We have illustrated in several settings the role of actuarially fair insurance contracts (b, p) (where b is the insurance benefit in the “bad state” and p is the insurance premium that has to be paid in either state). In this problem we will discuss it in a slightly different way that we will later use in Chapter 22.
A. Consider again the example, covered extensively in the chapter, of my wife and life insurance on me. The probability of me not making it is δ, and my wife’s consumption if I don’t make it will be 10 and her consumption if I do make it will be 250 in the absence of any life insurance.
(a) Now suppose that my wife is offered a full set of actuarially fair insurance contracts. What does this imply for how p is related to δ and b?
(b) On a graph with b on the horizontal axis and p on the vertical, illustrate the set of all actuarially fair insurance contracts.
(c) Now think of what indifference curves in this picture must look like. First, which way must they slope (given that my wife does not like to pay premiums but she does like benefits)?
(d) In which direction within the graph does my wife have to move in order to become unarm- bigamously better off?
(e) We know my wife willfully insure if she is risk averse (and her tastes are state-independent). What policy does that imply she will buy if δ = 0.25?
(f) Putting indifference curves into your graph from (b), what must they look like in order for my wife to choose the policy that you derived in (e).
(g) What would her indifference map look like if she were risk neutral? What if she were risk- loving?
B. Suppose u(x) = ln(x) allows us to write my wife’s tastes over gambles using the expected utility function. Suppose again that my wife’s income is 10 if I am not around and 250 if I am — and that the probability of me not being around is δ.
(a) Given her incomes in the good and bad state in the absence of insurance, can you use the expected utility function to arrive at her utility function over insurance policies (b, p)?
(b) Derive the expression for the slope of an indifference curve in a graph with b on the horizontal and p on the vertical axis.
(c) Suppose δ = 0.25 and my wife has fully insured under policy (b, p) = (240, 60). What is her MRS now?
(d) How does your answer to (c) compare to the slope of the budget formed by mapping out all actuarially fair insurance policies (as in A (b))? Explain in terms of a graph.