Question

Suppose a worker with a high-paying job faces uncertainty over whether they will still have
a job at the end of the year. With 90% probability, they will keep the job and make 150 this
year. With 10% probability they will lose the job, find another job, and make 50 in total
this year. (We might imagine these numbers are in thousands of dollars, but to keep the
numbers simple in this problem, we can express the possibilities as w1=150 and w2=50.)

(a) What is the expected value of the gamble faced by the worker?

b. Now suppose the worker's utility associated with each outcome is given by -1/w
w where w is either w1 or w2. This utility function may look a little odd because all utility values
are negative, but it remains the case that all higher (less negative) utilities are better.
What is the worker's expected utility from the gamble?

What is the certainty equivalent of this gamble for the worker?

(d) An insurance company is willing to offer a policy that will pay a benefit of 100 to the
worker if they lose their job. However, the worker has to pay a premium of r whether
they lose their job or not. They will buy the insurance of r is low enough. How low
does r have to be for the worker to buy the insurance?

(e) How does the threshold value of r relate to the certainty equivalent from part (c)?

Answer 1

a)

Probability of not loosing job=p₁=0.90

Income in case of no loss of job=w₁=150

Probability of loosing job=p₂=0.10

Income in case of loss of job=w₂=50

Expected value of gamble=p₁*w₁+p₂*w₂=0.90*150+0.10*50=140

b)

Utility in case of no loss of job=U₁=-1/w₁=-1/150=-0.006667 utils

Utility in case of loss of job=U₂=-1/w₂=-1/50=-0.02 utils

Expected utility=p₁*U₁+p₂*U₂=0.90*(-0.006667)+0.10*(-0.02)=-0.008 utils

c)

Let us find the amount of certain income that gives the same utility as expected utility.

U(w)=-1/w

-0.008=-1/w

w=1/.008=125

Agent would accept a certain income of 125 in exchange with given gamble.

d)

(we assume r is insurance premium not risk premium)

Expected utility in case of insurance=U(150-r)

Agent would pay a maximum of r such that

U(150-r)=-0.008

-1/(150-r)=-0.008

1=1.2-0.008r

r=0.2/0.008=25

e)

In case of full coverage, r is the difference between income/wealth in good state and Certainty equivalent.