The following section is intended for intermediate/upper intermediate 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, y_i) from consuming the public good x and private consumption y_i, taking contributions of others as given (x_-i). The consumer's problem can be then written as follows :

max uⁱ(x, y_i) = uⁱ(x_-i + _i, y_i) subject to constraints _i, y_i 0 and to p._i + y_i 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.

Suppose a unit of public good costs and the consumer i has a utility function of the following form:

uⁱ(x, y_i) = y_i + _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 x^m = ^* / , where x^m denotes the market outcome.

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