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| Handbook >> Public Goods >> The Individual's Public Good Choice Problem in the MarketThe following section is a more advanced discussion intended for students who are familiar with optimization techniques. It summarizes the differences between the market and Pareto efficient outcomes in mathematical form. The individual i chooses how much of the public good to buy on his own (i) to maximize his utility ui(x, yi) from consuming the public good x and private consumption yi, taking contributions of others as given (x-i). The consumer's problem can be then written as follows: max ui(x, yi) = ui(x-i + i, yi) subject to constraints i, yi 0 and to p.i + yi m, First order conditions: MRSi p, and MRSi = p if > 0. Graphical Illustration of First Order ConditionsNumerical ExampleSuppose a unit of public good costs and the consumer i has a utility function of the following form: ui(x, yi) = yi + ilog x for all i = 1,...,n. Pareto Efficiency: MRSi = p Market Outcome: MRSi p = , for all i Let's examine when an idividual purchases positive amount of public good Note that * << A and therefore, xm << x´, meaning that the market outcome is severly inefficient. | ||||||
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