Preference and Utility - UCLA Economics

1 Preference and UtilityIchiro ObaraUCLAO ctober 2, 2012 Obara (UCLA) Preference and UtilityOctober 2, 20121 / 20 Preference RelationPreference RelationObara (UCLA) Preference and UtilityOctober 2, 20122 / 20 Preference RelationPreferenceWe study a classical approach to consumer behavior: we assume thatconsumers choose the bundle of commodities/goods that they likemost given their need to make this like most more RelationPreference relationonXis a subset ofX X. When (x,y) is anelement of this set, we sayxis preferred toyand denotex usually use to denote a Preference be any set.

2 For consumer problems,Xis typically<L+.Obara (UCLA) Preference and UtilityOctober 2, 20123 / 20 Preference RelationPreferenceSome basic properties of Preference relations: onXiscompleteif eitherx yory xfor anyx,y X onXistransitiveifx yandy zimplyx zfor anyx,y,z they reasonable?Obara (UCLA) Preference and UtilityOctober 2, 20124 / 20 Preference RelationPreferenceSome critique of transitivity1 How much sugar do you need for a cup of coffee? You are indifferentbetween no sugar and one grain of sugar, one grain of sugar are you indifferent between no sugar and 10 spoons ofsugar?

3 2 Framing (UCLA) Preference and UtilityOctober 2, 20125 / 20 Preference RelationPreferenceWe almost always assume these properties. So let s give them some Preference onXisrationalif it is complete and now on, we only consider rational preferences most of the (UCLA) Preference and UtilityOctober 2, 20126 / 20 Preference RelationRemarkWe can derive two other Preference relations from a PreferenceStrict Preference relation is defined byx y {x yandy x}IndifferenceIndifference is defined byx y {x yandy x}.

4 From a rational Preference , we can derive a strict Preference thatsatisfiesasymmetryandnegative transitivity. On the other hand,we can derive a rational Preference from a strict Preference thatsatisfies these (UCLA) Preference and UtilityOctober 2, 20127 / 20 Preference RelationPreferenceThere are many other properties we assume from time to time. LetXbe asubset of<L+. onXislocally nonsatiatedif for everyx Xand >0, thereexistsy Xsuch that y x < andy x. onXismonotone( monotone) ifx y( >y) impliesx yfor anyx,y X.

5 OnXiscontinuousif both theupper contour setU(x) ={y X:y x}and thelower contour setL(x) ={y X:x y}are (relatively) closed for anyx X(equivalently, ifxn x X,yn y Xandxn yn, thenx y).Obara (UCLA) Preference and UtilityOctober 2, 20128 / 20 Preference RelationPreference onXisconvexifU(x) is convex for anyx X. onXisstrictly convexify xandz xandy6=zimply y+ (1 )z xfor any (0,1). onX=<L+ishomotheticifx y x yfor any (UCLA) Preference and UtilityOctober 2, 20129 / 20 Utility RepresentationUtility RepresentationObara (UCLA) Preference and UtilityOctober 2, 201210 / 20 Utility RepresentationUtility RepresentationIt is usually more convenient to work withutility functionsrather : Representation of Preference is represented by autility functionu.

6 X <ifx y u(x) u(y)for allx,y (UCLA) Preference and UtilityOctober 2, 201211 / 20 Utility RepresentationUtility RepresentationOnce a Preference is represented by a Utility function, then we canformulate the consumer problem as a constrained optimization problem:maxx Xu(x) x w,or equivalently,maxx B(p,w)u(x), which may be easily solved analytically or (UCLA) Preference and UtilityOctober 2, 201212 / 20 Utility RepresentationUtility RepresentationExamples of Utility FunctionsCobb-Douglas Utility function:u(x1,x2) =x 1x1 2for (0,1).

7 Quasi-linear Utility function:u(x,m) =v(x) + Utility function:u(x1,x2) = min{x1,x2}.Obara (UCLA) Preference and UtilityOctober 2, 201213 / 20 Utility RepresentationUtility RepresentationWhen can a rational Preference be represented by a Utility function?Consider the easiest case:Xis a finite set. Clearly every rationalpreference onXcan be represented by some Utility function (Try toprove thisformally).Obara (UCLA) Preference and UtilityOctober 2, 201214 / 20 Utility RepresentationUtility RepresentationWhen can a rational Preference be represented by a Utility function?

8 What ifXis a countable set? For example, this is the case if no goodis divisible (X=ZL+). We can still obtain a representation as {x1,..,xn}forn= 1,2,..IFor eachn, we can findunto satisfyx y un(x) un(y) for anyx,y Xn. In fact, we can keep the sameunin each step ( (x) =un+1(x) =..for anyx Xn).IFor eachx X, defineu(x) byu(x) :=un(x) by taking any largen. Itcan be easily verified that (1)uis well-defined and (2)urepresents .Obara (UCLA) Preference and UtilityOctober 2, 201214 / 20 Utility RepresentationUtility RepresentationYou can find a continuous Utility function when a Preference is (rationaland) (Debreu)LetX <L+be closed and convex.

9 A rational Preference onXiscontinuous if and only if there exists a continuous Utility functionu:X <that represents .Note: Closedness and convexity ofXcan be (UCLA) Preference and UtilityOctober 2, 201215 / 20 Utility RepresentationSketch of Proof if is trivial. We prove only if in the {x <n+| x n}. SinceBnis compact, there exists theleast preferred elementxnin a Utility function onU(xn) as (x) = miny U(x) y xn onU(xn).IThenunrepresents onU(xn) (use convexity).We can adjustu1,u2,..in such a way thatumcoincides withunonU(xn) for allm n.

10 DefineuonXbyu(x) := limun(x). Thenurepresents skip continuity (this follows from Gap Theorem ).Obara (UCLA) Preference and UtilityOctober 2, 201216 / 20 Utility if continuity is dropped? Can a plain rational Preference bealways represented by someu? following rational Preference is not continuous and cannot berepresented by any Utility Preference on<2+For anyx,y <2+,x yif and only if either (1)x1>y1or (2)x1=y1andx2 a functionffrom<+toQ(rational number) byassociating eachxwithf(x) Qsuch thatu(x,1)<f(x)<u(x,2).