Buying and selling prices for risky investments obviously arc related to CEs. Th
ID: 3226617 • Letter: B
Question
Buying and selling prices for risky investments obviously arc related to CEs. This problem, however, shows that the prices depend on exactly what is owned in the first place! Suppose that Peter Brown's utility for total wealth (A) can be represented by the utility function U(A) = In(A). He currently has $1, 000 in cash. A business deal of interest to him yields a reward of $100 with probability 0.5 and $0 with probability 0.5. a. If he owns this business deal in addition to the $1, 000, what is the smallest amount for which he would sell the deal? b. Suppose he does not own the deal. What equation must be solved to find the greatest amount he would be willing to pay for the deal? c. For part b, it turns our that the most he would pay is $48.75m, which is not exactly the same as the answer in part a. Can you explain why the amounts are different? d. (Extra credit for algebra hotshots.) Solve your equation in part b to verify the answer ($48.75) given in part c.Explanation / Answer
a. If he owns this deal in addition to $ 1000, what is the smallest amount for which he would seal the deal. Let say that selling value is X so
U ( 1100- X) < 0.5 * U( 1000 + 100) + 0.5 * U * ( 1000+0)
U ( 1100-X) < 0.5 [U(1100) + U(1000)]
ln (1100-X) < 0.5 [ln 1100 + ln 1000]
X = $ 51.25
(b) If he does not own the deal in addition to $ 1000, what is the greatest amount for which he would seal the deal. Let say that selling value is X so
U ( 1000 + X) > 0.5 * U( 1000 + 100) + 0.5 * U * ( 1000+0)
U ( 1100-X) > 0.5 [U(1100) + U(1000)]
ln (1000 +X) > 0.5 [ln 1100 + ln 1000]
ln ( 1000 + X) > 0.5 [ 7.00 + 6.9077]
X = $ 48.75
(c) The amounts are different because of the nature of utility function. The utility function of wealth is a slowly increasing function that means utility of wealth increase slowly with increase in wealth. the first penny have more utility than the last penny. So that's why he is willing to pay less for the deal in comparison to selling the deal in high price.
(d) Yes answer is $ 48.75 as i have solved earlier.
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