Question

Consider 500 people who each have a 12% chance of getting sick this year. If they get sick, they will have a $10,000 hospital bill. If they do not get sick, they will have no medical expenses. All 500 people are risk-averse, but they differ with respect to the degree of their risk- aversion. Half of them are very risk-averse and willing to pay a risk premium equal to 40% of the actuarially fair premium. The other half are less risk-averse and are only willing to pay a risk premium equal to the 25% of the actuarially fair premium.

a. What is the actuarially fair premium for a more risk-averse person? What is the actuarially far premium for a less risk-averse person?
b. What premium is a more isk-averse person willing to pay? What premium is a less risk- averse person willing to pay?
c. What does the demand curve for insurance look like? Plot it in a standard diagram with quantity of people who buy insurance on the horizontal axis and the price of insurance (i.e., premium) on the vertical axis. (Hint: Demand is not a smooth, straight line. The quantity demand abruptly jumps up when the insurance premium crosses certain thresholds.)
d. Suppose that the insurance market is competitive with free entry and exit. Competitive forces therefore lead insurers to sell insurance at the actuarially fair price. How many people buy insurance? What is the consumer surplus for the 250 people who are more risk-averse? What is it for the 250 people who are less risk-averse? What are insurers' expected profits? (Note: It may help to identify the consumer surplus in your diagram from partc.)
e. Now suppose insurers are successful at lobbying legislators to disallow entry from new insurance companies. Without as much competition, insurers are now able to charge a premium that is 30% more than actuarially fair price. How many people buy insurance now? What is the consumer surplus for the 250 people who are more risk-averse? What is it for the 250 people who are less risk-averse? What are insurers' expected profits? (Note: It may help to identify the new consumer and producer surplus in your diagram from partc.)
f. What is the effect of part e's barrier to entry on consumer surplus? What is the effect on producer surplus (i.e., insurers expected profits)? What is the effect on social surplus (i... the sum of consumer and producer surplus)?

Answer 1

a) The actuarially fair premium is the same for both group: $1,200

b) A very risk-averse's premium: $1,680

A less risk-averse's premium: $1,500

c) Downside curve

d) 500 people buy insurance.

Consumer surplus for 250 people who are more risk-averse: $480

Consumer surplus for 250 people who are more risk-averse: $300

Insurers' expected profit: 0

e) 250 people who are very risk-averse buy.

Consumer surplus for 250 people who are more risk-averse: $120

Consumer surplus for 250 people who are more risk-averse: -$60

Insurers' expected profit: $90,000

f) The barrier limits consumer surplus but makes insurers profitable.

The effect on social surplus: -$75,000

Step-by-step explanation:

a) $10,000*12%= $1,200

b) A very risk-averse's premium : $1,200 + 40% x $1,200 = $1,680

A less risk-averse's premium : $1,200 + 25% x $1,200 = $1,500

d) The insurance pricing is lower than the prices that people with different risk appetites are willing to pay.

Consumer surplus for 250 people who are more risk-averse: $1,680 - $1,200= $480

Consumer surplus for 250 people who are more risk-averse: $1,500 - $1,200= $300

Insurers' expected profit: 0 because there is no premium.

e) Premium: $1,200 x 130% = $1,560

The premium is still affordable to very risk-averse people who will buy.

Consumer surplus for 250 people who are more risk-averse: $1,680 - $1,560= $120

Consumer surplus for 250 people who are more risk-averse: $1,500 - $1,560= -$60

Insurers' expected profit: 250 x ($1,560 - $1,200) = $90,000

f) The effect on social surplus: 250 x $120 + 250 x (-$60) - $90,000 = -$75,000