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

Question

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

The decision maker now has $15,000 and two possible decisions. For decision 1, she loses $2,000 for certain. For decision 2, she loses $5,000 with probability 0.75, but gains $10,000 with probability 0.25. Which decision maximizes the expected utility of her net wealth?

a. She should choose option 2. Her expected utility is 9.43.

b. She should choose option 1. Her expected utility is 9.47.

c. She should choose option 1. Her expected utility is 9.55.

d. She is indifferent between the two choices.

Explanation / Answer

Utility as a function of wealth, x, is given by U(x) = ln x

For decision 1, she loses $2,000 for certain. So her wealth x becomes ($15,000-$2,000) = $13,000

Hence, Expected utility for decision 1 is, U(13000) = ln (13000) = 9.47

Now, for decision 2, she loses $5,000 with probability 0.75, but gains $10,000 with probability 0.25.

So, Expected wealth = (15,000-5,000)*0.75 + (15,000+10,000)*0.25 = $13,750

Hence, Expected utility for decision 2 is, U(13750) = ln (13750) = 9.53

Therefore, she should choose option 2. Her expected utility is 9.53. That means option (a) is correct.

Note: I think there is a typo in option (a). 9.53 is wrongly printed as 9.43

Thanks :)

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