The value x, of anindividual’s wealth, excluding the value of his car is£35,000.

His utility function is

U(x) = ln (x + 300)

There is a2%probability of theft or accident reducing the value of his car, worth£15,000, to a scrapvalue of zero.

An insurance company is considering insurance for the car up to amaximum payout of £5,000at apremium 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’sassets, £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