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COPYRIGHT NOTICE: Ariel Rubinstein: Lecture Notes in Microeconomic Theory is published by Princeton University Press and copyrighted, c 2006, by Princeton University Press. All rights reserved. No part of this book may be repro duced in any form by any electronic or mechanical means (including photocopying, recordin g, or information storage and retrieval) without permission in writing from the publisher , except for reading and browsing via the World Wide Web. Users are not permitted to mount this file on any network servers. Follow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu Lecture Notes in Microeconomic Theory Ariel Rubinstein Updates to the Printed Version The file you are viewing contains the prin ted version of the book. In relevant places throughout the text you will find small icons indicating the existence of updates to the text: A green icon indicates an addition to the text at this point.his line. The corrected and added text can be obtained from the author's homepage at http://arielrubinstein.tau.ac.il/ October 21, 2005 12:18 master Sheet number 1 Page number 1 October 21, 2005 12:18 master Sheet number 2 Page number 2 October 21, 2005 12:1master Sheet number 3 Page number i Lecture Notes in Microeconomic Theory October 21, 2005 12:18 master Sheet number 4 Page number ii October 21, 2005 master Sheet number 5 Page number iii Lecture Notes in Microeconomic Theory The Economic Agent Ariel Rubinstein PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD October 21, 2005 12:18 master Sheet number 6 Page number iv Copyright © 2006 by Princeton University Press. Requests for permission to reproduce material from this work should be sent to Permissions, Princeton University Press. Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock, Oxfordshire OX20 1SY All Rights Reserved Library of Congress Cataloging-in-Publication Data Rubinstein, Ariel. Lecture notes in microeconomic theory : the economic agent / Ariel Rubinstein. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-691-12030-0 (cl : alk. paper) ISBN-13: 978-0-691-12031-7 (pbk. : alk. paper) ISBN-10: 0-691-12030-7 (cl : alk. paper) ISBN-10: 0-691-12031-5 (pbk. : alk. paper) 1. Microeconomics. 2. Economics. I. Title. HB172.R72 2006 338.5 01–dc22 2005047631 British Library Cataloging-in-Publication Data is available This book has been composed in ITC Stone. Printed on acid-free paper. ∞ pup.princeton.edu Printed in the United States of America 10987654321 October 21, 2005 12:18 master Sheet number 7 Page number v Contents Preface vii Introduction ix Lecture 1. Preferences 1 Problem Set 1 10 Lecture 2. Utility 12 Problem Set 2 21 Lecture 3. Choice 24 Problem Set 3 37 Lecture 4. Consumer Preferences 40 Problem Set 4 50 Lecture 5. Demand: Consumer Choice 52 Problem Set 5 66 Lecture 6. Choice over Budget Sets and the Dual Problem 68 Problem Set 6 76 Lecture 7. Production 79 Problem Set 7 85 Lecture 8. Expected Utility 87 Problem Set 8 97 Lecture 9. Risk Aversion 100 Problem Set 9 112 Lecture 10. Social Choice 114 Problem Set 10 122 Review Problems 124 References 131 October 21, 2005 12:18 master Sheet number 8 Page number vi October 21, 2005 12:18 master Sheet number 9 Page number vii Preface This short book contains my lecture notes for the ﬁrst quarter of a microeconomics course for PhD or Master’s degree economics stu- dents. The lecture notes were developed over a period of almost 15 years during which I taught the course, or parts of it, at Tel Aviv, Princeton, and New York universities. I am publishing the lecture notes with some hesitation. Several superb books are already on the shelves. I most admire Kreps (1990), which pioneered the transformation of the game theoretic revolu- tion in economic theory from research papers into textbooks. His book covers the material in depth and includes many ideas for fu- ture research. Mas-Colell, Whinston, and Green (1995) continued this trend with a very comprehensive and detailed textbook. There are three other books on my short list: Bowles (2003), which brings economics back to its authentic, political economics roots; Jehle and Reny (1997), with its very precise style; and the classic Varian (1984). These ﬁve books constitute an impressive collection of textbooks for the standard advanced microeconomics course. My book covers only the ﬁrst quarter of the standard course. It does not aim to compete but to supplement these books. I had it published only because I think that some of the didactic ideas in the book might be beneﬁcial to students and teachers, and it is to this end that I insisted on retaining the lecture notes style. Throughout the book I use only male pronouns. This is my de- liberate choice and does not reﬂect the policy of the editors or the publishers. I believe that continuous reminders of the he/she issue simply divert readers’ attention. Language is of course very impor- tant in shaping our thinking and I don’t dispute the importance of the type of language we use. But I feel it is more effective to raise the issue of discrimination against women in the discussion of gender- related issues, rather than raising ﬂags on every page of a book on economic theory. A special feature of this book is that it is also posted on the Internet and access is entirely free. My intention is to update the book annu- ally (or at least in years when I teach the course). To access the latest October 21, 2005 12:18 master Sheet number 10 Page number viii viii Preface electronic version of the book, visit: http://arielrubinstein.tau.ac.il/ micro1/. I would like to thank all my teaching assistants, who contributed comments during the many years I taught the course: Rani Spiegler, Kﬁr Eliaz, Yoram Hamo, Gabi Gayer and Tamir Tshuva at Tel Aviv University; Bilge Yilmiz, Ronny Razin, Wojciech Olszewski, Attila Ambrus, Andrea Wilson, Haluk Ergin and Daisuke Nakajima at Princeton; and Sophie