# Thread: Utility functions and insurance

1. ## Utility functions and insurance

Hi, I think I've done this right, however, i need to know if I have as there is no point me thinking I am right if I'm not, obviously.

Dealer:
U(x) = 2ln(x)
Initial Wealth = 500,000
Books = 10,000
Total Wealth = 510,000
Pr(complete loss of books) = 0.02
Pr(no loss) = 0.98

Insurer:

U(x) = ln(x + 200,000)
Assets = 2,000,000

1 - The insurer initially offers the dealer a quote of £250. Find out if the dealer would accept this offer.

2 - Find the maximum offer the dealer would accept.

3 - The dealer offers to pay £201, find out if the insurer would accept this lower price.

1)

P:E[U(a-x)]=U(a-P)

E[U(a-x)] = (2ln(510000)*0.98)+(2ln(500000)*0.02)
E[U(a-x)] = 26.2835

26.2835 = 2ln(510000-P)
exp(13.14177) = 510000 - P
P = £201.948

As the initial offer is greater than P the initial offer would be rejected.

2)

The maximum quote the dealer would accept is £201.94 (rounding down as up would be more than the maximum)

3)

Insurer without insuring
U(x) = ln(2,000,000 + 200,000)
U(x) = ln(2,200,000)
U(x) = 14.60396792

Insurer insuring
U(x) = (ln(2,200,000 + 201) * 0.98) + (ln(2,200,000 + 201 - 10,000) * 0.02)
U(x) = 14.60396817

Utility function with insuring is greater than not insuring so the insurance company would accept the dealers quote of £201.

Does this method seem to be the right one? Are my conclusions correct?

2. Originally Posted by Rapid_W
Hi, I think I've done this right, however, i need to know if I have as there is no point me thinking I am right if I'm not, obviously.

Dealer:
U(x) = 2ln(x)
Initial Wealth = 500,000
Books = 10,000
Total Wealth = 510,000
Pr(complete loss of books) = 0.02
Pr(no loss) = 0.98

Insurer:
U(x) = ln(x + 200,000)
Assets = 2,000,000

1 - The insurer initially offers the dealer a quote of £250. Find out if the dealer would accept this offer.

2 - Find the maximum offer the dealer would accept.

3 - The dealer offers to pay £201, find out if the insurer would accept this lower price.

1)

P:E[U(a-x)]=U(a-P)

E[U(a-x)] = (2ln(510000)*0.98)+(2ln(500000)*0.02)
E[U(a-x)] = 26.2835

26.2835 = 2ln(510000-P)
exp(13.14177) = 510000 - P
P = £201.948

As the initial offer is greater than P the initial offer would be rejected.

2)

The maximum quote the dealer would accept is £201.94 (rounding down as up would be more than the maximum)

3)

Insurer without insuring
U(x) = ln(2,000,000 + 200,000)
U(x) = ln(2,200,000)
U(x) = 14.60396792

Insurer insuring
U(x) = (ln(2,200,000 + 201) * 0.98) + (ln(2,200,000 + 201 - 10,000) * 0.02)
U(x) = 14.60396817

Utility function with insuring is greater than not insuring so the insurance company would accept the dealers quote of £201.

Does this method seem to be the right one? Are my conclusions correct?
You leave too much for your reader to deduce, try to explain what you are doing or trying to do at each major stage of your solution. (In particlar don't use the same symbol for the dealers and insurers utility functions).

CB