Question

In: Economics

A individual has a utility function u(c) = √ c, where c is the individual’s consumption....

A individual has a utility function u(c) = √ c, where c is the individual’s consumption. (The individual consumes his entire wealth.) The individual’s wealth is $40,000 per year. However, there is a 2% chance that he will be involved in a catastrophic accident that will cost him $30,000.

PLEASE SHOW WORK

a. What is the individual’s utility from consumption if there is no accident? What is his utility if there is an accident? What is his expected utility?

b. What is the actuarially fair insurance premium if the individual purchases an insurance contract that pays out $30,000 in the event of an accident? What is his utility if there is no accident, and he purchases the actuarially fair insurance? What is his utility in the event of an accident if he purchases the actuarially fair insurance?

c. What is the most that he would be willing to pay for insurance given his utility function?

Solutions

Expert Solution

We are given

U(C)=C^0.5

a)
In case of accident=C=W=40000-30000=$10000

Utility in case of accident=U(10000)=10000^0.5=100 utils

In case of no accident=C=W=$40000

Utility in case of no accident=U(40000)=40000^0.5=200 utils

Probability of no accident=1-p=1-0.02=0.98

Probability of accident=p=0.02

Expected Utility=p*U(10000)+(1-p)*U(40000)=0.02*100+0.98*200=198 utils

Actuarially fair premium=probability of accident*loss in case of accident=0.02*30000=$600

Wealth in case of no accident=w=C=40000-600=$39400

So, utility in case of no accident and he purchases actuarially fair policy

U(39400)=39400^0.5=198.4943

Wealth in case of accident=w=C=10000-600+30000=$39400

So, utility in case of accident and he purchases actuarially fair policy

U(39400)=39400^0.5=198.4943

Let he is willing to pay a maximum sum of X for insurance

Expected utility=U(40000-X)=(40000-X)^0.5

Expected utility in case of insurance should be equal to utility in case of no insurance. So,

(40000-X)^0.5=198

40000-X=39204

X=40000-39204=$796

Agent would be willing to pay a maximum sum of $796 for insurance.

Rahul Sunny answered 1 year ago

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