Bade and Anna Ingster at NYU. Special thanks are due to Sharon Simmer and Raﬁ Aviav who helped me with the English editing and to Gabi Gayer and Daniel Wasserteil who pre- pared the ﬁgures. October 21, 2005 12:18 master Sheet number 11 Page number ix Introduction As a new graduate student, you are at the beginning of a new stage of your life. In a few months you will be overloaded with deﬁ- nitions, concepts, and models. Your teachers will be guiding you into the wonders of economics and will rarely have the time to stop to raise fundamental questions about what these models are sup- posed to mean. It is not unlikely that you will be brainwashed by the professional-sounding language and hidden assumptions. I am afraid I am about to initiate you into this inevitable process. Still, I want to use this opportunity to pause for a moment and alert you to the fact that many economists have strong and conﬂicting views about what economic theory is. Some see it as a set of theories that can (or should) be tested. Others see it as a bag of tools to be used by economic agents, and yet others see it as a framework through which professional and academic economists view the world. My own view may disappoint those of you who have come to this course with practical motivations. In my view, economic the- ory is no more than an arena for the investigation of concepts we use in thinking about economics in real life. What makes a theoretical model “economics” is that the concepts we are analyzing are taken from real-life reasoning about economic issues. Through the inves- tigation of these concepts we indeed try to understand reality better, and the models provide a language that enables us to think about economic interactions in a systematic way. But I do not view eco- nomic models as an attempt to describe the world or to provide tools for predicting the future. I object to looking for an ultimate truth in economic theory, and I do not expect it to be the foundation for any policy recommendation. Nothing is “holy” in economic theory and everything is the creation of people like yourself. Basically, this course is about a certain class of economic concepts and models. Although we will be studying formal concepts and mod- els, they will always be given an interpretation. An economic model differs substantially from a purely mathematical model in that it is a combination of a mathematical model and its interpretation. The names of the mathematical objects are an integral part of an eco- nomic model. When mathematicians use terms such as “ﬁeld” or October 21, 2005 12:18 master Sheet number 12 Page number x x Introduction “ring” which are in everyday use, it is only for the sake of conve- nience. When they name a collection of sets a “ﬁlter,” they are doing so in an associative manner; in principle, they could call it “ice cream cone.” When they use the term “good ordering” they are not making an ethical judgment. In contrast to mathematics, interpretation is an essential ingredient of any economic model. It is my hope that some of you will react and attempt to change what is currently called economic theory, and that some of you will acquire alternative ways of thinking about economic and social in- teractions. At the very least, the course should teach you to ask hard questions about economic models and in what sense they are rele- vant to the economic questions we are interested in. I hope that you walk away from this course with the recognition that the answers are not as obvious as they might appear. Microeconomics In this course we deal only with microeconomics, a collection of models in which the primitives are details about the behavior of units called economic agents. Microeconomic models investigate as- sumptions about economic agents’ activities and about interactions between these agents. An economic agent is the basic unit operat- ing in the model. Most often, we do have in mind that the eco- nomic agent is an individual, a person with one head, one heart, two eyes, and two ears. However, in some economic models, an economic agent is taken to be a nation, a family, or a parliament. At other times, the “individual” is broken down into a collection of economic agents, each operating in distinct circumstances and each regarded as an economic agent. We should not be too cheerful about the statement that an eco- nomic agent in microeconomics is not constrained to being an in- dividual. The facade of generality in economic theory might be misleading. We have to be careful and aware that when we take an economic agent to be a group of individuals, the reasonable as- sumptions we might impose on it are distinct from those we might want to impose on a single individual. In any case, with a particu- lar economic scenario in mind, the decision about how to think of that scenario in the framework of a microeconomic model involves a decision about whom we want to view as the primitives. October 21, 2005 12:18 master Sheet number 13 Page number xi Introduction xi An economic agent is described in our models as a unit that re- sponds to a scenario called a choice problem, where the agent must make a choice from a set of available alternatives. The economic agent appears in the microeconomic model with a speciﬁed delibera- tion process he uses to make a decision. In most of current economic theory, the deliberation process is what is called rational choice. The agent decides what action to take through a process in which he 1. asks himself “What is desirable?” 2. asks himself “What is feasible?” 