Question

Suppose my utility function for asset position ? is given by ?(?) = ( ? 1000) 2 . If I have $ 17,000 and I am considering the following two lotteries L1: With probability 1, I lose $1000. L2: With probability .78, I gain $0. With probability .22, I lose $10,000 a) Draw the lotteries and determine which lottery I select based in the utility value b) Determine which lottery I select based in the expected value

Answer 1

Given the utility function: U (x) = (x/1000)^2, where x would be the payoff

we will calculate utility value for both the cases

L1: the payoff is (-1000) with a probability of 1. That means it is sure to lose $1000 on it.

L2: the payoff is given as a 22% chance of losing $10000, i.e. (-10000) with the probability of 0.22 and a 78% chance of gaining $0, i.e. 0 with the probability of 0.78.

a).

Therefore Probabiltis are given as:: P(-1000) =1,

P(-10000)= 0.22,

p(0)= 0.78

U(L1) = P(-1000)*U(-1000) U(L2)

= P(-10000)*U(-10000) + P(0)*U(0)

= 1 * (-1000/1000)^2

= 0.22*(-10000/1000)^2 + 0.78 (0/1000)^2

= 1*(-1)^2

= 0.22*(100) + 0

U(L1) = 1 U(L2) = 22

U(L2)>U(L1)

That means the utility of L2 is higher for me. Therfore I would go with lottery L2 as per the utility value.

b).

The expected value is simply calculated by multiplying payoffs with their respective probabilities.

EV (L1) = (-1000)* p(-1000)

= (-1000) * 1

EV (L1) = (-1000)

EV (L2)= (-10000)*p(-10000) + 0 * P(0)

= (-10000)*0.22 + 0* 0.78

EV (L2) = (-2200)

EV(L1) > EV (L2)

THus as the expected value theorem states, choice with higher expected value should be chosen.

Thus as per elected value, Lottery L1 should be chosen.

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