On Lexicographic (Dictionary) Preference

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "On Lexicographic (Dictionary) Preference"

Transcription

1 MICROECONOMICS LECTURE SUPPLEMENTS Hajime Miyazaki File Name: lexico95.usc/lexico99.dok DEPARTMENT OF ECONOMICS OHIO STATE UNIVERSITY Fall 993/994/995 On Lexicographic (Dictionary) Preference This short note discusses a lexicographic preference over twocommodity bundles. The extension to the n-commodity case is straightforward. Let a = (a, a ) and b = (b, b ) be two consumption vectors. We say that a is lexicographically preferred to b, and write a f ~ b if and only if either (a > b ) or (a = b and a b ). From this definition, it follows that a f b if and only if either (a > b ) or (a = b and a > b ). It is straightforward to verify that the lexicographic preference is complete, transitive, and reflexive. It is also convex and strictly monotone. Nevertheless, the lexicographic preference cannot be represented by a utility function. An important observation is antisymmetry of the lexicographic preference. That is, (a b a = b) where a b means that a f ~ b and b f ~ a. As a result, a is indifferent to b (a ~ b), if and only if a = b and a = b. In other words, I(a), the indifference class of a, consists of "a" only. I(a) = {x x ~ a} = {x x = a} = {a}. The singleton indifference class already suggests a difficulty in constructing a utility function that represents the lexicographic preference. The singleton indifference class means that each

2 Fall 93/94/95/ Hajime Miyazaki consumption vector must have its own value, different from all others: that is, no two points in the plane R can have the same value assigned. But, R has a way too many more points than R from which the numerical values have to be assigned. It is just impossible to assign different numbers to all points in R plane from the real line R. There will not be enough real numbers assignable to all consumption vectors. This intuition can be formally validated by a reductio ad absurdum argument. It will become apparent that the existence of utility function representation and the existence of non-degenerate indifference curve are completely synonymous. The other important property of a lexicographic preference is that it is not a continuous preference. To see this, check whether the upper contour set A(a) = {x x f a } is closed. ~ a a A(a) 0 a Given a = (a, a ), any vector to the right of the vertical line at a is strictly preferred to a. Similarly, any vector directly above a is also strictly preferred to a. All other vectors are strictly less preferred to

3 Fall 93/94/95/ Hajime Miyazaki a. Thus, A(a) is the shaded half-space with the thick vertical half-line above the a vector included. The upper contour set A(a) is neither open nor closed. Since A(a) is not closed, the preference ordering of a converging sequence can be reversed in the limit. To illustrate, consider the following sequence. a x( n) = ( a +, + ) n n as n =,, 3,. For any n, a + (/n) > a. Thus, x( n) f a for any n. The limit of x(n) as n goes infinity is x = ( a, a ). But, x a a = (, ) p ( a, a) = a. a a a / 0 x x(n x( x( ) ) a a +(/) a + The diagram illustrates the following sequence and preference reversal.

4 Fall 93/94/95/ Hajime Miyazaki and x( ) f a x( ) f a x( 3) f a M M M x( n) f a M M M x p a. Proposition: A lexicographic preference on the two-commodity space cannot be represented by a utility function. Here I provide a heuristic proof that relies on reductio ad absurdum. Suppose that the lexicographic preference can be represented by a utility function u : R+ R. For each x, the lexicographic preference says that ( x, ) f ( x, 0) if and only if u(x, ) > u(x, 0). Then, for each x we can assign a nondegenerate interval R(x ) = [u(x, 0), u(x, )]. Next, take $x such that $x > x, so that u( x$, ) > u( x$, 0). Again, R( x$ ) = [ u( x$, 0), u( x$, )] is nondegenerate. Further, R( x$ ) and R(x ) are disjoint intervals, that is R( x$ ) R( x) = because $x > x implies that u( x$, 0) > u( x, ). Now, each nondegenerate interval contains a rational number, in fact, infinitely many rational numbers as well as infinitely many irrational numbers. Let Q(x ) be the set of all rational numbers contained in R(x ), and Q( x$ ) the set of all rational numbers contained in R( x$ ).