3. chooses the most desirable from among the feasible alterna- tives. Rationality in economics does not contain judgments about de- sires. A rational agent can have preferences which the entire world views as being against the agent’s interest. Furthermore, economists are fully aware that almost all people, almost all the time, do not practice this kind of deliberation. Nevertheless, we ﬁnd the investigation of economic agents who follow the rational process to be important, since we often refer to rational decision making in life as an ideal process. It is mean- ingful to talk about the concept of “being good” even in a society where all people are evil; similarly, it is meaningful to talk about the concept of a “rational man” and about the interactions between rational economic agents even if all people systematically behave in a nonrational manner. October 21, 2005 12:18 master Sheet number 14 Page number xii October 21, 2005 12:1master Sheet number 15 Page number xiii Lecture Notes in Microeconomic Theory October 21, 2005 12:18 master Sheet number 16 Page number xiv October 21, 2005 12:18 master Sheet number 17 Page number 1 LECTURE 1 Preferences Preferences Although we are on our way to constructing a model of rational choice, we begin the course with an “exercise”: formulating the no- tion of “preferences” independently of the concept of choice. We view preferences as the mental attitude of an individual (economic agent) toward alternatives.We seek to develop a “proper” formaliza- tion of this concept, which plays such a central role in economics. Imagine that you want to fully describe the preferences of an agent toward the elements in a given set X. For example, imagine that you want to describe your own attitude toward the universities you apply to before ﬁnding out to which of them you have been admit- ted. What must the description include? What conditions must the description fulﬁll? We take the approach that a description of preferences should fully specify the attitude of the agent toward each pair of elements in X. For each pair of alternatives, it should provide an answer to the question of how the agent compares the two alternatives. We present two versions of this question. For each version we formu- late the consistency requirements necessary to make the responses “preferences” and examine the connection between the two formal- izations. The Questionnaire Q Let us think about the preferences on a set X as answers to a “long” questionnaire Q which consists of all quiz questions of the type: October 21, 2005 12:18 master Sheet number 18 Page number 2 2 Lecture One Q(x,y) (for all distinct x and y in X) How do you compare x and y? Tick one and only one of the following three options: I prefer x to y (this answer is denoted as x ▯ y). I prefer y to x (this answer is denoted by y ▯ x). I am indifferent (this answer is denoted by I). A “legal” answer to the questionnaire is a response in which the respondent ticks exactly one of the boxes in each question. We do not allow the decision maker to refrain from answering a question or to tick more than one answer. Furthermore, we do not allow him to respond with answers that demonstrate a lack of ability to compare, such as: They are incomparable. I don’t know what x is. I have no opinion. Or a dependence on other factors such as: It depends on what my parents think. It depends on the circumstances (sometimes I prefer x but usu- ally I prefer y). Or the intensity of preferences such as: I somewhat prefer x. I love x and I hate y. Or confusion such as: I both prefer x over y and y over x. I can’t concentrate right now. The constraints that we place on the legal responses of the agents constitute our implicit assumptions. Particularly important are the assumptions that the elements in the set X are all comparable, that the individual has an opinion about all elements in the set X and that we do not allow him to specify the intensity of preferences. A legal answer to the questionnaire can be formulated as a func- tion f which assigns to any pair (x,y) of distinct elements in X exactly one of the three “values”: x ▯ y or y ▯ x or I, with the inter- pretation that f (x,y) is the answer to the question Q(x,y). (Alterna- tively, we can use the terminology of the soccer “betting” industry and say that f (x,y) must be 1, 2, or × with the interpretation that October 21, 2005 12:18 master Sheet number 19 Page number 3 Preferences 3 f (x,y) = 1 means that x is better than y, f (x,y) = 2 means that y is better than x and f (x,y) =× means indifference.) Not all legal answers to the questionnaire Q qualify as preferences over the set X. We will adopt two “consistency” restrictions: First, the answer to Q(x,y) must be identical to the answer to Q(y,x). In other words, we want to exclude the common “framing effect” by which people who are asked to compare two alternatives tend to prefer the “ﬁrst” one. Second, we require that the answers exhibit “transitivity.” In other words, the answers to Q(x,y) and Q(y,z) must be consistent with the answer to Q(x,z) in the following sense: If “x is preferred to y” and “y is preferred to z” then “x is preferred to z,” and if the answers to the two questions Q(x,y) and Q(y,z) are “indifference” then so is the answer to Q(x,z). To summarize, here is my favorite formalization of the notion of preferences: Deﬁnition 1 Preferences on a set X are a function f that assigns to any pair (x,y) of distinct elements in X exactly one of the three “values” x ▯ y, y ▯ x or I so that for any three different elements x, y and z in X, the following two properties hold: • No order effect: f (x,y) = f (y,x). • Transitivity: if f (x,y) = x ▯ y and f (y,z) = y ▯ z then f (x,z) = x ▯ z and if f (x,y) = I and f (y,z) = I then f (x,z) = I. Note again that I, x ▯ y, and y ▯ x are merely symbols representing verbal answers. Needless to say, the choice of symbols is not an arbitrary one. (Why do I use the notation I and not x ∼ y?) A Discussion of Transitivity The transitivity property is an appealing property of preferences. How would you react if somebody told you he prefers x to y, y to z and z to x? You would probably feel that his answers are “confused.” Furthermore, it seems that, when confronted with an intransitivity October 21, 2005 12:18 master Sheet number 20 Page number 4 4 