In: Economics
1. Jill has $200,000 of valuables, faces a probability of .02 of burglary, in which she would lose jewelry worth $50,000. She can buy full insurance for $15,000 that would fully reimburse the $50,000 loss. Her utility function is u(x) = 2*sqrt(x) What is the actuarially fair price for the insurance policy? Should Jill buy this insurance policy? What is the most that Jill is willing to pay for an insurance policy that fully covers her against loss?
Probability of burglary=p=0.02
Wealth in case of burglary=200000-50000=$150000
Utility in case of burglary=U(150000)=2*(150000)0.5=774.5967 utils
Probability of no burglary=1-p=1-0.02=0.98
Wealth in case of no burglary=$200000
Utility in case of no burglary=U(200000)=2*(200000)0.5=894.4272 utils
Expected utility=p*U(150000)+(1-p)*U(200000)=0.02*774.5967+0.98*894.4272=892.0306 utils
Actuarially fair price for the insurance for full coverage=Probability of loss*Loss amount
Actuarially fair price for the insurance for full coverage=0.02*50000=$1000
Let he buys the insurance at a price of $1000
Utility in case of burglary=U(200000-50000+50000-1000)=U(199000)=2*1990000.5=892.1883 utils
Utility in case of no burglary=U(200000-1000)=U(199000)=2*1990000.5=892.1883 utils
Expected utility in case he buys insurance=p*U(199000)+(1-p)*U(199000)=U(199000)=892.1883
We can observe expected utility has increased from 892.0306 utils to 892.1883 if agent buys insurance at actuarially fair premium. He should buy the insurance at this price.
Let X be Maximum willingness to pay for insurance.
So, U(200000-X)=Expected utility
2*(200000-X)0.5=892.0306
(200000-X)=(892.0306/2)2=198929.65
X=200000-198929.65=$1070.35
Maximum willingness to pay for insurance=$1070.35