Preferences and Utility - UCLA Economics

1 Preferences and UtilitySimon Board This Version: October 6, 2009 First Version: October, lectures examine the Preferences of a single agent. In Section1we analyse how theagent chooses among a number of competing alternatives, investigating when Preferences can berepresented by a Utility function. In Section2we discuss two attractive properties of Preferences :monotonicity and convexity. In Section3we analyse the agent s indifference curves and ask howshe makes tradeoffs between different goods. Finally, in Section4we look at some examples ofpreferences, applying the insights of the earlier The Foundation of Utility A Basic Representation TheoremSuppose an agent chooses from a set of goodsX={a, b, c, ..}. For example, one can think ofthese goods as different TV sets or two goods,xandy, the agentweakly prefersxoveryifxis at least as good asy. Toavoid us having to write weakly prefers repeatedly, we simply writex<y.

2 We now put somebasic structure on the agent s Preferences by adopting two Axiom:For every pairx, y X, eitherx<y,y<x, or both. Department of Economics , UCLA. Please email suggestions and typosto axiom is a foundational , Fall 2009 Simon BoardTransitivity Axiom:For every triplex, y, z X, ifx<yandy<zthenx< agent has complete Preferences if she can compare any two objects. An agent has transitivepreferences if her Preferences are internally consistent. Let s consider some , suppose that, given any two cars, the agent prefers the faster one. These Preferences arecomplete: given any two carsxandy, then eitherxis faster,yis faster or they have the samespeed. These Preferences are also transitive: ifxis faster thanyandyis faster thanz, thenxis faster , suppose that, given any two cars, the agent prefersxtoyif it is both faster and Preferences are transitive: ifxis faster and bigger thanyandyis faster and bigger thanz, thenxis faster and bigger thanz.

3 However, these Preferences are not complete: an SUVis bigger and slower than a BMW, so it is unclear which the agent prefers. The completenessaxiom says these Preferences are unreasonable: after examining the SUV and BMW, the agentwill have a preference between the , suppose that the agent prefers a BMW over a Prius because it is faster, an SUV over aBMW because it is bigger, and a Prius over an SUV, because it is more environmentally this case, the agent s Preferences cycle and are therefore intransitive. The transitivity axiomsays these Preferences are unreasonable: if environmental concerns are so important to theagent, then she should also take them into account when choosing between the Prius andBMW, and the BMW and the it is natural to think about Preferences , it is often more convenient to associate differentnumbers to different goods, and have the agent choose the good with the highest number.

4 Thesenumbers are calledutilities. In turn, autility functiontells us the Utility associated witheach goodx X, and is denoted byu(x) <. We say a Utility functionu(x)representsanagent s Preferences ifu(x) u(y) if and only ifx<y( )This means than an agent makes the same choices whether she uses her preference relation,<,or her Utility functionu(x).Theorem 1( Utility Representation Theorem).Suppose the agent s Preferences ,<, are com-plete and transitive, and thatXis finite. Then there exists a Utility functionu(x) :X <which represents<.2 Eco11, Fall 2009 Simon BoardTheorem1says that if an agent has complete and transitive Preferences then we can associatethese Preferences with a Utility function. Intuitively, the two axioms allow us to rank the goodsunder consideration. For example, if there are 10 goods, then we can say the best has a utilityu(x) = 9, the second best hasu(x) = 8, the third best hasu(x) = 7 and so on.

5 For a formalproof, see A Proof of Theorem12 The idea behind the proof is simple. For any goodx, letNBT(x) ={y X|x<y}be thegoods that are no better than x. The Utility ofxis simply given by the number of items inNBT(x). That is,u(x) =|NBT(x)|.( )If there are 10 goods, then the worst has a no better than set which is empty, so thatu(x) = second worst has a has a no better than set which has one element, sou(x) = 1. And now have to verify that this Utility function represents the agent s Preferences . We do thisin two steps: first, we show thatx<yimpliesu(x) u(y); second , we show thatu(x) u(y)impliesx< 1: Supposex<y. Pick anyz NBT(y);3by the definition ofNBT(y), we havey< Preferences are complete, we know thatzis comparable tox. Transitivity then tells usthatx<z, soz NBT(x). We have therefore shown that every element ofNBT(y) is alsoan element ofNBT(x); that is,NBT(y) NBT(x).

