Suppose Marilyn is 30 years old, plans to work for the next 35 years and has an 9% discount rate. Marilyn cares about the net present value of her lifetime income. (Ignore taxes.) Marilyn considers her even swaps between guaranteed annual incomes and uncertain annual incomes. She considers the following exchanges to be even swaps: $100,000 with probability 0.50 or $0 with probability 0.50 ~ guaranteed $30,000 $30,000 with probability 0.50 or $0 with probability 0.50 ~ guaranteed $10,000 $100,000 with probability 0.50 or $30,000 with probability 0.50 ~ guaranteed $60,000 You can assume she has piecewise linear utility, with risk-neutral even swaps for incomes below $10,000; with different risk neutral even swaps for incomes between $10,000 and $30,000; different risk neutral even swaps for incomes between $30,000 and $60,000; and different risk neutral even swaps for incomes above $60,000. (All dollar values are annual incomes.)

a) (25 points) Convert annual incomes into the net present value of lifetime incomes and draw a graph of Marilyn's utility for NPV of lifetime income. (Note: assuming the annual income starts from the next year, i.e., Year 1)

b) (25 points) There is approximately a 1/1000 chance that an adult will become permanently disabled during the coming year. Suppose that Marilyn is now earning $85000 per year and is considering buying a disability insurance policy that would pay 40% of her lost income if she becomes permanently disabled in the next year, for an up-front cost of $1000. Note that social security would pay $18000 per year if she becomes permanently disabled, which would either be her sole income if she does not have insurance or would be in addition to her insurance payouts if she does have insurance. Marilyn still plans to work for the next 35 years and still has an 9% discount rate, and all of the even swaps given in part a) still apply here. Should Marilyn buy disability insurance?