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 {x1,x2,,xn} denote the set of n goods in an economy. Let {p1,p2,,pn} 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:

p1x1+p2x2++pnxnm

Which we can simplify, using summation notation, as:

i=1npixim

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

i=1npixi=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.