Answer to Question #227457 in Macroeconomics for Mohamed

Question

Suppose you have a house worth $200,000 (wealth). Your utility of wealth is given by U(w) = ln(w). There is a small chance that a fire will damage your house causing a loss of $75,000. You estimate there is a 2% chance of fire.

a) What is your expected wealth?

b) What is your expected utility from owning the house?

c) Suppose you can add a fire detection/prevention system to your house. This would reduce the chance of a bad event to 0 but it would cost you $C to install. What is the most you are willing to pay for the security system? (Here is an identity you will find useful 

Answer 1

Given,

Utility function:

$U(w)=Ln(w)$

(a). The formula for expected wealth is given below:

Expected wealth=probability of fire×wealth in case of fire+probability of

 no fire×wealth in case of no fire

$Expected \space wealth=2100×(200,000−75,000)+\frac{98}{100}×200,000\\Expected\space wealth=2,500+196,000\\Expected \space wealth=\$198,500$

The expected wealth is $198,500.

(b). The value of expected utility is given below:

$Expected\space utility=\frac{2}{100}ln(125,000)+\frac{98}{100}ln(200,000)\\Expected\space utility=\frac{2}{100}×11.7361+\frac{98}{100}×12.2061\\Expected \space utility=0.2347+11.9620\\Expected\space utility=12.1967$

The value of expected utility is 12.1967 approx.

(c). The maximum amount that can be paid is the difference between the certainty equivalent (CE) and the current value of wealth.

$U(w)=expected\space utility\\ln(w)=12.1967\\w=e^{12.1967}w=198,134.2282\\Maximum \space payment=200,000−198,134.2282\\Maximum \space payment=\$1865.77 approx$

The most that can be paid is $1865.77 approx.