Economics Problem $1$: Insurance Markets. Every person has $$2,500$ to spend. There are two

groups of people: $150$ sick people and $250$ healthy people. A sick person visits the hospital

with probability $p=0\mathrm{.}2.$ A healthy person visits the hospital with probability $p=0\mathrm{.}1.$ Any

hospital visit costs $$1,200.$ All individuals have the following expected utility function:

$EU=(1-p)\sqrt[\phantom{2}]{C_g}+p\sqrt[\phantom{2}]{C_b}$

First, assume that the insurance company can distinguish between sick and healthy

individuals and can charge each individual their actuarially fair premium.

a$)$ What is the actuarially fair price for a sick person?

b$)$ What is the actuarially fair price for a healthy person?

Next, assume the insurance company cannot distinguish between sick people and

healthy people. In a pooling equilibrium, they will charge one price to everyone for

full insurance.

a$)$ If they charge one price to everyone for full insurance, what price $x$ do they need

to charge to make zero expected profits when everyone buys?

b$)$ Will the sick buy full insurance at the price $x?$

c$)$ Will the healthy buy full insurance at the price $x?$

d$)$ Is this a successful pooling equilibrium? Briefly explain your reasoning.

Now, suppose the insurance company offers two insurance packages:

full insurance at a price of $$240$

partial insurance with $$100$ of coverage at a price of $$10$

The insurance company still cannot distinguish between sick and healthy people.

a$)$ Will the sick buy full or partial insurance?

b$)$ Will the healthy buy full or partial insurance?

c$)$ Is this a successful separating equilibrium? Briefly explain your reasoning.

Repeat part $3.$ with the following two insurance packages:

full insurance at a price of $$240$

partial insurance with $$400$ of coverage at a price of $$40$

Explain any difference in your results.