1.2 — Budget Constraint — Appendix

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.