We were able to compute the optimum level of the public good because we knew each individual’s utility function. In the real world each individual’s utility function is private information. Thus, to reach the social optimum we must solve 2 problems. The information problem and the free rider problem. The information problem is that no one knows the optimal quantity of the public good. The free rider problem is that no one is willing to pay for the public good. The government used the contribution levels under the top-up mechanism as a signal to decide how much of the public good to produce and almost magically the top-up function is such that citizens will voluntarily contribute exactly the amount that correctly signals how much society as a whole values the public good. Amazing!

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