Lecture One in their responses, people are embarrassed and want to change their answers. Before the lecture, students in Tel Aviv had to ﬁll out a question- naire similar to Q regarding a set X that contains nine alternatives, each specifying the following four characteristics of a travel pack- age: location (Paris or Rome), price, quality of the food, and quality of the lodgings. The questionnaire included only thirty six ques- tions since for each pair of alternatives x and y, only one of the questions, Q(x,y) or Q(y,x), was randomly selected to appear in the questionnaire (thus the dependence on order of an individual’s re- sponse could not be checked within the experimental framework). In the 2004 group, out of eighteen MA students, only two had no intransitivities in their answers, and the average number of triples in which intransitivity existed was almost nine. Many of the viola- tions of transitivity involved two alternatives that were actually the same, but differed in the order in which the characteristics appeared in the description. “A weekend in Paris at a four-star hotel with food quality Zagat 17 for $574,” and “A weekend in Paris for $574 with food quality Zagat 17 at a four-star hotel.” All students expressed indifference between the two alternatives, but in a comparison of these two alternatives to a third alternative—“A weekend in Rome at a ﬁve-star hotel with food quality Zagat 18 for $612”—half of the students gave responses that violated transitivity. In spite of the appeal of the transitivity requirement, note that when we assume that the attitude of an individual toward pairs of alternatives is transitive, we are excluding individuals who base their judgments on “procedures” that cause systematic violations of tran- sitivity. The following are two such examples. 1. Aggregation of considerations as a source of intransitivity. In some cases, an individual’s attitude is derived from the aggregation of more basic considerations. Consider, for example, a case where X ={ a,b,c} and the individual has three primitive considerations in mind. The individual ﬁnds one alternative better than the other if a majority of considerations support the ﬁrst alterna- tive. This aggregation process can yield intransitivities. For ex- ample, if the three considerations rank the alternatives as follows: a ▯1b ▯ c1 b ▯ c2▯ a a2d c ▯ a ▯ 3, the3 the individual de- termines a to be preferred over b, b over c, and c over a, thus violating transitivity. October 21, 2005 12:18 master Sheet number 21 Page number 5 Preferences 5 2. The use of similarities as an obstacle to transitivity. In some cases, the decision maker expresses indifference in a comparison be- tween two elements that are too “close” to be distinguishable. For example, let X =▯ (the set of real numbers). Consider an indi- vidual whose attitude is “the more the better”; however, he ﬁnds it impossible to determine whether a is greater than b unless the difference is at least 1. He will assign f (x,y) = x ▯ y if x ≥ y − 1 and f (x,y) = I if |x − y| < 1. This is not a preference relation since 1.5 ∼ 0.8 and 0.8 ∼ 0.3, but it is not true that 1.5 ∼ 0.3. Did we require too little? Another potential criticism of our deﬁni- tion is that our assumptions might have been too weak and that we did not impose some reasonable further restrictions on the concept of preferences. That is, there are other similar consistency require- ments we may impose on a legal response to qualify it as a descrip- tion of preferences. For example, if f (x,y) = x ▯ y and f (y,z) = I, we would naturally expect that f (x,z) = x ▯ z. However, this ad- ditional consistency condition was not included in the above def- inition since it follows from the other conditions: If f (x,z) = I, then by the assumption that f (y,z) = I and by the no order effect, f (z,y) = I, and thus by transitivity f (x,y) = I (a contra- diction). Alternatively, if f (x,z) = z ▯ x, then by no order effect f (z,x) = z ▯ x, and by f (x,y) = x ▯ y and transitivity f (z,y) = z ▯ y (a contradiction). Similarly, note that for any preferences f , we have if f (x,y) = I and f (y,z) = y ▯ z, then f (x,z) = x ▯ z. The Questionnaire R A second way to think about preferences is through an imaginary questionnaire R consisting of all questions of the type: R(x,y) (for all x,y ∈ X, not necessarily distinct). “Is x at least as preferred as y?” Tick one and only one of the following two options: Yes No October 21, 2005 12:18 master Sheet number 22 Page number 6 6 Lecture One By a “legal” response we mean that the respondent ticks exactly one of the boxes in each question. To qualify as preferences a legal response must also satisfy two conditions: 1. The answer to at least one of the questions R(x,y) and R(y,x) must be Yes. (In particular, the “silly” question R(x,x) which appears in the questionnaire must get a Yes response.) 2. For every x,y,z ∈ X, if the answers to the questions R(x,y) and R(y,z) are Yes, then so is the answer to the question R(x,z). We identify a response to this questionnaire with the binary rela- tion on the set X deﬁned by x y if the answer to the question R(x,y) is Yes. n (Reminder:An n-ary relation on X is a subset of X . Examples: “Being a parent of” is a binary relation on the set of human beings; “being a hat” is an unary relation on the set of objects; “x + y = z”is a 3-nary relation on the set of numbers; “x is better than y more than ▯ ▯ x is better than y ” is 4-nary relation on a set of alternatives, etc. An n-ary relation on X can be thought of as a response to a questionnaire regarding all n-tuples of elements of X where each question can get only a Yes/No answer.) This brings us to the “traditional” deﬁnition: Deﬁnition 2 A preference on a set X is a binary relation on X satisfying: • Completeness: For any x,y∈X, x y or y x. • Transitivity: For any x,y,z∈X,if x y and y z, then x z. The Equivalence of the Two Deﬁnitions We have presented two