Consider a population of four people, each of whom has an initial income level of $100. Each faces the possibility of being laid off and losing his/her job and having no income. However, the probability of being laid off varies across the individuals. Assume the four individuals have the following probabilities of being laid off:
Person 1: p = 0.10
Person 2: p = 0.20
Person 3: p = 0.30
Person 4: p = 0.40
All of the individuals have the same utility function, though, given by U(w) = Vw, where w is the individual's income.
a) If the insurer could observe individual risks, what is the actuarially fair premium that it would charge each individual to fully insure each one against his/her income loss if laid off?
b) For each individual, determine if he/she would buy the full insurance if the premium is set at the actuarially fair level for that individual. Illustrate your answer for each individual numerically, i.e., by calculating (expected) utility with and without insurance.
c) In practice, insurers typically cannot observe individual risks and instead base premiums on average risk. What is the average risk across this population? If it had to charge the same premium to all individuals and based that premium on the average population risk, what premium would it charge to fully insure individuals against their income loss if they are laid off?
d) Repeat part (b) for the case where the premium is based on average population risk. Will any of the individuals drop out of the insurance market, i.e., choose not to buy the insurance? If so, which type of individual(s) drop out? Does the insurer face an adverse selection problem? Explain.
f) If part of the population chooses not to buy insurance when the premium is based on average risk, will the insurer adjust its premium from the level in (c)? If so, what would the new premium be?
g) Given the premium in (f), who will choose to buy insurance and who will not? Again, illustrate your answer both graphically and numerically.
h) Does the insurance market "unravel" in this example? Explain why or why not.