2. Structural Properties of Preferences and Utility Functions

Transcription

1 2. Structural Properties of Preferences and Utility Functions Daisuke Oyama Microeconomics I May 9, 2018

2 Throughout this chapter, X = R k +, and is a preference relation on X that is complete and transitive. 1 / 17

3 Monotonicity Definition 2.1 is monotone (or nondecreasing) if, for all x, y X, x y implies x y. is strictly monotone (or strictly increasing) if, for all x, y X, x y and x y implies x y. is strictly monotone for strict increases in the bundle or semi-strictly monotone, if, for all x, y X, x y implies x y. A function u: X R is nondecreasing if, for all x, y X, x y implies u(x) u(y). u is strictly increasing if, for all x, y X, x y and x y implies u(x) > u(y). 2 / 17

4 Strictly monotone monotone. Strictly monotone strictly monotone for strict increases in the bundle. Strictly monotone for strict increases in the bundle + continuous monotone. 3 / 17

5 Proposition If is represented by some nondecreasing (strictly increasing) utility function, then it is monotone (strictly monotone). 2. If is monotone (strictly monotone), then any utility function that represents is nondecreasing (strictly increasing). 4 / 17

6 For many important results, we will need to assume that the consumer is insatiable. Definition 2.3 is globally insatiable if for any x X, there exists y X such that y x. is locally insatiable if for any x X and any ε > 0, there exists y X such that y x ε and y x. 5 / 17

7 Monotone + locally insatiable strictly monotone for strict increases in the bundle. Strictly monotone for strict increases in the bundle locally insatiable. 6 / 17

8 Convexity Definition 2.4 is convex if, for all x, y X, x y implies ax + (1 a)y y for all a [0, 1]. is semi-strictly convex if it is convex, and for all x, y X, x y implies ax + (1 a)y y for all a (0, 1). is strictly convex if, for all x, y X, x y and x y implies ax + (1 a)y y for all a (0, 1). 7 / 17

9 Strictly convex semi-strictly convex convex. 8 / 17

10 Proposition 2.5 is convex if and only if for all x X, NWT(x) = {y X y x} is a convex set. Problem / 17

11 For A X, c (A) = {x A x y for all y A}. Proposition If is convex, then c (A) is convex for any convex set A X. 2. If is strictly convex, then c (A) contains at most one point for any convex set A X. 10 / 17

12 Definition 2.7 f : X R is: concave if for all x, y X, f(ax + (1 a)y) af(x) + (1 a)f(y) for all a [0, 1]; strictly concave if for all x, y X with x y, f(ax + (1 a)y) > af(x) + (1 a)f(y) for all a (0, 1); quasi-concave if for all x, y X such that f(x) f(y), f(ax + (1 a)y) f(y) for all a [0, 1]; semi-strictly quasi-concave if it is quasi-concave, and for all x, y X such that f(x) > f(y), f(ax + (1 a)y) > f(y) for all a (0, 1); and strictly quasi-concave if for all x, y X with x y such that f(x) f(y), f(ax + (1 a)y) > f(y) for all a (0, 1). 11 / 17

13 Concave quasi-concave Strictly concave strictly quasi-concave Strictly quasi-concave semi-strictly quasi-concave quasi-concave Proposition 2.8 Suppose that is represented by a utility function u. is convex/strictly convex/semi-strictly convex if and only if u is quasi-concave/strictly quasi-concave/semi-strictly quasi-concave. 12 / 17

14 Continuity Proposition 2.9 If is complete, transitive, and continuous, then there exists a continuous utility function that represents. 13 / 17

15 Indifference Curves Let X = R 2 +. Indifferent class for x X: I(x) = {x X x x}. 1. The indifferent classes do not contain a ball with a positive radius, if is locally insatiable. 2. The indifference curves are strictly decreasing, if is strictly monotone. 3. The indifference curves are continuous in the sense that if x x x, then any continuous curve that connects x and x has a point y such that y x, if is monotone and continuous. 4. The indifference curves are strictly convex towards the origin, if is strictly convex. 14 / 17

16 Quasi-linearity Definition 2.15 on R k + is quasi-linear in the kth commodity if it is represented by a utility function of the form u(x) = v(x 1,..., x k 1 ) + x k. Proposition 2.16 is quasi-linear in the kth commodity if and only if the following properties hold: 1. For all z R k 1 + and m, m R +, (z, m) (z, m ) if and only if m m. 2. For all z, z R k 1 + and m, m R +, (z, m) (z, m ) if and only if (z, m + m ) (z, m + m ) for all m R For all z, z R k 1 +, (z, m) (z, m ) for some m, m R / 17

17 Homotheticity Definition 2.17 is homothetic if, for all x, y X, x y implies λx λy for all λ 0. Definition 2.18 A function f : X R is homogeneous of degree α 0 if f(λx) = λ α f(x) for all x X and all λ 0. We often say that f is homogeneous if it is homogeneous of degree / 17

18 Proposition 2.19 is continuous and homothetic if and only if there exists a continuous and homogeneous utility function that represents. 17 / 17