deﬁnitions of preferences on the set X.W e now proceed to show their equivalence. There are many ways to construct “a one-to-one correspondence” between the objects satis- fying the two deﬁnitions. But, when we think about the equivalence of two deﬁnitions in economics we are thinking about much more than the existence of a one-to-one correspondence: the correspon- dence has to preserve the interpretation. Note the similarity to the notion of an isomorphism in mathematics. For example, an iso- morphism between two topological spaces X and Y is a one-to-one October 21, 2005 12:18 master Sheet number 23 Page number 7 Preferences 7 Table 1.1 A response to Q(x,y) and Q(y,x) A response to R(x,y) and R(y,x) x ▯ y Yes, No I Yes, Yes y ▯ x No, Yes function from X onto Y that is required to preserve the open sets. In economics, the one-to-one correspondence is required to preserve the more informal concept of interpretation. We will now construct a one-to-one and onto correspondence, Translation, between answers to Q that qualify as preferences by the ﬁrst deﬁnition and answers to R that qualify as preferences by the second deﬁnition, such that the correspondence preserves the meaning of the responses to the two questionnaires. In other words, Translation is a “bridge” between the responses to Q that qualify as preferences and the responses to R that qualify as preferences. To illustrate the correspondence imagine that you have two books. Eachpageintheﬁrstbookisaresponsetothequestionnaire Q which qualiﬁes as preferences by the ﬁrst deﬁnition. Each page in the sec- ond book is a response to the questionnaire R which qualiﬁes as preferences by the second deﬁnition. The correspondence matches each page in the ﬁrst book with a unique page in the second book, so that a reasonable person will recognize that the different responses to the two questionnaires reﬂect the same mental attitudes towards the alternatives. Since we assume that the answers to all questions of the type R(x,x) are “Yes,” the classiﬁcation of a response to R as a preference only requires the speciﬁcation of the answers to questions R(x,y), where x ▯= y. Table 1.1 presents the translation of responses. This translation preserves the interpretation we have given to the responses, that is, “I prefer x to y” has the same meaning as the statement “I ﬁnd x to be at least as good as y, but I don’t ﬁnd y to be at least as good as x.” The following observations complete the proof that Translation is indeed a one-to-one correspondence from the set of preferences, as given by deﬁnition 1, onto the set of preferences as given by deﬁnition 2. October 21, 2005 12:18 master Sheet number 24 Page number 8 8 Lecture One By the assumption on Q of a no order effect, for any two alterna- tives x and y, one and only one of the following three answers was received for both Q(x,y) and Q(y,x): x ▯ y, I and y ▯ x. Thus, the responses to R(x,y) and R(y,x) are well deﬁned. Next we verify that the response to R that we have constructed with the table is indeed a preference relation (by the second deﬁni- tion). Completeness: In each of the three rows, the answers to at least one of the questions R(x,y) and R(y,x) is afﬁrmative. Transitivity: Assume that the answers to R(x,y) and R(y,z) are afﬁrmative. This implies that the answer to Q(x,y) is either x ▯ y or I, and the answer to Q(y,z) is either y ▯ z or I. Transitivity of Q implies that the answer to Q(x,z) must be x ▯ z or I, and therefore the answer to R(x,z) must be afﬁrmative. To see that Translation is indeed a one-to-one correspondence, note that for any two different responses to the questionnaire Q there must be a question Q(x,y) for which the responses differ; there- fore, the corresponding responses to either R(x,y) or R(y,x) must differ. It remains to be shown that the range of the Translation function includes all possible preferences as deﬁned by the second deﬁnition. Let be preferences in the traditional sense (a response to R). We have to specify a function f , a response to Q, which is converted by Translation to . Read from right to left, the table provides us with such a function f . By the completeness of , for any two elements x and y, one of the entries in the right-hand column is applicable (the fourth option, that the two answers to R(x,y) and R(y,x) are “No,” is excluded), and thus the response to Q is well deﬁned and by deﬁnition satisﬁes no order effect. We still have to check that f satisﬁes the transitivity condition. If F(x,y) = x ▯ y and F(y,z) = y ▯ z, then x y and not y x and y z and not z y. By transitivity of , x z. In addition, not z x since if z x, then the transitivity of would imply z y. If F(x,y) = I and F(y,z) = I, then x y, y x, y z and z y.B y transitivity of , both x z and z x, and thus F(x,z) = I. Summary From now on we will use the second deﬁnition, that is, a preference on X is a binary relation on a set X satisfying Completeness and October 21, 2005 12:18 master Sheet number 25 Page number 9 Preferences 9 Transitivity. For a preference relation , we will use the notation x ∼ y when both x y and y x; the notation x ▯ y will stand for if x y and not y x. Bibliographic Notes Recommended readings: Kreps 1990, 17–24; Mas-Colell et al. 1995, chapter 1, A–B. Fishburn (1970) contains a comprehensive treatment of prefer- ence relations. October 21, 2005 12:18 master Sheet number 26 Page number 10 Problem Set 1 Problem 1. (Easy) Let be a preference relation on a set X. Deﬁne I(x) to be the set of all y ∈ X for which y ∼ x. Show that the set (of sets!) {I(x)|x ∈ X} is a partition of X, i.e., • For all x and y, either I(x) = I(y) or I(x) ∩ I(y) =∅ . • For every x ∈ X, there is y ∈ X such that x ∈ I(y). Problem 2. (Standard) Kreps (1990) introduces another formal deﬁnition for preferences. His prim- itive is a binary relation P interpreted as “strictly preferred.” He requires P to satisfy: • Asymmetry: For no x and y do we have both xPy and