Your current disposable income is $10,000.There is a 10% chance you will get in a serious car accident,incurring damage of $1,900.(There is a 90% chance that nothing will happen.) Your utility function is ,where I is income.What is the most you would be willing to pay for this policy (rather than no insurance)?

Question

Your current disposable income is $10,000.There is a 10% chance you will get in a serious car accident,incurring damage of $1,900.(There is a 90% chance that nothing will happen.) Your utility function is  ,where I is income.What is the most you would be willing to pay for this policy (rather than no insurance)?

Quizplus · Accepted Answer

To calculate the expected utility without insurance, we can use the formula:
EU(no insurance) = 0.9*u(10000) + 0.1*u(8100)
where u(10000) = 11,000 and u(8100) = 9,390. Therefore, EU(no insurance) = 0.9*11,000 + 0.1*9,390 = 10,791.

To calculate the expected utility with insurance, we can use the formula:
EU(with insurance) = u(10000 - x) - p*u(10000 - x - 1900) - (1 - p)*u(10000 - x)
where x is the amount paid for insurance and p is the probability of getting in a serious car accident (p = 0.1 in this case). We want to find the value of x that maximizes EU(with insurance).

We can use calculus to find the optimal value of x. Taking the derivative of EU(with insurance) with respect to x and setting it equal to zero, we get:
-190p*u'(10000 - x - 1900) + u'(10000 - x) = 0
Simplifying, we get:
19u'(8100 - x) = u'(10000 - x)
Dividing by u'(8100 - x), we get:
19 = u'(10000 - x)/u'(8100 - x)
Using the given utility function, we can calculate that u'(I) = 11/I^2. Therefore, we can substitute and solve for x:
19 = (11/(10000 - x)^2)/(11/(8100 - x)^2)
Simplifying, we get:
19 = (8100 - x)^2/(10000 - x)^2
Solving for x, we get:
x = 199.33

Therefore, the most you would be willing to pay for this policy (rather than no insurance) is $199.33, which is closest to option C ($199).

Your Current Disposable Income Is $10,000