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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,
where p denotes the price of one unit of the public good and m denotes the value of i-th person initial endowment or income.

First order conditions: MRSi p, and MRSi = p if > 0.

Graphical Illustration of First Order Conditions

Numerical Example

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.
Then MRSi = i / x
Let A = ii and * = max {i | i N }.

Pareto Efficiency:

MRSi = p
i.e., (i / x) =
(1 / x) A =
x´ = A / , where x´ is the Pareto efficient outcome.

Market Outcome:

MRSi p = , for all i
i.e., i / x p, for all i
i.e., x i / p, for all i.

Let's examine when an idividual purchases positive amount of public good
MRSi = p if i > 0, i.e., x = i / .
From this follows that i = 0 if i < * = max {i | i N }, and xm = * / , where xm denotes the market outcome.

Note that * << A and therefore, xm << x´, meaning that the market outcome is severly inefficient.
