Structural Properties of Utility Functions

Transcription

1 Structural Properties of Utility Functions Econ 2100 Fall 2015 Lecture 4, September 14 Problem Set 2 due now. Outline 1 Monotonicity 2 Convexity 3 Quasi-linearity 4 Walrasian Demand

2 Definition From Last Class The utility function u : X R represents the binary relation on X if x y u(x) u(y). Theorem (Debreu) Suppose X R n. A binary relation on X is complete, transitive, and continuous if and only if it admits a continuous utility representation u : X R. Today we study connections between a utility function and the underlying preference relation it represents. Structural Properties of Utility Functions The main idea is to understand the connection between properties of preferences and characteristics of the utility function that represents them. NOTATION: From now on, assume X = R n. If x i y i for each i, we write x y.

3 Monotonicity Monotonicity says more is better. Definitions A preference relation is weakly monotone if x y implies x y. A preference relation is strictly monotone if x y and x y imply x y. In our notation, x y and x y imply x i > y i for some i. Question What does monotonicity imply for the utility function representing?

4 Monotonicity: An Example Example Suppose is the preference relation on R 2 defined by x y if and only if x 1 y 1 is weakly monotone, because if x y, then x 1 y 1. It is not strictly monotone, because (1, 1) (1, 0) and (1, 1) (1, 0) but not [(1, 1) (1, 0)] since (1, 0) (1, 1).

5 Strict Monotonicity: An Example The lexicographic preference on R 2 is strictly monotone Proof: Suppose x y and x y. Then there are two possibilities: (a) : x 1 > y 1 and x 2 y 2 or (b) : x 1 y 1 and x 2 > y 2. If (a) holds, then x y because x 1 y 1, and not (y x) because neither y 1 > x 1 (excluded by x 1 > y 1) nor y 1 = x 1 (also excluded by x 1 > y 1). If (b) holds, then x y because either x 1 > y 1 or x 1 = y 1 and x 2 y 2, and not (y x) because not (y 1 > x 1) (excluded by x 1 y 1) and not (y 2 x 2) (excluded by x 2 > y 2). In both cases, x y and not (y x) thus x y.

6 Monotonicity and Utility Functions Definitions A function f : R n R is nondecreasing if x y implies f (x) f (y); strictly increasing if x y and x y imply f (x) > f (y). Monotonicity is equivalent to the corresponding utility function being nondecreasing or increasing. Proposition If u represents, then: 1 is weakly monotone if and only if u is nondecreasing; 2 is strictly monotone if and only if u is strictly increasing. Proof. Question 1a. Problem Set 3, due next Monday.

7 Local Non Satiation Definition A preference relation is locally nonsatiated if for all x X and ε > 0, there exists some y such that y x < ε and y x. For any consumption bundle, there is always a nearby bundle that is strictly preferred to it. Definition A utility function u : X R is locally nonsatiated if it represents a locally nonsatiated preference relation ; that is, if for every x X and ε > 0, there exists some y such that y x < ε and u(y) > u(x). Example: The lexicographic preference on R 2 is locally nonsatiated Fix (x 1, x 2 ) and ε > 0. Then (x 1 + ε 2, x 2) satisfies (x 1 + ε 2, x 2) (x 1.x 2 ) < ε and (x 1 + ε 2, x 2) (x 1, x 2 ).

8 Local Non Satiation and Strict Monotonicity Proposition If is strictly monotone, then it is locally nonsatiated. Proof. Let x be given, and let y = x + ε n e, where e = (1,..., 1). Then we have y i > x i for each i. Strict monotonicity implies that y x. Note that Thus is locally nonsatiated. y x = n ( ε ) 2 ε = n < ε. n i=1

9 Shapes of Functions Definitions Suppose C is a convex subset of X. A function f : C R is: concave if f (αx + (1 α)y) αf (x) + (1 α)f (y) for all α [0, 1] and x, y C; strictly concave if f (αx + (1 α)y) > αf (x) + (1 α)f (y) for all α (0, 1) and x, y X such that x y; quasiconcave if f (x) f (y) f (αx + (1 α)y) f (y) for all α [0, 1]; strictly quasiconcave if f (x) f (y) and x y f (αx + (1 α)y) > f (y) for all α (0, 1).

