Rational Choice Under Uncertainty

1. Application of the Von Neumann-Morgenstern Expected Utility Model. Let us assume that John is deciding between buying or not buying health insurance. In a typical year, he expects a 90 percent chance of being healthy (and thus, a 10 percent chance of getting sick). If John gets sick without health insurance, he will have to shoulder his entire medical costs estimated at $30,000. Lets also assume that the annual premium for the health insurance is $4,000 and that the insurance company will pay 85 percent of Johns medical expenses (i.e., John will only pay 15% X $30,000 = $4,500 instead of the entire $30,000.) If we calculate the expected value of the two options, we will have the following results:

Consumption Choices	Outcome 1: Stay Healthy	Outcome 2: Get Sick	Expected Value
Option 1: Do not buy health insurance	p1 = 0.90 pay-off = $0 (zero medical expense)	p1 = 0.10 pay-off = -$30,000 (John will pay the entire medical costs	EV1 = 0.90 *$0 + 0.10* -$30,000 = -$3,000
Option 2: Buy Health Insurance ($4,000 annual premium)	p1 = 0.90 pay-off = -$4,000 (annual insurance premium)	p1 = 0.10 pay-off = -$4,000 + -$4,500 = -$8,500 (annual insurance premium + Johns share of the medical expenses)	EV2 = 0.90 * -$4,000 + 0.10* -$8,500 = -$4,450

If we go by the expected values of the two options (which are both negative), John should not buy health insurance (-$3,000 > -$4,450). Is not buying health insurance in order to save $4,000 in annual insurance premium an attractive gamble in this case? Explain through the Von Neumann-Morgenstern Expected Utility Model that not purchasing health insurance may not be the rational decision here. Calculating the expected utilities of the two options will help in your discussion. Assume that Johns initial wealth is $30,000 and that his utility function is U = . Is John risk-averse, risk-neutral, or risk-loving?