Thread: Insurance doesn't give full payout

The value x, of an individual’s 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 company’s 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

I'm not quite seeing it, either. If you leave out the 35,000 not at risk, the result is different.