Oskar is an expected utility maximiser with utility function
𝑢
(
𝑐
)
=
𝑐
. A gamble that gives him consumption of $9 with probability 1/2, and consumption of $25 with probability 1/2 is exactly as good for Oskar as being sure to have an income of:

Question

Oskar is an expected utility maximiser with utility function 
𝑢
(
𝑐
)
=
𝑐
. A gamble that  gives him consumption of $9 with probability 1/2, and consumption of  $25 with probability 1/2 is exactly as good for Oskar as being sure to have an income of:

BlackTom AI · Accepted Answer

We start by restating the scenario to ensure clarity: Oskar is an expected utility maximiser with u(c) = c, so his utility is simply consumption itself. The gamble offers $9 with probability 1/2 and $25 with probability 1/2. The task is to find the certain amount that would give the same utility as this lottery.

Option by option analysis:
- Option 4: $4. Here, the certain amount would yield utility u(4) = 4. The lottery has expected payoff E[c] = 0.5×9 + 0.5×25 = 4.5 + 12.5 = 17. Since 4 ≠ 17, this option does not match the certain amount that would be as good as the gamble. It undervalues the risk and the potential payoff.

- Option 17: $17. For a linear utility function u(c) = c, the expected utility of the gamble is E[u(c)] = 0.5×u(9) + 0.5×u(25) = 0.5×9 + 0.5×25 = 4.5 + 12.5 = 17. So the certain amount with utility 17 is u(17) = 17, which matches the lottery’s expected utility precisely. This makes the gamble and the sure income equivalent for Oskar under risk neutrality.

- Option 12.5: $12.50. The utility would be u(12.5) = 12.5. Since the lottery’s expected utility is 17, this amount provides a lower level of satisfaction than the gamble, so it is not equivalent.

- Option 16: $16. Here, u(16) = 16, which again is less than the gamble’s expected utility of 17, so this does not equate to the lottery.

- Option 9: $9. The sure amount has utility 9, which is also below the lottery’s expected utility of 17, making it an inappropriate match for equivalence in this risk-neutral setting.

In summary, because the utility function is linear, the equivalence point between the gamble and a sure amount is set by the lottery’s expected payoff, which is 17. Each option was evaluated against that benchmark to determine whether it yields the same utility as the gamble.

Oskar is an expected utility maximiser with utility function 𝑢 ( 𝑐 ) = 𝑐 . A gamble that gives him consumption of $9 with probability 1/2, and consumption of $25 with probability 1/2 is exactly as good for Oskar as being sure to have an income of:

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