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:

$$p_1{x}_1+p_2{x}_2+\cdots +p_n{x}_n\le m$$

Which we can simplify, using summation notation, as:

$$\sum _{i=1}^n{p}_i{x}_i\le m$$

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

$$\sum _{i=1}^n{p}_i{x}_i=m$$

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.