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There's a question in my ECON notes that I don't understand, any help would be greatly appreciated. Here are the definitions used about VNM Utility Theory. The question is posted after the definitions.

In this question, I'm really confused about why U(4) = 0.6. Shouldn't U(4) = 0.5 because 10(0.5) -2(0.5) = 4?

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Utility of the expected value and the expected utility of a random value are not the same things.

The usual example I give, is suppose you can enter a lottery where there is a 50% chance of winning \$10 billion. Would you forfeit this chance for getting a certain \$1 billion? Most people would, as the first \$1 billion has much more effect on their lives than the later ones, even though the expected value of the lottery is \$5 billion. This means that the expected utility they assign to the lottery is lower than the utility they assign to the expected value, as they would even prefer a certain sum lower than the expected value to the uncertain lottery.

A simple mathematical example in the expected utility framework is provided here.

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  • $\begingroup$ Thank you so much for your response! I understand your example regarding 10 billion vs 1 billion. However, I don't quite yet how to get U(4) = 0.6 from the above definition. if that was a problem on a test, how would you solve it mathematically? Is it even possible? $\endgroup$ Sep 15, 2022 at 19:57
  • $\begingroup$ @DracoBlitz I had to reread the question several times. I am not quite sure what your quiz is going for, but it seems like asking the question "Would you prefer to get 4 dollars for sure or to participate in a lottery where you get 10 or -2 dollars with 60\%-40\% probability respectively?" and getting the answer "I am indifferent" would elicit this utility. $\endgroup$
    – Giskard
    Sep 16, 2022 at 5:45

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