Introduction to Choice Theory Stanford University

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are to any individual person Some of the greatest thinkers of the era were both. philosophers and economists, Could the utilitarian and economic approaches be combined That question. suggests several others How can we tell if the Smithian model of choice is right. that is that individuals make choices in their own interests What does that mean. precisely How can we use data to tell whether the proposition is true What are. all the empirical implications of rational choice What kind of data do we need. to make the test Even if the Smithian model is true can the utility function we. need for policy making be recovered from choice data If we can recover utilities. is simply adding up utilities really the best way to use that information for public. decisions What is the best way to use that information. The utility maximization approach to choice has several characteristics that. help account for its long and continuing dominance in economic analysis First. from its earliest development it has been deeply attached to principles of govern. ment policy making The original utilitarian program proved to be too ambitious. but the idea that welfare criteria could be derived from choice data has proved. to be workable in practice Moreover because this approach incorporates the. principle that people s own choices should determine the government s welfare cri. terion it is well aligned with modern democratic values Second many of the. comparative statics predictions of the choice theory the qualitative predictions. concerning the ways in which choices change as people s environments change. tend to be confirmed in empirical studies Third the optimization approach in. cluding utility maximization and profit maximization has a spectacularly wide. scope It has been used to analyze not only personal and household choices about. traditional economic matters like consumption and savings but also choices about. education marriage child bearing migration crime and so on as well as business. decisions about output investment hiring entry exit etc Fourth the optimiza. tion approach provides a compact theory that makes empirical predictions from a. relatively sparse model of the choice problem just a description of the chooser s. objectives and constraints In contrast for example psychological theories with. strong support from laboratory experiments predict that many choices depend. systematically on a much wider array of factors such as the way information is. presented to the subjects the noise level in the laboratory and other variables that. might influence the subject s psychological state, Despite the attractions of the rational choice approach its empirical failings in. economics and psychology experiments have promoted an intense interest in new. approaches A wide range of alternative models have been advocated Learning. models in which individuals make choices like those that have worked well for. them in the past have attracted particular attention from economic theorists and. experimenters Bounded rationality models in which decision makers adopt rules. that evolve slowly have had some empirical successes For example a model in. which department stores use standard mark ups to set retail prices appears to give. a better account of those prices than does a simple profit maximization model. according to which mark ups vary sensitively according to price elasticities Other. models assume that people seek acceptance by imitating their peers rely on in. tuition on heuristics or make choices that are heavily influenced by their current. emotional state Recent research try to identify parts of the brain involved in. decision making and model how brain processes a ect decisions Still others give. up on modeling choice mechanisms at all and simply concentrate on measuring. and describing what people choose, Although these various alternatives appear to have advantages for some pur. poses in this class we will focus on the decidedly useful and still dominant model. of rational choice,2 Preferences and Choice, Rational choice theory starts with the idea that individuals have preferences and. choose according to those Our first task is to formalize what that means and. precisely what it implies about the pattern of decisions we should observe. Let X be a set of possible choices In consumer choice models one might. specify that X Rn meaning for instance that there are n di erent goods beer. tortilla chips salsa etc and if x X then x x1 xn specifies quantities of. each type of good In general however the abstractness of the choice set X allows. enormous flexibility in adapting the model to various applications Some of the. controversies about the scope of economic theory concern whether the assumptions. we will make below to describe consumption choices are likely to work equally well. to describe choices about whether or whom to marry how many children to have. or whether to join a particular religious sect, Now consider an economic agent We define the agent s weak preferences over.
the set X as follows,x y x is at least as good as y. We say that x is strictly preferred to y or x y if x y but not y x We say. the agent is indi erent between x and y or x y if x y and y x. Two fundamental assumptions describe what we mean by rational choice These. are the assumptions that preferences are complete and transitive. Definition 1 A preference relation on X is complete if for all x y X either. x y or y x or both, Completeness means that if we face an agent with two choices she will neces. sarily have an opinion on which she likes more She may be indi erent but she. is never completely clueless Also because this definition does not excludes the. possibility that y x completeness implies that x x that is that the relation. is reflexive, Definition 2 A preference relation on X is transitive if whenever x y and. y z then x z, Transitivity means that an agent s weak preferences can cycle only among. choices that are indi erent That is if she weakly prefers beer to wine wine. to tequila and tequila to beer then she must be indi erent among all three. wine tequila beer The assumption that preferences are transitive is inconsistent. with certain framing e ects as the following example shows. Example Consider the following three choice problems this example is due to. Kahneman and Tversky 1984 see also MWG You are about to buy a. stereo for 125 and a calculator for 15, You learn there is a 5 calculator discount at another store branch ten.
minutes away Do you make the trip, You learn there is a 5 stereo discount at another store branch ten minutes. away Do you make the trip, You learn both items are out of stock You must go to the other branch. but as compensation you will get a 5 discount Do you care which item is. discounted, Many people answer yes to the first question but no to the second Yet it. would seem that only the goods involved and their total price should matter. If we assume that the choice set consists only of goods and total cost then. this example suggests that the framing of the choice also matters which. contradicts the rational choice framework, Casual evidence further suggests that the answer to the third question is. indi erence So even if we formulate the problem to distinguish choices ac. cording to which item is discounted this collection of choices violates tran. sitivity To see why let x be traveling to the other store to get a calculator. discount y be traveling to get a stereo discount and let z be staying at the. first store The first two choices say that x z and z y But the last says. Given preferences how will an economic agent behave We assume that given. a set of choices B X the agent will choose the element of B she prefers most. To formalize this we define the agent s choice rule. C B x B x y for all y B, to be the set of items in B the agent likes as much as any of the other alternatives.
