1. ## Utility question

This problem requires the von Neumann-Morgenstern theorem to solve.

Let L = {A, B, C, D} be a set of lotteries. You are indifferent between receiving A for sure and a lottery that gives you B with probability 0.9 and C with probability 0.1. You are also indifferent between receiving A for sure and a lottery that gives you B with probability 0.6 and D with probability 0.4. You preferences satisfy the vonn Neumann-Morgenstern axioms.

(i) What do you prefer most, C or D?
(ii) Calculate the (relative) difference in utility between B and C, and between B and D.
(iii) If we stipulate that your utility of B is 1 and your utility of C is 0, what are your utilities of A and D?

I did the problem like this:

For (i), it was obvious that D is preferred to D, because in the lottery with D I was willing to accept a lower chance of receiving B than I was in the lottery with C.

u(A) = 0.9u(B) + 0.1u(C)

u(A) = 0.6u(B) + 0.4u(D)

Therefore,

0.9u(B) + 0.1u(C) = 0.6u(B) + 0.4u(D)

Subtract both sides by 0.6u(B) and get

0.3u(B) + 0.1u(C) = 0.4u(D)

Let u(B) = 1, and u(C) = 0. Therefore, by the above equation:

0.3 = 0.4u(D)

Divide both sides by 0.4 and get

0.75 = u(D)

Now for the answer to (ii), if u(B) = 1, u(C) = 0, and u(D) = 0.75, then

u(B) - u(c) = 1
and
u(B) - u(d) = 0.25

Therefore, the difference between B and C is 4 times the difference between B and D.

Finally, for (iii),

Since u(A) = 0.9u(B) + 0.1u(C)

u(A) = 0.9

We can confirm this by looking at the other equation too:

u(A) = 0.6u(B) + 0.4u(D) = 0.6 + 0.4*0.75 = 0.6 + 0.3 = 0.9

So, u(A) = 0.9, and u(D) = 0.75.

I looked at the answer in the book, and this is exactly what the author put. But then I looked at the errata list for the book on his website (http://www.martinpeterson.org/errata.pdf)

and it says the following:

p. 115,... in the solution to 5.7 all that can be concluded is that 0.9B + 0.1C = 0.6B + 0.4D.

Did I make the same mistake as the author? If so, what is the mistake?

2. your answer to part ii is not plausible because you relied on the information given in a subsequent part of the question. That information was for a special case of the problem where U(B) = 1 and U(C)=0. There is no reason to suppose that your ratio holds in general.

try recalculating your ratio for different values of u(B) and U(C). you will probably find the ratio is not constant