Suppose that a decision maker’s utility as a function of her wealth, x , is give

Question

Suppose that a decision maker’s utility as a function of her wealth, x, is given by U(x) = ln (2x) (where ln is the natural logarithm).

The decision maker now has $10,000 and two possible decisions. For Alternative 1, she loses $1000 for certain. For Alternative 2, she loses $0 with probability 0.9 and loses $5,000 with probability 0.1. Which alternative should she choose and what is her expected utility (rounded to 2 decimals)?

a) she is indifferent between both alternatives

b) alternative 2, expected utility is 9.83

c) alternative 1, expected utility is 9.10

d) alternative 1, expected utility is 9.80

Explanation / Answer

For alternative 1, the expected utility is computed as:

E(U1) = Ln( 2*(10000-1000)) note that as there is a loss of 1000 for certain therefore x = -1000

E(U1) = Ln( 2*(9000)) = Ln(18000) = 9.7981

Therefore the expected utility of first alternative is 9.7981

Now for our second alternative, the expected utility is computed as:

E(U2) = 0.9* Ln( 2*(10000-0)) + 0.1*Ln( 2*(10000-5000))

E(U2) = 0.9* Ln(20000) + 0.1*Ln(10000) = 8.9131 + 0.9210 = 9.8341

Therefore the expected utility of second alternative is 9.8341

Therefore she should go with alternative 2 with an expected utility of 9.83

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