research into the topics

covered in today’s lecture:

• Fair (break-even) price

• Expected payouts

• Adverse selection

• Market unravelling

• Universal health care

• Signalling

• Signalling theory of education

• Grade inflation

• Moral hazard

• Incentive contract

• Participation constraint

• Incentive compatibility constraint

Quiz solutions

Quiz 1 (d)

Quiz 2 (b)

Quiz 3 (b)

Quiz 4 (a)