There are several things to note about C B,C B may contain more than one element. If B is finite then C B is non empty, If B is infinite then C B might be empty To see why suppose B. x x 0 1 If the agent feels that more is better so x y if x y. If an agent s preferences are complete and transitive then her choice rule will. not be completely arbitrary as the following result shows. Proposition 1 Suppose is complete and transitive Then 1 for every finite. non empty set B C B 6 and 2 if x y A B and x C A and. y C B then x C B and y C A, Proof For 1 we proceed by mathematical induction on the number of elements. of B First suppose the number of elements is one so B x By completeness. x x so x C B Hence for all sets B with just one element C B 6. Next fix n 1 and suppose that for all sets B with exactly n elements C B 6. Let A be a set with exactly n 1 elements and let x A Then there is a set. B with exactly n elements such that A B x By the induction hypothesis. C B 6 so let y C B If y x then by definition y C A so. C A 6 By completeness the only other possibility is that x y In that. case for all z B x y z so transitivity implies that x z Since x x it. follows that x C A and hence that C A 6 Hence for every set A with. exactly n 1 elements C A 6 By the principle of mathematical induction. it follows that for every finite set A with any number of elements C A 6. which proves 1, For 2 if x y A and x C A then x y The condition y C B. means that for all z B y z Then by transitivity for all z B x z From. that and x B we conclude that x C B A symmetric argument implies. that y C A Q E D, A similar proof establishes that if is complete and transitive then any finite.
choice set has a worst element We will use that variation of the proposition to. construct a utility function below,3 Choice and Revealed Preferences. While economic theories tend to begin by making assumptions about people s. preferences and then asking what will happen it is interesting to turn this process. around Indeed much empirical work reasons in the reverse way it looks at. people s choices e g how much money they ve saved what car they bought. and tries to rationalize those choices that is figure out whether the choices are. compatible with optimization and if so what the choices imply about the agent s. preferences, What are the implications of optimization Can we always rationalize choices. as being the result of preference maximization Or does the model of preference. maximization have testable restrictions that can be violated by observed choices. In the preceding section we derived a choice rule from a given preference re. lation writing C B to emphasize the derivation In empirical data however. the evidence comes in the form of choices so it is helpful to make the choice rule. the primitive object of our theory, Definition 3 Let B be the set of subsets of X A choice rule is a function C. B B with the property that for all B B C B B, In principle we can learn an agent s choice rule by watching her in action Of. course we have to see her choose from all subsets of X but more on this in the. homework Suppose we are able to learn an agent s choice rule Can we tell if her. choice behavior is consistent with her maximizing some underlying preferences. Definition 4 A choice function C B B satisfies Houthaker s Axiom of. Revealed Preference if whenever x y A B and x C A and y C B. then x C B and y C A, Proposition 2 Suppose C B B is non empty Then there exists a complete.
and transitive preference relation on X such that C C if and only if C. satisfies HARP, That is C could be the result of an agent maximizing complete transitive. preferences if and only if C satisfies HARP, Proof First suppose that C C is the result of an agent maximizing. complete transitive preferences From the previous proposition we know that C. must satisfy HARP, Conversely suppose C satisfies HARP Define the revealed preference relation. c as follows if for some A X y A and x C A then say that x c y We. need to show three things namely that c is complete and transitive and that. For completeness pick any x y X Because C is non empty then either. C x y x in which case x c y or C x y y in which case y c x. or C x y x y in which case x c y and y c x, For transitivity suppose x c y and y c z and consider C x y z which. by hypothesis is non empty If y C x y z then by HARP x C x y z. If z C x y z then by HARP y C x y z So in every possibility. If x C A and y A then by the definition of c x c y So x C A c. This implies that C A C A c Also since C A is non empty there is some. y C A If x C A c then x c y so by HARP x C A This implies. that C A c C A Q E D, The preceding problem develops the properties of rational choice for the case.
when the entire choice function C A is observed Real data is usually less com. prehensive than that For example in consumer choice problems the relevant sets. A may be only the budget sets which is a particular subcollection of the possible. To develop a theory based on more limited observations economists have devel. oped the weak axiom of revealed preference WARP Some aspects of the theory. using WARP are developed in homework problems A more detailed treatment is. found in the recommended textbooks, So far we have a pretty abstract model of choice As a step toward having a more. tractable mathematical formulation of decision making we now introduce the idea. of utility which assigns a numerical ranking to each possible choice For example. 2 Preferences and Choice Rational choice theory starts with the idea that individuals have preferences and chooseaccordingtothose Our rst task is to formalize what that means and precisely what it implies about the pattern of decisions we should observe Let Xbe a set of possible choices In consumer choice models one might specify that X Rn meaning for instance that there are

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