Final answer:
The question involves calculating an insurance premium and payout for a therapist using expected utility theory where the utility function is U(I) = ln(I). Premiums and payouts are determined based on equalizing expected utility with and without insurance and ensuring the insurance company breaks even.
Step-by-step explanation:
The student's question about determining the insurance premium, r, and the payout, q, under a full and fair insurance contract involves applying the concepts of expected utility and actuarial fairness. Jean is a therapist earning $600 with a 20% chance of earning zero due to sickness. To calculate the premium and payout, we need to equalize Jean's expected utility without insurance to her utility with insurance. With a utility function U(I) = ln(I), the expected utility without insurance is 0.8 * ln(600) + 0.2 * ln(0). Since the utility of zero income is undefined, we consider the expected utility with a small income amount instead, such as $1, making the calculation 0.8 * ln(600) + 0.2 * ln(1).
To find the insurance premium r that Jean would pay on days when she is not sick, we need to solve for r where the expected utility with insurance equals her expected utility without insurance. The utility with insurance when Jean is not sick would then be U(600 - r). On a day when Jean is sick, the utility would be U(q). The insurance company will want to break even on average, meaning the premium paid by clients should cover the payouts, which can be expressed as 0.8 * r = 0.2 * q. To solve for r and q, we would set up and solve a system of equations based on these considerations.