Question
A freelance lighting technician has been promised an event bonus: $0 with probability 1/4 or $200 with probability 3/4. Before the event, an insurer offers to replace this uncertain bonus with a guaranteed $150, provided the technician pays a fixed fee \(f\) up front. All other wealth is unchanged.
The technician values final wealth \(w\) according to \(u(w)=-e^{-w/200}\), with \(w\) measured in dollars. What is the largest fee they would pay to accept the guarantee rather than the uncertain bonus? Give the exact expression and an approximate dollar amount. Does the answer depend on their initial wealth, and what would happen to the fee as risk aversion approached zero?
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Hint 1
Set the expected exponential utility of initial wealth plus the random bonus equal to the utility of initial wealth plus the guaranteed $150 minus the fee.
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Solution
Let initial wealth be \(W\) and the random bonus be \(X\in\{0,200\}\). The expected utility of retaining the bonus is
\[E[u(W+X)]=-e^{-W/200}\left(\tfrac14+\tfrac34e^{-1}\right).\]
The certainty equivalent \(C\) of \(X\) is defined so that a sure $C produces this same utility: \(-e^{-(W+C)/200}=E[u(W+X)]\). Cancelling \(-e^{-W/200}\) and taking logarithms gives
\[e^{-C/200}=\tfrac14+\tfrac34e^{-1},\qquad C=-200\ln\!\left(\tfrac14+\tfrac{3}{4e}\right)\approx\$128.5252.\]
Replacing \(X\) with a sure $150 net of fee \(f\) is preferred precisely when \(150-f\ge C\), because exponential utility is strictly increasing in wealth. Hence the largest acceptable fee is
\[\boxed{f_{\max}=150-C=150+200\ln\!\left(\tfrac14+\tfrac{3}{4e}\right)\approx\$21.4748}.\]
At equality, the technician is indifferent; a larger fee makes the risky bonus preferable. The result is independent of \(W\): exponential utility with constant absolute risk aversion allows the \(e^{-W/200}\) factor to cancel. For a risk-aversion coefficient \(a>0\), more generally \(C(a)=-a^{-1}\ln E[e^{-aX}]\). As \(a\to0\), the logarithm's first-order expansion gives \(C(a)\to E[X]=0.25(0)+0.75(200)=\$150\). The willingness to pay for a sure $150 therefore tends to zero in the risk-neutral limit. The positive fee here is a genuine risk premium, not an expected-value improvement.
Assumptions
The technician has exponential utility \(u(w)=-\exp(-w/200)\) over total dollar wealth.
The bonus probabilities are known, and insurance replaces the whole bonus with $150 while charging fee f regardless of what the bonus would have been.
There are no other policy costs, distortions or constraints affecting the two choices.
Final Answer
Maximum fee \(f=150+200\ln(\frac14+\frac3{4e})\approx\$21.47\); the risky bonus has certainty equivalent \(\approx\$128.53\). Initial wealth cancels; the maximum fee approaches $0 in the risk-neutral limit.