yPx. • Negative-Transitivity: For all x, y, and z ∈ X,if xPy, then for any z either xPz or zPy (or both). Explain the sense in which Kreps’ formalization is equivalent to the tra- ditional deﬁnition. Problem 3. (Standard) In economic theory we are often interested in other types of binary rela- tions, for example, the relation xSy:“ x and y are almost the same.” Suggest properties that would correspond to your intuition about such a concept. Problem 4. (Difﬁcult. Based on Kannai and Peleg 1984.) Let Z be a ﬁnite set and let X be the set of all nonempty subsets of Z. Let be a preference relation on X (not Z). Consider the following two properties of preference relations on X: a. If A B and C is a set disjoint to both A and B, then A ∪ C B ∪ C, and if A ▯ B and C is a set disjoint to both A and B, then A ∪ C ▯ B ∪ C. b. If x ∈ Z and {x}▯{ y} for all y ∈ A, then A ∪{ x}▯ A, and if x ∈ Z and {y}▯{ x} for all y ∈ A, then A ▯ A ∪{ x}. October 21, 2005 12:18 master Sheet number 27 Page number 11 Preferences 11 Discuss the plausibility of the properties in the context of interpreting as the attitude of the individual toward sets from which he will have to make a choice at a “second stage.” Provide an example of a preference relation that • Satisﬁes the two properties. • Satisﬁes the ﬁrst but not the second property. • Satisﬁes the second but not the ﬁrst property. Show that if there are x,y, and z ∈ Z such that {x}▯{ y}▯{ z}, then there is no preferene relation satisfying both properties. Problem 5. (Fun) Listen to the illusion called the Shepard Scale. (Currently, it is avail- able at http://www.sandlotscience.com/Ambiguous/ShpTones1.htm and http://asa.aip.org/demo27.html.) Can you think of any economic analogies? October 21, 2005 12:18 master Sheet number 28 Page number 12 LECTURE 2 Utility The Concept of Utility Representation Think of examples of preferences. In the case of a small number of alternatives, we often describe a preference relation as a list arranged from best to worst. In some cases, the alternatives are grouped into a small number of categories and we describe the preferences on X by specifying the preferences on the set of categories. But, in my experience, most of the examples that come to mind are similar to: “I prefer the taller basketball player,” “I prefer the more expensive present,” “I prefer a teacher who gives higher grades,” “I prefer the person who weighs less.” Common to all these examples is that they can naturally be spec- iﬁed by a statement of the form “x y if V(x) ≥ V(y)” (or V(x) ≤ V(y)), where V : X →▯ is a function that attaches a real number to each element in the set of alternatives X. For example, the prefer- ences stated by “I prefer the taller basketball player” can be expressed formally by: X is the set of all conceivable basketball players, and V(x) is the height of player x. Note that the statement x y if V(x) ≥ V(y) always deﬁnes a pref- erence relation since the relation ≥ on ▯ satisﬁes completeness and transitivity. Even when the description of a preference relation does not in- volve a numerical evaluation, we are interested in an equivalent nu- merical representation. We say that the function U : X →▯ represents the preference if for all x and y ∈ X, x y if and only if U(x) ≥ U(y). If the function U represents the preference relation , we refer to it as a utility function and we say that has a utility representation. It is possible to avoid the notion of a utility representation and to “do economics” with the notion of preferences. Nevertheless, we usually use utility functions rather than preferences as a means of de- scribing an economic agent’s attitude toward alternatives, probably October 21, 2005 12:18 master Sheet number 29 Page number 13 Utility 13 because we ﬁnd it more convenient to talk about the maximization of a numerical function than of a preference relation. Note that when deﬁning a preference relation using a utility func- tion, the function has an intuitive meaning that carries with it addi- tional information. In contrast, when the utility function is formed in order to represent an existing preference relation, the utility func- tion has no meaning other than that of representing a preference relation. Absolute numbers are meaningless in the latter case; only relative order has meaning. Indeed, if a preference relation has a utility representation, then it has an inﬁnite number of such repre- sentations, as the following simple claim shows: Claim: If U represents , then for any strictly increasing function f :▯→▯ , the function V(x) = f (U(x)) represents as well. Proof: a b iff U(a) ≥ U(b) (since U represents ) iff f (U(a)) ≥ f (U(b)) (since f is strictly increasing) iff V(a) ≥ V(b). Existence of a Utility Representation If any preference relation could be represented by a utility function, then it would “grant a license” to use utility functions rather than preference relations with no loss of generality. Utility theory inves- tigates the possibility of using a numerical function to represent a preference relation and the possibility of numerical representations carrying additional meanings (such as, a is preferred to b more than c is preferred to d). We will now examine the basic question of “utility theory”: Under what assumptions do utility representations exist? Our ﬁrst observation is quite trivial. When the set X is ﬁnite, there is always a utility representation. The detailed proof is pre- sented here mainly to get into the habit of analytical precision. We October 21, 2005 12:18 master Sheet number 30 Page number 14 14 Lecture Two start with a lemma regarding the existence of minimal elements (an element a ∈ X is minimal if a x for any x ∈ X). Lemma: In any ﬁnite set A ⊆ X there is a minimal element (similarly, there is also a maximal element). Proof: By induction on the size of A.I f A is a singleton, then by complete- ness its only element is minimal. For the inductive step, let A