The Brouwer Fixed-Point Theorem is a profound and powerful result. It turns out to be essential in proving the existence of general equilibrium. We have already seen that it is convenient (in Chapter 5), but it can be shown to be indispensable (Chapter 18).

The Brouwer Fixed-Point Theorem says that a continuous function from a compact convex set into itself has a fixed point. There is at least one point that is left unchanged by the mapping. Note that the convexity is essential. For example, the fixed point property is not true (and the theorem is inapplicable) for a function mapping the circumference of a circle into itself. Indeed, typical of well-constructed mathematical results, all of the assumptions are essential. The fixed-point property will not hold for a discontinuous function or on an open or unbounded set.

In R, the Brouwer Fixed-Point Theorem takes a particularly simple form, equivalent to the Intermediate Value Theorem. Let f map the closed interval [a, b] into itself. Then the theorem is equivalent to the assertion that every continuous curve y = f(x) from one side of the square [a, b] × [a, b] to the opposite side must intersect the diagonal (the line y = x). See Figure 9.1.

Economic applications do not require that the economist know or understand the proof of the Brouwer Theorem.