Question

Show that if a preference relation is locally nonsatiated then it is not monotone

Answer 1

While there are clearly preference relations that are locally nonsatiated but not monotonic, from the point of view of competitive demand theory, local nonsatiation is no more general than monotonicity. By this I mean that if ≽ is a locally nonsatiated upper semicontinuous preference, then there is a monotonic preference ≽∗ that generates the same demand. Let there be n commodities, so the neoclassical consumption set is Rn +, the set of nonnegative vectors in Rn . A preference is a reflexive, transitive, total binary relation ≽ on Rn +. The corresponding strict preference is denoted ≻. The preference ≽ is locally nonsatiated if for each x ∈ Rn + and ε > 0, there is some y ∈ Rn + such that d(x, y) < ε 1 and y ≻ x. The preference ≽ is monotonic if x ≫ y implies x ≻ y. 2 Clearly every monotonic preference relation is locally nonsatiated. A preference is upper semicontinuous if for each x, the upper contour set {y ∈ Rn + : y ≽ x} is closed, or equivalently the strict lower contour set {y ∈ Rn + : x ≻ y} is relatively open in Rn +. The preference ≽ is convex if for each y, the upper contour set {x ∈ Rn + : x ≽ y} is convex. Given a price vector p ∈ Rn ++ and income level w > 0, the budget set β(p, w) is the compact convex set {x ∈ Rn + : p · x ⩽ w}. It is a fact that if ≽ is upper semicontinuous, then every compact set has a ≽-greatest element; and if ≽ is locally nonsatiated, then every ≽-greatest element x ∗ in β(p, w) satisfies p · x ∗ = w. To simplify the explicit description of the desired monotonic relation, let us introduce the following notation. Given a set A, x ≽ A means ( ∀z ∈ A ) [ x ≽ z ] and let D(x) = {y ∈ Rn + : 0 ≦ y ≦ x}. Note that D(x) is nonempty for x ∈ Rn + as it always contains x