Adverse Selection and Market Efficiency

The assignment content discusses adverse selection and the concept of equilibrium in economic systems. It presents three solutions to demonstrate how equilibrium can be achieved in different scenarios. Solution 1 shows how workers can achieve rational expectations and equilibrium by taking a costless test to verify their abilities. Solution 2 demonstrates how firms can pay wages based on worker productivity, leading to efficient allocation of resources. Solution 3 discusses the concept of health insurance and how expected healthcare costs affect the purchase of insurance, ultimately achieving equilibrium premiums for young and old populations.

Adverse selection 0
adverse selection

Adverse selection 1
Solution 1)
Quality of worker = which is distributed in interval
Worker produce quantity worth $
Worker can produce at home
For every
Equilibrium exist when
Since for all ,
Thus, if
W*= 0 nobody work
W*=1/2 everybody work
Thus, the rational expectation equilibrium for the economy is:
Yes, all above equilibrium is efficient, because
Everybody work since they earn at home is less than given wages and
everybody work at home and can earn higher wages.
Solution 2)
Workers have access to a costless test through which they can verify their ability to prospective
employers so that they can be paid according to their true ability. A worker who takes the test is
paid a wage equal to her productivity

Adverse selection 2
Given that is increasing in , which means higher the productivity higher will be the
gap between wage worker can earn at home and wages he can work from working outside.
a)
Given that Ø is the set of workers types that don’t take the test
Expected revenue from set of workers that don’t take the test = and wages for these
worker will be w.
Since there is perfect competition, thus firm will earn zero profit
Thus, the firms will pay a wage
b)
Given that the firms offer a wage w to those employers that don’t take the test.
Thus worker who will choose to work has efficiency greater than and are not a part of Set Ø
(set of workers types that don’t take the test)
Symbolically,
Thus, worker those has higher productivity and can earn higher wage given to worker that don’t
take the test.
c)
From above solution we know that there is a set of worker who don’t take test
i.e. for some
Since is distributed uniformly
Half of the worker are belong to set Ø set of workers types that don’t take the test and half who
take test.
The wage offered to workers who don’t take the test

Adverse selection 3
Since there is perfect competition, thus firm will earn zero profit
Hence proved, the wage offered to workers who don’t take the test is
d)
Use your answer in part (c) to show that ◊ = 0, so that all workers take the test. Hint: By part (b),
there exists ◊ such that workers with type below ◊ don’t take the test. If ◊ > 0, what wage would
a worker with type ◊ ≠ ‘ (with ‘ very small) get if he takes the test? What wage would he get if
he doesn’t take the test?
Worker who will choose to work has efficiency greater than and are not a part of Set Ø (set of
workers types that don’t take the test)
Symbolically,
Thus if , wage would a worker with type worker get if he take the test
Because wage will increase only if increases.
And if worker take test, wage will be zero because is the minimum productivity of labour
From part c) we know the profit equation
And also know equilibrium wage

Adverse selection 4
Thus, , to make part c possible
i.e for equilibrium to be achieved
Hence proved, , so that all workers take the test
Solution 3)
Expected healthcare costs is v which is distributed uniformly over the interval [0, 10] in the
young population, and uniformly over the interval [10, 30] in the old population.
A person with expected healthcare cost v is willing to pay up to v + 4 to purchase (complete)
health insurance
Identify the set of young people who purchase health insurance at that premium, and find
expected insurance-company profits as a function of p. Do the same for the old, when all
companies set premium q for them
a)
Insurance company set premium for young is p
Set of young people who purchase health insurance at that premium p are those whose benefit
from insurance is greater than cost.
Expected insurance-company profits as a function of p.
Similarly, Set of old people who purchase health insurance at that premium q are those whose
benefit from insurance is greater than cost.
Expected insurance-company profits as a function of p.

Adverse selection 5
b)
Equilibrium premium for young
Premium= Benefit (probability of loss)
Here in this, since probability of using insurance is not given assume probability that person need
insurance is ½
Thus,
Similarly, Equilibrium premium for old
Premium= Benefit (probability of loss)
Here in this, since probability of using insurance is not given assume probability that person need
insurance is ½
Thus,

Adverse selection 6