(Solution) - Bill owns an outdoor amphitheater in which he stages rock -(2025 Original AI-Free Solution)

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Academic Level: Undergrad. (yrs 3-4)

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Bill owns an outdoor amphitheater in which he stages rock concerts. Because audiences like loud music, his profits rise with the decibel level, D, of the concerts, at least up to a point. More specifically, his profits are
P = 120,000D - 500D2
Unfortunately, the noise from the concerts bothers 100 residents who live nearby. The cost to each resident (expressed in dollar terms) is
C = 100D + D2/2
How loud will the concerts be without regulation or a corrective tax (assuming the problem is not resolved through private negotiation)? What is the efficient noise limit? What is the efficient Pigouvian tax?