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Mathematical methods for economic theory: 3.3 Concave and convex functions of many variables

mjo.osborne.economics.utoronto.ca · 4,047 words · saved by 1 readers

For n = 1, the definition coincides with the definition of an interval: a set of numbers is convex if and only if it is an interval. For n = 2, two examples are given in the following figures. The set in the first figure is convex, because every line segment joining a pair of points in the set lies entirely in the set. The set in the second figure is not convex, because the line segment joining the points x and x' does not lie entirely in the set. Convex set x x' A set that is not convex The following property of convex sets (which you are asked to prove in an exercise) is sometimes useful. More precisely, we can make the following definition (which is again essentially the same as the corresponding definition for a function of a single variable). Note that only functions defined on convex sets are covered by the definition. First note that the domain of f is a convex set, so the definition of concavity can apply. The functions g and f are illustrated in the following figures. (

Convex sets To extend the notions of concavity and convexity to functions of many variables we first define the notion of a convex set. Definition A set S of n -vectors is convex if (1−λ) x + λ x ' ∈ S whenever x ∈ S , x ' ∈ S , and λ ∈ [0,1]. We call (1 − λ) x + λ x ' a convex combination of x and x '. Geometrically, the set of all convex combinations of two points x and x ' is the line segment connecting x and x '. For n = 1, the definition coincides with the definition of an interval : a set of numbers is convex if and only if it

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