Joe’s wealth is $100 and he is an expected utility maximizer with utility function
𝑢
(
𝑐
)
=
𝑐
. Joe is afraid of oversleeping his economics exam. He figures there is only a 1 in 10 chance that he will, but if he does, it will cost him $100 in fees to the university for taking an exam late. Joe’s neighbour, Mary, never oversleeps. She offers to wake him one hour before the test, but he must pay her for this service. What is the most that Joe would be willing to pay for this wake-up service?

Question

Joe’s wealth is $100 and he is an expected utility maximizer with utility function 
𝑢
(
𝑐
)
=
𝑐
. Joe is afraid of oversleeping his economics exam. He figures there is only a 1 in 10 chance that he will, but if he does, it will cost him $100 in fees to the university for taking an exam late. Joe’s neighbour, Mary, never oversleeps. She offers to wake him one hour before the test, but he must pay her for this service. What is the most that Joe would be willing to pay for this wake-up service?

BlackTom AI · Accepted Answer

We start by outlining the scenario and the math involved to compare options.
Joe has wealth 100 and an expected utility framework where u(c) = c. Without any wake-up service, his expected utility is:
- If he oversleeps (probability 0.1), he pays 100 in fees, leaving wealth 0, so utility 0.
- If he does not oversleep (probability 0.9), wealth remains 100, so utility 100.
Thus, E[u] = 0.9 * 100 + 0.1 * 0 = 90.
Now suppose Joe pays p dollars upfront for Mary’s wake-up service. If he pays, he will be awakened and avoid oversleeping, so his wealth becomes 100 − p with certainty, and his utility is simply 100 − p.
He will be willing to pay up to p such that his certain utility from waking equals or exceeds his current expected utility without the service (i.e., 90): 100 − p ≥ 90 implies p ≤ 10. Therefore, the maximum he would pay is $10.
With this framework in mind, evaluate each option:
- Option "$100": If Joe paid $100 for the wake-up, his wealth would drop to 0, yielding utility 0, which is far below his current expected utility of 90. This is not worthwhile.
- Option "$10": Paying $10 leaves him with wealth 90, giving utility 90, which matches his baseline expected utility. This is exactly the threshold at which the service is worthwhile.
- Option "$0": Paying nothing leaves him at the same expected utility of 90 as before; while not better, it is not worse, but it’s not exploiting the potential gain from the service beyond the baseline.
- Option "$19": Paying $19 leaves wealth 81, utility 81, which is below his 90 expected utility without the service, so this is not worth it.
- Option "$81": Paying $81 leaves wealth 19, utility 19, which is far worse than the 90 he could achieve without the service, so this is clearly not worth it.
In summary, the viability of the wake-up service hinges on whether 100 − p is at least as good as the 90 expected utility from the risk of oversleeping; only p = 10 achieves that boundary conditions, while all larger payments reduce utility below 90 and all smaller payments leave him with less certainty than his optimized baseline.

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