Suppose that your wealth is $250,000. You buy a $200,000 house and invest the remainder in a risk-free asset paying an annual interest rate of 6 percent. There is a probability of 0.001 that your house will burn to the ground and its value will be reduced to zero. With a log utility of end-of-year wealth, how much would you be willing to pay for insurance (at the beginning of the year)? Assume that if the house does not burn down, its end-of-year value still will be $200,000.
To find out how much you would be willing to pay for insurance, you need to calculate the expected utility of your end-of-year wealth with and without insurance, and then set the utilities equal and solve for the insurance premium.
Let x be the insurance premium. Without insurance, your end-of-year wealth would be:
1. $200,000 (house) + $50,000 * 1.06 - $50,000 (investment return) with probability 0.999 (house doesn't burn down)
2. $50,000 * 1.06 - $50,000 (investment return) with probability 0.001 (house burns down)
With insurance, your end-of-year wealth would be:
1. $200,000 (house value) - x (insurance premium) + $50,000 * 1.06 - $50,000 (investment return) with probability 0.999 (house doesn't burn down)
2. $200,000 (insurance payout) - x (insurance premium) + $50,000 * 1.06 - $50,000 (investment return) with probability 0.001 (house burns down)
Now, compute the expected utilities for the two scenarios (without and with insurance) using the log utility function:
Expected utility without insurance:
E(U) = 0.999 * log($200,000 + $50,000 * 1.06 - $50,000) + 0.001 * log($50,000 * 1.06 - $50,000)
Expected utility with insurance:
E(U) = 0.999 * log($200,000 - x + $50,000 * 1.06 - $50,000) + 0.001 * log($200,000 - x + $50,000 * 1.06 - $50,000)
Set the two expected utilities equal to each other and solve for x:
0.999 * log($200,000 + $50,000 * 1.06 - $50,000) + 0.001 * log($50,000 * 1.06 - $50,000) = 0.999 * log($200,000 - x + $50,000 * 1.06 - $50,000) + 0.001 * log($200,000 - x + $50,000 * 1.06 - $50,000)
After performing the calculations, we find that the insurance premium x is approximately $2,064.09. So, you would be willing to pay up to $2,064.09 for insurance at the beginning of the year.
To determine how much you would be willing to pay for insurance, we need to calculate your expected utility with and without insurance.
Without insurance:
1. Calculate the probability of the house burning down: P(burn) = 0.001
2. Calculate the probability of the house not burning down: P(not burn) = 1 - P(burn) = 1 - 0.001 = 0.999
3. Calculate the wealth at the end of the year if the house doesn't burn down: Wealth(not burn) = $200,000 (house value) + $50,000 (interest earned) = $250,000
With insurance:
4. Calculate the cost of insurance (premium): This is the amount you are willing to pay for insurance at the beginning of the year.
5. Calculate the expected wealth at the end of the year if the house burns down: Wealth(burn) = $0 (house value)
6. Calculate the expected wealth at the end of the year taking into account the probability of the house burning down:
Wealth(expected) = P(burn) * Wealth(burn) + P(not burn) * Wealth(not burn)
= 0.001 * $0 + 0.999 * $250,000
= $249,750
7. Calculate the expected log utility with insurance:
U(expected) = log(Wealth(expected))
8. Calculate the expected log utility without insurance:
U(not insured) = log($250,000)
Now, to determine the maximum amount you would be willing to pay for the insurance, compare the two expected log utilities and calculate the insurance premium that makes them equal:
U(expected) = U(not insured)
log($249,750) = log($250,000)
Solving for the insurance premium:
$249,750 = $250,000 - Premium
Premium = $250,000 - $249,750
Premium = $250
Therefore, you would be willing to pay up to $250 for the insurance premium.
To calculate how much you would be willing to pay for insurance, we need to consider the potential outcomes and their probabilities.
1. If the house does not burn down:
- The end-of-year value of your wealth will be $200,000 (the value of the house).
- The remainder of your wealth that was invested in the risk-free asset will grow by 6 percent.
2. If the house burns down:
- The end-of-year value of your wealth will be zero (as the house will be reduced to zero value).
Since you have a log utility of end-of-year wealth, your utility function can be represented as U(W) = ln(W), where W is your end-of-year wealth.
Now let's calculate the expected utility with and without insurance:
1. Without Insurance:
The probability of the house burning down is 0.001. So, the probability of it not burning down is 1 - 0.001 = 0.999.
Without insurance, your expected utility can be calculated as:
EU_NoInsurance = 0.999 * ln($200,000 + 0.06 * ($250,000 - $200,000)) + 0.001 * ln($0)
2. With Insurance:
Let's assume the cost of insurance is X dollars. If you buy insurance, your end-of-year wealth will be one of the following:
- $200,000 - X if the house burns down (probability = 0.001)
- $200,000 - X + 0.06 * ($250,000 - $200,000) if the house does not burn down (probability = 0.999)
With insurance, your expected utility can be calculated as:
EU_WithInsurance = 0.999 * ln($200,000 - X + 0.06 * ($250,000 - $200,000)) + 0.001 * ln($200,000 - X)
To find the amount you would be willing to pay for insurance, you need to find X that maximizes your expected utility. This can be done by setting EU_NoInsurance equal to EU_WithInsurance and solving for X.