Question

3. The economy is made up of two types of agents: There are 4000 people in group 1 and each of them have the utility function U₁ = W^3/4 The income for everyone in group 1 is such that they have a 75% chance of making $50 and a 25% chance of making 300. Group 2 has 6000 people and each have the utility U₂ = W^1/3. The income for these people is that they have a 30% chance of making 95 and a 70% chance of making 120.

a) An insurance company, who cannot tell one agent from another offers to sell everyone an insurance policy of the following form: In exchange for the consumer giving the insurance company all their income, no matter what it is, the insurance company will give the consumer $112.25.

i) Show that this insurance is a good deal for each type of consumer.

ii) Show that the insurance company makes money on average with each type of consumer.

iii) For each group find how many people can earn a low income and still have the insurance company make money. Use this to explain why

insuring group 2 is much less risky.

b) Now assume that things are different, now all 4000 of the type 1 people have the same utility as before, but all get their income like the type 2 people used to, and all 6000 type 2 people still have the same utility, but get their income as the type 1 people did.

i) Show that now, only type 2 people buy insurance.

ii) Calculate the relative risk aversion for each consumer, and explain why the previous result makes sense.

Answer 1