Joe’s wealth is $100 and he is an expected utility maximizer with utility function u\left(c\right)=\sqrt{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\left(c\right)=\sqrt{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 baseline numbers. Joe has wealth 100 and uses a utility function u(c) = sqrt(c). Without any wake-up service, there is a 0.1 chance he oversleeps and must pay 100 in late fees, leaving him with wealth 0 in that event; otherwise he keeps his wealth at 100. Therefore, the expected utility without the service is:
EU_no_service = 0.1 * u(0) + 0.9 * u(100) = 0.1 * sqrt(0) + 0.9 * sqrt(100) = 0 + 0.9 * 10 = 9.

Now suppose Joe buys the wake-up service and pays a price p. If he buys it, oversleeping becomes impossible, so his wealth after paying the price is 100 − p with certainty, and his utility is u(100 − p) = sqrt(100 − p).
We want the maximum p such that buying the service is at least as good as not buying it, i.e. u(100 − p) ≥ EU_no_service. Setting them equal gives:
sqrt(100 − p) = 9.
Squaring both sides yields 100 − p = 81, so p = 19.

Therefore, the most Joe would be willing to pay for the wake-up service is 19 dollars.

Note on structure: the reasoning compares the guaranteed outcome under the service to the expected outcome without it, uses the given utility function, and resolves the equality to find the indifference price. If the price exceeded 19, the guaranteed utility would fall below the current expected utility; if it were below 19, the service would be worth more to him.

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