10 Convex Preferences Definitions A preference relation is convex if x y αx + (1 α)y y for all α (0, 1) strictly convex if x y and x y αx + (1 α)y y for all α (0, 1) Convexity says that taking convex combinations cannot make the decision maker worse off. Strict convexity says that taking convex combinations makes the decision maker better off. Question What does convexity imply for the utility function representing?

11 Convex Preferences: An Example Let on R 2 be defined as x y if and only if x 1 + x 2 y 1 + y 2 is convex Proof: Suppose x y, i.e. x 1 + x 2 y 1 + y 2, and fix α (0, 1). Then So, αx + (1 α)y = (αx 1 + (1 α)y 1, αx 2 + (1 α)y 2 ) [αx 1 + (1 α)y 1 ] + [αx 2 + (1 α)y 2 ] = α[x 1 + x 2 }{{} y 1+y 2 ] + (1 α)[y 1 + y 2 ] proving αx + (1 α)y y. α[y 1 + y 2 ] + (1 α)[y 1 + y 2 ] = y 1 + y 2, This is not strictly convex, because (1, 0) (0, 1) and (1, 0) (0, 1) but 1 2 (1, 0) (0, 1) = (1 2, 1 ) (0, 1). 2

12 Convexity and Quasiconcave Utility Functions Convexity is equivalent to quasi concavity of the corresponding utility function. Proposition If u represents, then: 1 is convex if and only if u is quasiconcave; 2 is strictly convex if and only if u is strictly quasiconcave. Proof. Convexity of implies that any utility representation is quasiconcave, but not necessarily concave. Question 1b. Problem Set 3, due next Monday.

13 Quasiconcave Utility and Convex Upper Contours Proposition Let be a preference relation on X represened by u : X R. Then, the upper contour set is a convex subset of X if and only if u is quasiconcave. Proof. Suppose that u is quasiconcave. Fix z X, and take any x, y (z). Wlog, assume u(x) u(y), so that u(x) u(y) u(z), and let α [0, 1]. By quasiconcavity of u, u(z) u(y) u(αx + (1 α)y), so αx + (1 α)y z. Hence αx + (1 α)y belongs to (z), proving it is convex. Now suppose that (z) is convex for all z X. Let x, y X and α [0, 1], and suppose u(x) u(y). Then x y and y y, and so x and y are both in (y). Since (y) is convex (by assumption), then αx + (1 α)y y. Since u represents, Thus u is quasiconcave. u(αx + (1 α)y) u(y)

14 Proposition Convexity and Induced Choices If is convex, then C (A) is convex for all convex A. If is strictly convex, then C (A) has at most one element for any convex A. Proof. Let A be convex and x, y C (A). By definition of C (A), x y. Since A is convex: αx + (1 α)y A for any α [0, 1]. Convexity of implies αx + (1 α)y y. By definition of C, y z for all z A. Using transitivity, αx + (1 α)y y z for all z A. Hence, αx + (1 α)y C (A) by definition of induced choice rule. Therefore, C (A) is convex for any convex A. Now suppose there exists a convex A for which C (A) 2. Then there exist x, y C (A) with x y. Since x y and x y, strict convexity implies αx + (1 α)y y for all α (0, 1). Since A is convex, αx + (1 α)y A, but this contradicts the fact that y C (A).

15 Definition Quasi-linear Utility The function u : R n R is quasi-linear if there exists a function v : R n 1 R such that u(x, m) = v(x) + m. We usually think of the n-th good as money (the numeraire). Proposition The preference relation on R n admits a quasi-linear representation if and only 1 (x, m) (x, m ) if and only if m m, for all x R n 1 and all m, m R; 2 (x, m) (x, m ) if and only if (x, m + m ) (x, m + m ), for all x R n 1 and m, m, m R; 3 for all x, x R n 1, there exist m, m R such that (x, m) (x, m ). 1 Given two bundles with identical goods, the consumer always prefers the one with more money. 2 Adding (or subtracting) the same monetary amount does not chage rankings. 3 Monetary transfers can always be used to achieve indifference. Proof. Question 1c. Problem Set 3, due next Monday.

