Question

Bob's farm harvests corns worth 106 thousand dollars (and nothing else). In each year, there is a 24% chance that a storm will attack and leaves him with only 26 thousand dollars worth of the corns. Bob's preferences over wealth are represented by U=ln{w} . What is the maximum that Bob is willing to pay for full insurance (in unit of thousand dollars)? Please round your answer to the nearest integer.

Answer 1

Solution:

Maximum willingness to pay for full insurance can be found where Bob is indifferent in buying the insurance and not buying the insurance. In other words, that value of insurance where Bob's expected utility with insurance equals his expected utility without insurance.

Expected utility of Bob without insurance = 24%*ln(26000) + (1 - 24%)*ln(106000)

Expected utility of Bob without insurance = 0.24*10.1659 + 0.76*11.5712

Expected utility of Bob without insurance = 2.4398 + 8.7941 = 11.2339

Expected utility with insurance (which is simply the utility with insurance, as no probabilities) = ln(106000 - P); where P is the price of insurance

So, ln(106000 - P) = 11.2339

On taking anti-log on both sides, 106000 - P = e11.2339

106000 - P = 75652.0638

P = 106000 -  75652.0638 = 30347.9362 or 30.348 thousand dollars

So, the maximum that Bob is willing to pay for full insurance is 30 thosand dollars (approximating to the nearest integer).