The Individual's Public Good Choice Problem in the Market
The 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 Conditions
Suppose 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.
MRSi = p
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.