16 Proposition Quasi-linear Preferences and Utility Suppose that the preference relation on R n admits two quasi-linear representations: v(x) + m, and v (x) + m, where v, v : R n 1 R. Then there exists c R such that v (x) = v(x) c for all x R n 1. Proof. Let v(x) + m and v (x) + m both represent. Given any x 0 R n 1, for all x R n 1 there exists n x such that (x, 0) (x 0, n x ). (by (2) and (3) of the previous Proposition.) Since the utility functions both represent we have v(x) + 0 = v(x 0 ) + n x and v (x) + 0 = v (x 0 ) + n x Thus n x = v (x) v (x 0 ). Therefore Hence, for all x R n 1, v(x) v (x) = v(x) = v(x 0 ) + v (x) v (x 0 ) }{{} =n x a constant c {}}{ v(x 0 ) v (x 0 ) v (x) = v(x) c.

17 Homothetic Preferences and Utility Homothetic preferences are also useful in many applications, in particular for aggregation problems. Definition The preference relation on X is homothetic if for all x, y X, x y αx αy for each α > 0 Proposition The continuous preference relation on R n is homothetic if and only if it is represented by a utility function that is homogeneous of degree 1. Proof. A function is homogeneous of degree r if f (αx) = α r f (x) for any x and α > 0. Question 1d. Problem Set 3, due next Monday.

18 Main Idea Walrasian Demand The consumer uses income to purchase goods (commodities) at the exogenously given prices. Definition The Budget Set B p,w R n at prices p and income w is the set of all affordable consumption bundles and is defined by B p,w = {x R n + : p x w}. Definition Given a preference relation, the Walrasian demand correspondence x : R n ++ R + R n + is defined by x (p, w) = {x B p,w : x y for any y B p,w }. By definition, for any x x (p, w) x x for any x B p,w. Walrasian demand equals the induced choice rule given the preference relation and the available set B p,w : x (p, w) = C (B p,w ).

19 Walrasian Demand With Utility Although we do not need the utility function to exist to define Walrasian demand, if a utility function exists there is an equivalent definition. Definition Given a utility function u : R n + R, the Walrasian demand correspondence x : R n ++ R + R n + is defined by x (p, w) = arg max x B p,w u(x) where B p,w = {x R n + : p x w}. As before, and for any x x (p, w) x (p, w) = C (B p,w ). x x for any x B p,w. We can derive some properties of Walrasian demand directly from assumptions on preferences and/or utility.

20 Walrasian Demand Is Homogeneous of Degree Zero Proposition Walrasian demand is homogeneous of degree zero: for any α > 0 x (αp, αw) = x (p, w) Proof. For any α > 0, B αp,αw = {x R n + : αp x αw} = {x R n + : p x w} = B p,w because α is a scalar Since the constraints are the same, the utility maximizing choices must also be the same.

21 The Consumer Spends All Her Income This is sometimes known as Walras Law for individuals Proposition (Full Expenditure) If is locally nonsatiated, then p x = w for any x x (p, w) Proof. Suppose not. Then there exists an x x (p, w) with p x < w Find some y such that y x < ε with ε > 0 and p y w why does such a y exist? By local non satiation, this implies y x contradicing x x (p, w).

22 Problem Set 3: Partial List Due Monday, 21 September, at the beginning of class 1 Prove that if u represents, then: 1 is weakly monotone if and only if u is nondecreasing; is strictly monotone if and only if u is strictly increasing. 2 is convex if and only if u is quasiconcave; is strictly convex if and only if u is strictly quasiconcave. 3 u is quasi-linear if and only 1 (x, m) (x, m ) if and only if m m, for all x R n 1 and all m, m R; 2 (x, m) (x, m ) if and only if (x, m + m ) (x, m + m ), for all x R n 1 and m, m, m R; 3 for all x, x R n 1, there exist m, m R such that (x, m) (x, m ). 4 a continuous is homothetic if and only if it is represented by a utility function that is homogeneous of degree 1.