Although the concept of utility is a relatively abstract one, this problem quantifies your level of utility at a given level of wealth for you to complete the analysis. For simplicity, assume that you are in a world with only two states: one where you get into a car accident and one where you don’t. If you do not get into a car accident, your wealth is 24 and your utility is 20, so u(24) = 20. However, if you do get into a car accident, you suffer a loss; your wealth declines to 12, and your utility falls to 16, so u(12) = 16.
• Draw these two possible outcomes on a graph with wealth on the horizontal axis and utility on the vertical axis.
• Suppose that before the COVID-19 pandemic, there’s a 50% probability you get into a car accident. (Although that might not be the most realistic assumption, economists often make simplifying assumptions like this that don’t affect the result but do make the math a little easier.) In this case, your expected level of wealth is 18, because there’s a 50% probability your wealth is 12 and a 50% probability your wealth is 24. What is your expected level of utility in this case? Add this point to your graph.
• If you are risk averse, how does this value compare to your actual utility at a wealth of 18? What does this say about the shape of the utility curve between these two points? How would that shape change if you were not risk averse?
• Because of consumers’ desire to reduce risk, most insurance companies charge a fee, called the premium, to take on this risk. If you are risk averse, what’s the highest premium you would be willing to pay to avoid the gamble between these two states of the world? In other words, what premium results in you receiving the same utility as the expected level of utility from the gamble?
• Given this analysis, how do you think the stay-at-home orders that occurred during the COVID-19 pandemic affected the probability that you would experience a car accident? How did this change the premium you would be willing to pay? Is this consistent with the behavior of insurance companies in April of 2020? Why or why not?