be of cardinality n + 1 and let x ∈ A. The set A−{x} is of cardinality n and by the inductive assumption has a minimal element denoted by y.I f x y, then y is minimal in A.I y x, then by transitivity z x for all z ∈ A−{x} and thus x is minimal. Claim: If is a preference relation on a ﬁnite set X, then has a utility representation with values being natural numbers. Proof: We will construct a sequence of sets inductively. Let X be 1he sub- set of elements that are minimal in X. By the above lemma, X 1 is not empty. Assume we have constructed the sets X ,... ,X1.fI k X = X ∪ X ∪ ... ∪ X we are done. If not, deﬁne X to be the 1 2 k k+1 set of minimal elements in X − X − X 1···−2X . By the kemma X k+1▯=∅ . Since X is ﬁnite we must be done after at most |X| steps. Deﬁne U(x) = k if x ∈ X . khus, U(x) is the step number at which x is “eliminated.” To verify that U represents , let a b. Then b / X − X − X −···− X and thus U(a) ≥ U(b). 1 2 U(a) Without any further assumptions on the preferences, the exis- tence of a utility representation is guaranteed when the set X is October 21, 2005 12:18 master Sheet number 31 Page number 15 Utility 15 countable (recall that X is countable and inﬁnite if there is a one-to- one function from the natural numbers to X, namely, it is possible to specify an enumeration of all its members {x } n ). n=1,2,... Claim: If X is countable, then any preference relation on X has a utility representation with a range (−1,1). Proof: Let {x n be an enumeration of all elements in X. We will construct the utility function inductively. Set U(x ) = 0. Assume that you 1 have completed the deﬁnition of the values U(x ),... ,U(x 1 n−1 ) so that x k x ifl U(x ) ≥ k(x )f.I l xn is indifferent to x fok some k < n, then assign U(x ) = n(x ). If kot, by transitivity, all num- bers in the set {U(x )|kx ≺ k }∪{−n1} are below all numbers in the set {U(x k| x n x }∪k 1}. Choose U(x ) to benbetween the two sets. This guarantees that for any k < n we have x x iffnU(x )k≥ U(x ).n k Thus, the function we deﬁned on {x ,... ,x } represents the prefer- 1 n ence on those elements. To complete the proof that U represents , take any two elements, x and y ∈ X. For some k and l we have x = x and y = k . The abovel applied to n = max{k,l} yields x x kff U(l ) ≥ U(xk). l Lexicographic Preferences Lexicographic preferences are the outcome of applying the follow- ing procedure for determining the ranking of any two elements in a set X. The individual has in mind a sequence of criteria that could be used to compare pairs of elements in X. The criteria are applied in a ﬁxed order until a criterion is reached that succeeds in distinguish- ing between the two elements, in that it determines the preferred alternative. Formally, let ( ) k k=1,...,Ke a K-tuple of orderings over the set X. The lexicographic ordering induced by those orderings is deﬁned by x y if (1) there is k such that for all k < k we have ∗ L x ∼ k and x ▯ k∗ y or (2) x ∼ k for all k. Verify that isLa preference relation. October 21, 2005 12:18 master Sheet number 32 Page number 16 16 Lecture Two Example: Let X be the unit square, i.e., X =[ 0,1]×[ 0,1]. Let x y if x ≥ y . k k k The lexicographic ordering indLced from and i1: (a ,a2) 1 2 L (b1,b2) if 1 > b 1r both a =1b and1a ≥ b 2 (Thu2, in this example, the left component is the primary criterion while the right compo- nent is the secondary criterion.) We will now show that the preferences do not Lave a utility representation. The lack of a utility representation excludes lexico- graphic preferences from the scope of standard economic models in spite of the fact that they constitute a simple and commonly used procedure for preference formation. Claim: The preference relation onL[0,1]×[ 0,1], which is induced from the relations x k if x ≥ky (kk= 1,2), does not have a utility rep- resentation. Proof: Assume by contradiction that the function u : X →▯ represents . L For any a ∈[ 0,1], (a,1) ▯ (L,0) we thus have u(a,1)> u(a,0). Let q(a) be a rational number in the nonempty interval I = (u(a,0), a u(a,1)). The function q is a function from X into the set of ra- tional numbers. It is a one-to-one function since if b > a then (b,0) ▯ (a,1) and therefore u(b,0)> u(a,1). It follows that the in- L tervals a and Ibare disjoint and thus q(a) ▯= q(b). But the cardinality of the rational numbers is lower than that of the continuum, a con- tradiction. Continuity of Preferences In economics we often take the set X to be an inﬁnite subset of a Euclidean space. The following is a condition that will guarantee the existence of a utility representation in such a case. The basic in- tuition, captured by the notion of a continuous preference relation, October 21, 2005 12:18 master Sheet number 33 Page number 17 Utility 17 Figure 2.1 Two deﬁnitions of continuity of preferences. is that if a is preferred to b, then “small” deviations from a or from b will not reverse the ordering. Deﬁnition C1: A preference relation on X is continuous if whenever a ▯ b (namely, it is not true that b a), there are neighborhoods (balls) B and a b around a and b, respectively, such that for all x ∈ B aad y ∈ B , xb▯ y (namely, it is not true that y x). (See ﬁg. 2.1.) Deﬁnition C2: A preference relation on X is continuous if the graph of (that is, the set {(x,y)|x y}⊆ X × X) is a closed set (with the product topology); that is, if {(a ,b )} is a sequence of pairs of elements in X n n satisfying a n b fnr all n and a → anand b → b, tnen a b. (See ﬁg. 2.1.) Claim: The preference relation on X satisﬁes C1 if and only if it satisﬁes C2. October 21, 2005 12:18 master Sheet number 34 Page number 18 18 Lecture Two Proof: Assume that on X is continuous according to C1. Let {(a ,b )} n n be a sequence of pairs satisfying a bnfor anl n and a → a andn b n b. If it is not true that a b (that is, b ▯ a), then there exist two balls B aad B arobnd a and b, respectively, such that for all y ∈ B bnd x ∈ B , a ▯ x. There