High School

An agent is considering a risky option that will pay $484 with a probability of 0.6 and $1,681 with a probability of 0.4. If the agent has an expected utility function [tex]u(x) = x^{0.5}[/tex], what is the expected utility of this option?

Options:
A. 29.6
B. 32.3
C. 36.6
D. 39.0

Answer :

To determine the expected utility of the given option, we'll make use of the provided utility function [tex]u(x) = x^{0.5}[/tex] and the probabilities for each outcome.

The concept of expected utility involves calculating the sum of the utilities of each possible outcome, each weighted by its probability.

Given:

  • Option 1 pays $484 with probability 0.6
  • Option 2 pays $1,681 with probability 0.4

First, we calculate the utility for each outcome:

  1. Utility for $484:

    [tex]u(484) = 484^{0.5}[/tex]

    [tex]u(484) = 22[/tex]

  2. Utility for $1,681:

    [tex]u(1,681) = 1,681^{0.5}[/tex]

    [tex]u(1,681) = 41[/tex]

Next, we calculate the expected utility using the formula:

[tex]\text{Expected Utility} = (\text{Probability of Outcome 1} \times \text{Utility of Outcome 1}) + (\text{Probability of Outcome 2} \times \text{Utility of Outcome 2})[/tex]

[tex]= (0.6 \times 22) + (0.4 \times 41)[/tex]

[tex]= 13.2 + 16.4[/tex]

[tex]= 29.6[/tex]

The expected utility of the option is 29.6.

Therefore, the correct multiple-choice answer is 29.6.

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