Insurance doesn't give full payout
The value x, of an individualís wealth, excluding the value of his car is £35,000.
His utility function is
U(x) = ln (x + 300)
There is a 2% probability of theft or accident reducing the value of his car, worth £15,000, to a scrap value of zero.
An insurance company is considering insurance for the car up to a maximum payout of £5,000 at a premium of £200. Show that this insurance arrangement would be acceptable both to the company and the individual.
The company's utility function is,
U(x) = 2ln(x),
where x is the companyís assets, £2,000,000.
I can work out that the insurance company would accept the deal, however I don't know how to work it out for the individual.
P: E[U(a-x)] = U(x - P)
as E[U(a-x)] is the part without insurance i assume i have to do something with the U(x - P) part
If it repayed the full amount i would just do
ln(50,000+300 - P)
Do i have to incorporate the fact that 98% of the time i won't use insurance, however if the insurance is used 2% of the time there is still a loss of 10,000?
my logical step is to do
E[U(a-x)]=10.81867791=(ln(50000+300 - P) * 0.98) + (ln(50000+300-10000 - P)
However this works P out to be 154.98846, which is less than 200, so must be wrong as the question states that the individual accepts the 200 offer.
Please help :)