Question

Bob has utility over hammers (h) and dollars (m). U = v(3ch − 3rh) + v(cd − rd) where v(x) = x for x ≥ 0 and v(x) = 2x for x ≤ 0.

(a) Assume that Bob’s reference point is 0 hammers and 0 dollars. For each of the following choices, show Bob’s expected utility for each option, and state which choice he would make.

i. Would Bob choose Option A: 50% chance to win 16 hammers and 50% chance to win 4 hammers or Option B: definitely winning 8 hammers?

ii. Would Bob choose Option A: 50% chance to lose 16 hammers and 50% chance to lose 4 hammers or Option B: definitely losing 12 hammers?

iii. Would Bob choose Option A: 50% chance to gain 8 hammers and 50% chance to lose 4 hammers or Option B: gain 1 hammer.

Answer 1

i. Bob would choose Option A, with a 50% chance to win 16 hammers and 50% chance to win 4 hammers, as it offers higher expected utility.

ii. Bob would choose Option A, with a 50% chance to lose 16 hammers and 50% chance to lose 4 hammers, as it provides a higher expected utility than definitely losing 12 hammers in Option B.

iii. Bob would choose Option B, gaining 1 hammer, as it yields a higher expected utility compared to the 50% chance of gaining 8 hammers and 50% chance of losing 4 hammers in Option A.

Step-by-step explanation:

i. Calculating the expected utility for Option A:
`\[U_(A) = (1)/(2)v(3(16)h - 3(0)h) + (1)/(2)v(16d - 0d) = (1)/(2)v(48h) + (1)/(2)v(16d)\]\[= (1)/(2)(48h) + (1)/(2)(32d) = 24h + 16d\]Calculating the expected utility for Option B: \[U_(B) = v(3(8)h - 3(0)h) + v(8d - 0d) = v(24h) + v(8d)\]\[= (24h) + (16d) = 24h + 16d\]`

Since both options yield the same expected utility, Bob would choose Option A due to the variability it offers.

ii. For Option A:
`\[U_(A) = (1)/(2)v(3(-16)h - 3(0)h) + (1)/(2)v(-16d - 0d) = (1)/(2)v(-48h) + (1)/(2)v(-16d)\]\[= (1)/(2)(-96h) + (1)/(2)(-32d) = -48h - 16d\]For Option B: \[U_(B) = v(3(-12)h - 3(0)h) + v(-12d - 0d) = v(-36h) + v(-12d)\]\[= (-72h) + (-24d) = -72h - 24d\]`

Option A has a higher expected utility, so Bob would choose Option A.

iii. For Option A:
`\[U_(A) = (1)/(2)v(3(8)h - 3(0)h) + (1)/(2)v(8d - 0d) = (1)/(2)v(24h) + (1)/(2)v(8d)\]\[= (1)/(2)(24h) + (1)/(2)(16d) = 12h + 8d\]`

For Option B:
`\[U_(B) = v(3(1)h - 3(0)h) + v(1d - 0d) = v(3h) + v(1d)\]\[= (3h) + (2d) = 3h + 2d\]`

Option B has a higher expected utility, so Bob would choose Option B.