There is a long tradition of graphical analysis in economics, but this method has some severe limitations. The first is that it can only be used in problems that have two or three dimensions. Adding another commodity to our graphical model of the exchange economy would be difficult, and adding a fourth would be impossible. The second is that it is difficult to determine the circumstances under which a result will hold. Perhaps a different result would have been obtained if a line had shifted a little bit further, or if the bend in a curve had been a little bit sharper. The third is that graphical arguments have to be relatively simple. Only so many lines can be placed in a graph before it becomes an unreadable jumble.

Mathematics allows economists to circumvent these restrictions. There are only three physical dimensions, but mathematics allows us to imagine infinitely many dimensions. Mathematics also allows us to quantify things like the shift of a line or the sharpness of a curve. And finally, mathematics is a language designed for extended logical arguments.

This chapter returns to the exchange economy in which George and Harriet trade ale and bread, and to the two fundamental theorems, formulating them in mathematical terms. It is your introduction to a methodology upon which we will increasingly rely.