Question

Suppose that half of the population is healthy and the other half is unhealthy. Both types have probability 0.4 of getting sick, but a healthy person pays $1000 of medical expenses if he gets sick, while an unhealthy person pays $10000 of medical expenses if he gets sick. The insurance company cannot tell whether a customer is healthy. Suppose that all customers start with wealth of $30000 and that their utility functions are U (W) =W1/2

1. What is the actuarially fair price for the insurance if everyone participates?

Answer 1

The actuarially fair price of insurance for both healthy and unhealthy individuals, each with a 0.4 probability of getting sick, would be $2200 per person. This is calculated by averaging the expected expenses of sickness for both groups in a population where half are healthy and half are unhealthy.

Step-by-step explanation:

To calculate the actuarially fair price of insurance given a population with healthy and unhealthy people who both have a 0.4 probability of getting sick, we consider the expected medical expenses for each group. Since half of the population is healthy, and half is unhealthy, we calculate the expected expense for each subgroup and then take the average to find the fair premium that should be charged if everyone participates in the insurance scheme.

For the healthy population (50%), the expected expense if sick is $1000, with a probability of 0.4. Therefore, the expected expense for a healthy individual is 0.4 * $1000 = $400.

For the unhealthy population (50%), the expected expense if sick is $10000, with a probability of 0.4. Therefore, the expected expense for an unhealthy individual is 0.4 * $10000 = $4000.

The actuarially fair premium would be the average of these expected expenses. As such, it's (0.5 * $400) + (0.5 * $4000) = $200 + $2000 = $2200. So, the actuarially fair premium with everyone participating would be $2200 per person.