6 As a result,u(x) =|NBT(x)| |NBT(y)|=u(y)as 2: Supposeu(x) u(y). By completeness, we know that eitherx<yory<x. UsingStep 1, it must then be the case that eitherNBT(y) NBT(x) orNBT(x) NBT(y), sothe no better than sets cannot partially overlap or be disjoint. By the definition of utilities( ) we know that there are more elements inNBT(x) than inNBT(y), which implies that2 More NBT(y) means thatzis an element ofNBT(y).3 Eco11, Fall 2009 Simon BoardNBT(y) NBT(x). Completeness means that a good is weakly preferred to itself, so thaty NBT(y). SinceNBT(y) NBT(x), we concludey NBT(x). Using the definition ofthe no better than set, this implies thatx<y, as Increasing TransformationsA number system isordinalif we only care about the ranking of the numbers. It iscardinalif we also care about the magnitude of the numbers. To illustrate, Usain Bolt and RichardJohnson came 1st and 2nd in the 2008 Olympic final of the 100m sprint.

7 The numbers 1 and2 are ordinal: they tell us that Bolt beat Johnson, but do not tell us that he was 1% faster or10% faster. The actual finishing times were for Bolt and for Johnson. These numbersare cardinal: the ranking tells us who won, and the magnitudes tells us about the margin ofthe ordinal: when comparing two goods, all that matters is the ranking of the utilities;the actual numbers themselves carry no significance. This is obvious from the construction:when there are 10 goods, it is clearly arbitrary that we give Utility 9 to the best good, 8 to thesecond best, and so on. This idea can be formalised by the following result:Theorem (x)represents the agent s Preferences ,<, andf:< <is a strictlyincreasing function. Then the new Utility functionv(x) =f(u(x))also represents the agent spreferences<.The proof of Theorem2is simply a rewriting of definitions.

8 Supposeu(x) represents the agent spreferences, so that equation ( ) holds. Ifx<ythenu(x) u(y) andf(u(x)) f(u(y)), sothatv(x) v(y). Conversely, ifv(x) v(y) then, sincef( ) is strictly increasing,u(x) u(y)andx<y. Hencev(x) v(y) if and only ifx<yandv(x) represents<.Theorem2is important when solving problems. Suppose an agent has Utility functionu(x) = 15(x1/21+x1/22+ 10)3 Solving the agent s problem with this Utility function may be be algebraically messy. Using4 Eco11, Fall 2009 Simon BoardTheorem2, we can rewrite the agent s Utility asv(x) =x1/21+x1/22 Sinceu(x) andv(x) preserve the rankings of the goods, they represent the same a result, the agent will make the same choices with utilityu(x) andv(x). This is usefulsince it is much simpler to solve the agent s choice problem usingv(x) thanu(x).Theorem2is also useful for cocktail parties. For example, some people dislike the way I rankmovies of a 1-10 scale.

9 They claim that a movie is a rich artistic experience, and cannot besummarised by a number. However, Theorem1tells us that, if my Preferences are complete andtransitive, then I can represent my Preferences over movies by a number. Moreover, Theorem2tells us that I can rescale the numbers to put them on a 1-10 from Budget SetsTheorem1assumes that the consumer chooses from a finite number of goods. While this isrealistic, it is more mathematically convenient to allow consumers to choose from a continuumof goods. For example, if the agent has $10 and a hamburger costs $2, it is easier to allow theconsumer to any number between 0 and 5, rather than forcing her to choose an the choice set is given byX <n+. A typical element isx= (x1, .. , xn), wherexiisthe number of theithgood the agent consumes. In order to prove a representation theorem forthis larger set of choices, we need one more (rather technical) Axiom:Supposex1, x2, x3.

10 Is a sequence of feasible choices, so thatxi Xfor eachi, and suppose the sequence converges tox X. Ifxi<yfor eachi, thenx< 3(Representation Theorem for Budget Sets).Suppose the agent s Preferences ,<,are complete, transitive and continuous, and thatX <n+. Then there exists a continuousutility functionu(x) :X <which represents<.We will not prove this result. The following example examines a case where the continuityaxiom does not hold and no Utility representation there are two goods and the agent haslexicographic Preferences :when faced withtwo bundles the agent prefers the bundle with the most ofx1; if the two bundles have the same5 Eco11, Fall 2009 Simon Boardx1then she prefers the bundle with the most ofx2. To verify that this does not satisfy thecontinuity axiom, consider a sequence of bundlesxi= (1 +1i,1) which converges tox= (1,1)asi , and lety= (1,2).