ASSIGNMENT 2 (Econ 357)

Winter 2022

Problem 1:

Ann purchases insurance against damage to her phone wires at 45 cent a month, even though the probability that she incurs a repair cost of ✩60 is only 0.4%.

Is such behavior consistent with the rational behavior (it means, does it yield the highest payoff for Ann among all of her feasible choices), given that Ann’s utility function is u = √w , and w = 60 is her initial wealth? What about Ann’s choice if her wealth is w = 70?

Suppose that Bob’s wealth is w = 60, and that his utility function is u = w

Suppose that he also purchases the same insurance. Is his purchase of insurance(.)

consistent with the rational behavior? Compare your answer, assuming w = 60,

with Ann’s utility, and explain the difference.

Answer to problem 1: The decision to not purchase insurance is equivalent to having a lottery which pays -60 with probability 0.004, and zero otherwise. A decision to purchase insurance can be thought instead as choosing a lottery which pays -0.45 with certainty.

Therefore, Ann is rational (it means it is optimal for her to purchase insur- ance) if the following is satisfied

u(w − 0.45) = √w − 0.45 ≥ 0.004 √w − 60 + 0.996 √w,

or

√w − 0.45 − (0.004 √w − 60 + 0.996 √w) ≥ 0.

Assuming w  =  60, we get √60 − 0.45 − (0.004 √60 − 60 + 0.996 √60)  =

therefore it is not rational for Ann to purchase insurance.

Now, suppose w = 60, and assume the utility function w (Bob).  Then, the difference in utility between purchasing, and not purchasing the insurance, becomes:

(w − 0.45) − 0.004(w − 60) + 0.996(60) = 0.0058.

First, this is positive, and therefore it is rational for Bob to purchase the in- surance.  Second, it is strictly larger than 0.0019, which means that Bob has a bigger interest in purchasing insurance than Ann, assuming the same level of wealth w = 60. This is because Bob dislikes risk more than Ann.