is an N large enough such that for all n > N, both b ∈ n andba ∈ B .nThereaore, for all n > N, we have b n a ,nwhich is a contradiction. Assume that is continuous according to C2. Let a ▯ b. Denote by B(x,r) the set of all elements in X distanced less than r from x. Assume by contradiction that for all n there exist a ∈ Bna,1/n) and b n B(b,1/n) such that b an. Thensequence (b ,a ) convernes to (b,a); by the second deﬁnition (b,a) is within the graph of , that is, b a, which is a contradiction. Remarks 1. If on X is represented by a continuous function U, then is continuous. To see this, note that if a ▯ b then U(a)> U(b). Let ε = (U(a) − U(b))/2. By the continuity of U, there is a δ> 0 such that for all x distanced less than δ from a, U(x)> U(a) − ε, and for all y distanced less than δ from b, U(y)< U(b) + ε. Thus, for x and y within the balls of radius δ around a and b, respectively, x ▯ y. 2. The lexicographic preferences which were used in the counterex- ample to the existence of a utility representation are not contin- uous. This is because (1,1) ▯ (1,0), but in any ball around (1,1) there are points inferior to (1,0). 3. Note that the second deﬁnition of continuity can be applied to any binary relation over a topological space, not just to a prefer- ence relation. For example, the relation = on the real numbers (▯ ) is continuous while the relation ▯= is not. Debreu’s Theorem Debreu’s theorem, which states that continuous preferences have a continuous utility representation, is one of the classic results in October 21, 2005 12:18 master Sheet number 35 Page number 19 Utility 19 economic theory. For a complete proof of Debreu’s theorem see Debreu 1954, 1960. Here we prove only that continuity guarantees the existence of a utility representation. Lemma: n If is a continuous preference relation on a convex set X ⊆▯ , and if x ▯ y, then there exists z in X such that x ▯ z ▯ y. Proof: Assume not. Construct a sequence of points on the interval that connects the points x and y in the following way. First deﬁne x = x 0 and y 0 y. In the inductive step we have two points, x and y ,on t t the line that connects x and y, such that x x tnd y y . Consiter the middle point between x and ytand dentte it by m. According to the assumption, either m x or y m. In the former case deﬁne x = m and y = y , and in the latter case deﬁne x = x and t+1 t+1 t t+1 t yt+1 = m. The sequences {x } ant {y } aretconverging, and they must converge to the same point z since the distance between x and y t t converges to zero. By the continuity of we have z x and y z and thus, by transitivity, y x, contradicting the assumption that x ▯ y. Comment on the Proof: Another proof could be given for the more general case, in which the assumption that the set X is convex is replaced by the assumption that it is a connected subset of ▯ . Remember that a connected set cannot be covered by two disjoint open sets. If there is no z such that x ▯ z ▯ y, then X is the union of two disjoint sets {a|a ▯ y} and {a|x ▯ a}, which are open by the continuity of the preference relation. Recall that a set Y ⊆ X is dense in X if in every open subset of X n there is an element in Y. For example, the set Y ={ x ∈▯ | x is a k rational number for k = 1,..,n} is a countable dense set in ▯ . n October 21, 2005 12:18 master Sheet number 36 Page number 20 20 Lecture Two Proposition: Assume that X is a convex subset of ▯ that has a countable dense subset Y.fI is a continuous preference relation, then has a (continuous) utility representation. Proof: By a previous claim we know that there exists a function v : Y → [−1,1], which is a utility representation of the preference relation restricted to Y. For every x ∈ X, deﬁne U(x) = sup{v(z)|z ∈ Y and x ▯ z}. Deﬁne U(x) =− 1 if there is no z ∈ Y such that x ▯ z, which means that x is the minimal element in X. (Note that for z ∈ Y it could be that U(z)< v(z).) If x ∼ y, then x ▯ z iff y ▯ z. Thus, the sets on which the supre- mum is taken are the same and U(x) = U(y). If x ▯ y, then by the lemma there exists z in X such that x ▯ z ▯ y. By the continuity of the preferences there is a ball around z such that all the elements in that ball are inferior to x and superior to y. Since Y is dense, there exists z 1 Y such that x ▯ z ▯ 1. Similarly, there exists z2∈ Y such that z ▯1z ▯ 2. Finally, U(x) ≥ v(z 1 (by the deﬁnition of U and x ▯ z ), 1 v(z )> v(z ) (since v represents on Y and z ▯ z ), and 1 2 1 2 v(z 2 ≥ U(y) (by the deﬁnition of U and z ▯ y)2 Bibliographic Notes Recommended readings: Kreps 1990, 30–32; Mas-Colell et al. 1995, chapter 3, C. Fishburn (1970) covers the material in this lecture very well. The example of lexicographic preferences originated in Debreu (1959) (see also Debreu 1960, in particular Chapter 2, which is available online at http://cowles.econ.yale.edu/P/cp/p00b/p0097.pdf.) October 21, 2005 12:18 master Sheet number 37 Page number 21 Problem Set 2 Problem 1. (Easy) The purpose of this problem is to make sure that you fully understand the basic concepts of utility representation and continuous preferences. a. Is the statement “if both U and V represent then there is a strictly monotonic function f :▯→▯ such that V(x) = f (U(x))” correct? b. Can a continuous preference be represented by a discontinuous func- tion? c. Show that in the case of X =▯ , the preference relation that is rep- resented by the discontinuous utility function u(x) =[ x] (the largest integer n such that x ≥ n) is not a continuous relation. d. Show that the two deﬁnitions of a continuous preference relation (C1 and C2) are equivalent to Deﬁnition C3: For any x ∈ X, the upper and lower contours {y| y x} and {y| x y} are closed sets in X, and to Deﬁnition C4: For any x ∈ X, the sets {y| y ▯ x} and {y| x ▯ y} are open sets in X. Problem 2. (Moderate) Giveanexampleofpreferencesoveracountablesetinwhichthepreferences cannot be represented

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