Public Good Provision under Locally Private Signals

We study public-good provision when a planner observes agents' preferences only through a fixed local-privacy channel that randomizes each report before it reaches the planner. We characterize the optimal reduced-form allocation: the project is implemented when an aggregate posterior score is positive, where each agent's score combines the posterior expected valuation and posterior virtual value. Privacy enters through these posterior objects, muting the responsiveness of provision to private preferences and, under weak monotone likelihood ratios, potentially generating pooling. We then distinguish the optimal reduced-form allocation from its implementation through signal-measurable transfers: the required transfers solve a Fredholm integral equation whose solution is unique under completeness when it exists, while existence requires a separate range condition. Maximum reduced-form revenue exhibits three population regimes: it is asymptotically linear, of square-root order, or exponentially small according as the lower endpoint of the valuation distribution is positive, zero, or negative. Finally, welfare comparisons depend on the privacy calibration. At a common noise scale, Laplace Blackwell-dominates logistic noise, while under a common tight $\mu$-GDP calibration the ordering reverses for the maximally separated binary endpoint experiment. Thus the preferred privacy channel depends on the standard used to hold privacy fixed.