## Budget Constraint for $$n$$ Goods

While we can derive a lot of useful propositions and testable claims using a simple model with just 2 goods, obviously there are many orders of magnitude more goods in an economy. Economics at the graduate and professional level begins with abstract models of $$n$$ number of goods in an economy. Naturally, we cannot graph this, so we must deal in abstract equations and set theory:

Let $$\{x_1, x_2, \cdots, x_n\}$$ denote the set of $$n$$ goods in an economy. Let $$\{p_1,p_2, \cdots, p_n\}$$ denote the set of market prices affiliated with each good. Let $$m$$ again denote an individual’s income.

For $$n$$ goods, the budget set is defined as:

$p_1x_1 + p_2x_2 + \cdots + p_n x_n \leq m$

Which we can simplify, using summation notation, as:

$\sum^{n}_{i=1}p_ix_i \leq m$

To get the limit of this set, the budget constraint, make it an equality:

$\sum^{n}_{i=1}p_ix_i = m$

As usual, this simply says that one’s total expenditures (on all goods) on the left-hand side, must be equal to one’s income on the right-hand side.