(This question will not be graded)

You are considering getting health insurance. The insurance company knows there are two types of individuals seeking this type of insurance, the healthy and the unhealthy, each group accounting for one half of the population. The insurance company offers a uniform premium and thus wants to avoid insuring the unhealthy people because they give the company an expected disutility of 15. Striking a deal with a healthy individual gives them a utility of 10. Not offering the insurance leaves them with zero utility.

A healthy individual gets a utility of 10 from insurance (no insurance gives a utility of 0). An unhealthy individual gets a utility of 20 from insurance (no insurance gives a utility of 0).

In order to signal to the insurance company that you are healthy, you can choose to provide proof of a check-up that states you are in good condition. Providing this proof implies a disutility of 5 for the healthy type and a disutility of 25 for the unhealthy type (this proof is much more difficult to achieve for the unhealthy type, who would need to find a doctor who could be bribed).