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. 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?

2 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?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}.

3 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. 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.

4 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: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).

5 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?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) =.)

6 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. 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).

7 IThenunrepresents onU(xn) (use convexity).We can adjustu1,u2,..in such a way thatumcoincides withunonU(xn) for allm n. 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).Then a different rational number is assigned to differentx, (UCLA) Preference and UtilityOctober 2, 201217 / 20 Utility RepresentationProperties of Preferences in terms of Utilities onXislocally nonsatiated for everyx Xand >0, there existsy Xsuch that y x < andu(y)>u(x).

8 OnXismonotone( monotone) x y( >y)impliesu(x)>u(y) for anyx,y X. onXisconvex uis quasi-concave, (y) u(x) andu(z) u(x)implyu( y+ (1 )z) u(x) for any [0,1]. onXisstrictly convex uis strictly quasi-concave, (y) u(x)andu(z) u(x) withy6=zimplyu( y+ (1 )z)>u(x) for any (0,1). ishomothetic u( x) =u( y) for any 0 andx,y Xsuch thatu(x) =u(y). of = quasi-concavity (UCLA) Preference and UtilityOctober 2, 201218 / 20 Utility RepresentationOrdinal Property and Cardinal PropertyLetf:< <be anystrictly increasingfunction. Thenu(x) andf(u(x)) represents the same Preference becauseu(x) u(y) f(u(x)) f(u(y)). (x) = logu(x) = properties (of utilities) that are preserved under any suchmonotone transformation areordinal properties. The properties thatare not arecardinal and quasi-concavity are ordinal and decreasing marginal Utility are cardinal (UCLA) Preference and UtilityOctober 2, 201219 / 20 Utility RepresentationOrdinal Property and Cardinal we regard a Utility function as merely one convenient representationof the underlying Preference , then we should be careful to make surethat results do not depend on a particular representation andparticular cardinal the other hand, if we know that representation does not affect aresult (ex.)

9 Optimization), then we had better use a more convenientrepresentation with nicer cardinal properties(Example. Somequasi-concave Utility function can be transformed into a concave utilityfunction. This doesn t change the Preference , whereas concave functions areeasier to use than quasi-concave functions).Obara (UCLA) Preference and UtilityOctober 2, 201220 / 20