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 u^{i}(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 u^{i}(x, y_{i}) = u^{i}(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: MRS^{i} p, and MRS^{i} = 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:

u^{i}(x, y_{i}) = y_{i} + _{i}log x for all i = 1,...,n.

Then MRS^{i} = _{i} / x

Let A = _{i}_{i} and ^{*} = max {_{i} | i N }.

**Pareto Efficiency**:

MRS^{i} = p

i.e., (_{i} / x) =

(1 / x) A =

x´ = A / , where x´ is the Pareto efficient outcome.

**Market Outcome**:

MRS^{i} 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

MRS^{i} = 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.