6. You are indifferent between receiving A for sure and a lottery that gives you

Question

6. You are indifferent between receiving A for sure and a lottery that gives you B with a probability of 0.9 and C with a probability of 0.1. You are also indifferent between receiving A for sure and a lottery that gives you B with a probability of 0.6 and d with a probability of 0.4. Finally, you prefer B to A and A to D. All of your preferences satisfy the von Neumann-Morgenstern axioms a) What do you prefer most, C or D? b) Calculate the relative difference in utility between B and C, and between B and D. c) If we stipulate that your utility of B is 1 and your utility of C is 0, what are then your utilities of A and D?

Explanation / Answer

(a) D is preferred to C, because in the lottery with D I am 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)
(b) 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.

(c) 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.

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