FixedPoint - Econ 511 Professor John Nachbar December 2009...

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Econ 511 Professor: John Nachbar December 2009 Fixed Point Theorems 1 Overview Definition 1. Given a set W and a function f : W W , x * W is a fixed point of f iff f ( x * ) = x * . Many existence problems in economics – for example existence of competitive equilibrium in general equilibrium theory, existence of Nash in equilibrium in game theory – can be formulated as fixed point problems. Because of this, theorems giving sufficient conditions for existence of fixed points have played an important role in economics. My treatment is schematic, focusing on only a few representative theorems and omitting many proofs. An excellent introduction to fixed point theory is Border (1985). McLennan (2008) is a recent concise survey that provides a treatment more sophisticated than the one here. The remainder of these notes is divided into three sections. Section 2 focuses on fixed point theorems where the goal is to find restrictions on the set W strong enough to guarantee that every continuous function on W has a fixed point. The prototype of theorems in this class is the Brouwer Fixed Point Theorem, which states that a fixed point exists provided W is a compact and convex subset of R N . In contrast, the Contraction Mapping Theorem (Section 3) imposes a strong continuity condition on f but only very weak conditions on W . Finally, the Tarski Fixed Point Theorem (Section 4) requires that f be weakly increasing, but not necessarily continuous, and that W be, loosely, a generalized rectangle (possibly with holes). 2 The Brouwer Fixed Point Theorem and its Relatives 2.1 The Brouwer Fixed Point Theorem If W = [ a, b ] R and f : W W is continuous then the Intermediate Value theorem implies existence of a fixed point. Explicitly, if f ( a ) = a or f ( b ) = b then we are done. Otherwise, f ( a ) > a and f ( b ) < b . Define g ( x ) = f ( x ) - x . Then g ( a ) > 0 while g ( b ) < 0. Moreover, g is continuous since f is continuous. Therefore, by the Intermediate Value Theorem, there is an x * ( a, b ) such that g ( x * ) = 0, hence f ( x * ) = x * . The Brouwer Fixed Point Theorem and its relatives generalize this result. Theorem 1 (Brouwer Fixed Point Theorem) . If W R N is non-empty, compact and convex, then every continuous function f : W W has a fixed point. 1
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Proof. Omitted. See Scarf (1982) for an argument that is self-contained. The proof in Border (1985) is the same. While the one-dimensional Brouwer argument given above was trivial, the N - dimensional proof is not. It is worth considering why this is so. If W = [0 , 1] then a fixed point occurs where the graph of f crosses the 45 line. Since the 45 line bisects the square [0 , 1] × [0 , 1] R 2 it is “obvious” that if f is continuous then its graph must cross this line; the proof based on the Intermediate Value theorem formalizes exactly this intuition. In contrast, if W = [0 , 1] 2 R 2 , then the graph of f lies in the 4-dimensional cube [0 , 1] 2 × [0 , 1] 2 , and the analog of the 45 line is a 2-dimensional plane in this cube. A 2-dimensional plane cannot bisect a 4-dimensional cube (just as a 1- dimensional line cannot bisect a 3-dimensional cube). Brouwer must, among other
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