Question
In: Economics
A person's utility function is U = C1/2 . C is the amount of consumption they...
A person's utility function is U = C1/2 . C is the amount of consumption they have in a given period. Their income is $40,000/year and there is a 2% chance that they'll be involved in a catastrophic accident that will cost them $30,000 next year.
a. What is their expected utility?
b. Calculate the actuarially fair insurance premium.
c. What would their expected utility be if they purchased the actuarially fair insurance premium?
Solutions
Expert Solution
U=C1/2
a)
In case of no insurance
Probability of accident=p=0.02
Consumption in case of accident=C=40000-30000=$10000
Utility in case of accident=U(10000)=100001/2=100 utils
Probability of no accident=1-p=1-0.02=0.98
Consumption in case of no accident=C=40000
Utility in case of no accident=U(40000)=400001/2=200 utils
Expected utility=p*U(10000)+(1-p)*U(40000)=0.02*100+0.98*200=198 utils
b)
Actuarially fair insurance premium=Probability of accident*Loss in case of accident
Actuarially fair insurance premium=0.02*30000=$600
c)
In case of insurance purchased at $600
Probability of accident=p=0.02
Consumption in case of accident=C=40000-30000+30000-600=$39400
Utility in case of accident=U(10000)=100001/2=198.49 utils
Probability of no accident=1-p=1-0.02=0.98
Consumption in case of no accident=C=40000-600=$39400
Utility in case of no accident=U(39400)=394001/2=198.49 utils
Expected utility=p*U(39400)+(1-p)*U(39400)=0.02*198.49+0.98*198.49=198.49 utils
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