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September 2022 - Q1

B

Bernadette Pieterse

Active Member
Hello,

For question 1 the max price paid is calculated as
E(U(w-P+X)) = U(w)
In the notes the max “premium” paid for insurance is
E(U(w-X)) = U(w-P)

Why do we calculate it differently in this question? It seems like the same concept as the insurance premium paid.

Thanks in advance.
 
This is the kind of problem you can get if you try to apply a set equation as THE method for doing something, when it’s really the logic behind where the equation came from that we need to apply.

I have a washing machine and I don’t have insurance. My total wealth is £1000. I am choosing not to buy insurance against this washing machine breaking, which means I am facing a random loss X over the next year. So my happiness is

U(1000 – X)

Now, I could get some insurance, which (ignoring excesses and the infiinte hassle involved in all forms of insurance) would eliminate the uncertainty of the position. My guaranteed level of wealth would be 1000 – p, and my guranateed level of happiness is…

U(1000 – p)

And my equation is E[U(1000 – X)] = U(1000 – p)

Note the LHS is only my expected happiness (crucially not my happiness of my expected position!) since I don’t know what X is going to be.

Now, let’s forget about washing machines and think about the lottery.

Unlike washing machines, I am not facing any random loss (or gain) since I have not entered the lottery yet. So my guaranteed level of happiness at the moment is U(1000).

If I enter the lottery, I will lose the price of the ticket t but I will then be facing a random win. I will have 1000 – t + X, which gives me a happiness of U(1000 – t + X).

And my equation is E[U(1000 – t + X)] = U(1000)

Again, note the LHS is my expected happiness since I don’t know what X is going to be.

So, when setting up the equation, try to think about the final position you will be in in the two different scenarios – you are trying to equate the happiness derived from each of those final positions.

I think this will work much better than taking “the” equation from “the” notes and trying to work out which bit is which,

Good luck!
John
 
Hi John,

Thank you for that explanation!

Is it correct to say that with the washing machine - I am limiting my loss to P instead of X?

And with the lottery there is no “Loss” so it should not be included in my utility function only in the expected utility?

Many Thanks!
 
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