Mathematical Economics - Expected Utility

Probability Level 3

Mr. X had just purchased a new car for $10,000. His utility from the car can be expressed as :-

U(Car)=log(Car) where Car is the value of the car. For simplicity, we assume the purchase price=value of car.

In general, new car owners have 25% chance of getting into an accident resulting in an average loss of $1,000. Therefore, they will purchase insurance to protect the loss of utility .

Mr. X would like to guard against this loss by purchasing insurance. Determine if Mr. X will/will not purchase insurance if the probability of meeting an accident increases from 25% to 30%.

My head hurts No Let me toss a coin to decide Yes

1 solution

Venture Hi
Apr 8, 2014

Mr. X utility in buying a new car is: U(Car) = log(10000) = 4

At a 25% probability of getting into an accident with a potential loss of $1000, the expected utility is now:

     E(U) = 0.25log(Car - 1000) + 0.75log(Car)

          = 0.25log(10000-1000) + 0.75log(10000)

          = 3.988560627

The actuarially fair premium to guard against this loss is calculated as:

0.25(1000) + 0.75(0) = $250

With this insurance, Mr. X expected utility is now calculated as:

     E(U) = log(Car - 250)

          = log(10000 - 250)

          = 3.9890004616

In comparing, Mr. X expected utilities without insurance versus with this $250 insurance, we find that his utility with insurance is 3.9890004616 > 3.988560627 (without insurance). Therefore, it is worthwhile for Mr. X to purchase the $250 insurance to guard against the 25% probability of getting into an accident.

Now if the probability of getting into an accident is now 30% instead of 25%, then the actuarially fair premium to guard against such loss is now:

0.3(1000) + 0.7(0) = $300

If the insurance now costs $300, then the new expected utility with a probability of 30% is calculated as:

     E(U) = log(Car - 300)

          = log(10000 - 300)

          = 3.986771734

Since the utility with insurance at $300 and 30% probability - 3.986771734 < 3.988560627 (without insurance). Therefore, it is NOT worthwhile for Mr. X to purchase the $300 insurance to guard against the 30% probability of getting into an accident.

Mathematical Economics - Expected Utility

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