5 Fall 93/94/95/ Hajime Miyazaki Because R(x ) and R( x$ ) are disjoint, Q(x ) and Q( x$ ) are also disjoint. Observe that we can define R(x ) for any nonnegative real number, and all such R(x ) s are disjoint from each other. There are as many nondegenerate R(x ) intervals as the number of all nonnegative real numbers. Since each R(x ) contains Q(x ), it follows that there are as many such Q(x ) sets as the number of all nonnegative real numbers. Since each Q(x ) contains rational numbers, the upshot is that the total number of rational numbers collected over all Q(x ) s is at least as great as the number of all nonnegative real numbers. u( x$, ) u(x, ) R( x$, 0 ) (0, ) R(x, 0) u(x, 0) u( x$, 0 ) ( x$, 0 ) 0 (x, 0) Mathematicians have the notion called "cardinality" to measure the size of a given by counting the number of elements in the set. Using this terminology, we have just argued that the cardinality of the

6 Fall 93/94/95/ Hajime Miyazaki rational-number set is at least as great as the cardinality of the nonnegative real-number set. But, it is an established fact that the cardinality of the nonnegative real-number set is the same as the cardinality of the set of all real numbers. It is also an established fact that the cardinality of the set of all rational numbers is strictly (far) less than the cardinality of the set of all real numbers. We have thus reached a conclusion that is contrary to this established mathematical fact by supposing that a utility function exists representing the lexicographic preference. Q.E.D. The upshot is that there is no utility function representation of the lexicographic preference. A rigorous proof of the nonexistence proposition for the lexicographic case can be found in Theory of Value: An Axiomatic Analysis of Economic Equilibrium by Gerald Debreu, (959) Cowles Foundation Monograph, ISBN In the same monograph, Debreu provides a most general existence proof of a utility function for any continuous preference preordering. Representation Function The absence of a utility function does not imply the absence of a decision function for a lexicographic consumer. Suppose that a consumer chooses the most preferred bundle from a budget set of the form B(p, p m) = {(x x ) p x + p x m},, where prices are strictly positive. Consider the decision function v( B( p, p m)) = ( max { x p x + p x m for some x }, ) 0 = ( m / x, 0 ). A consumer with this decision function will always choose the same consumption bundle that a consumer with lexicographic preferences

7 Fall 93/94/95/ Hajime Miyazaki would choose on all budget sets of the form B( p, p m). In this sense, this ν function serves as an adequate behavioral rule for the lexicographic consumer facing these budget sets. To put it more bluntly, a consumer with the utility function u( x, x ) = x behaves exactly the same as the lexicographic consumer on all budget sets of the form B( x, x m) with p >> 0. But neither this u nor v above is a utility function for the lexicographic preference, because neither can be used to rank consumption vectors of the form ( x, ). Integer Consumption The impossibility of utility representation has crucially depended on the idea that all real vectors were consumption bundles. If one commodity can only be consumed in nonnegative integer units, a utility presentation is possible even for the lexicographic consumer. a 0 a a+ a+ a+3

8 Fall 93/94/95/ Hajime Miyazaki Suppose that the consumer has a lexicographic preference over two-commodity bundles, but that the first commodity x can only be consumed in nonnegative integer amounts. x = 0,,, 3, The relevant consumption set is Ζ + R + where Ζ + is the set of nonnegative integers and R + the set of nonnegative real numbers. The upper contour set for a is A(a) = {x in Ζ + R + x f ~ a }. For each nonnegative integer "a", A(a) is a closed set. It is then possible to define a numerical representation of the lexicographic preference defined on Ζ + R +. u x x 3 0 Consider the function u( x, x ) = ( x + ) x + where x is a nonnegative integer. Note that for each integer x, x u( x, ) < x +

9 Fall 93/94/95/ Hajime Miyazaki and u(x, ) increases asymptotically in x to the value x + (as x ). Since x + u( x +, ) < x +, we have u( x, ) < u( x +, ). As a result, 0 = u(0, 0) < u(0, ) < u(, ) < u(, ) <. Once again, we have disjoint intervals R(x ). But, the previous conundrum does not arise. The cardinality of Ζ + is decisively far less than the cardinality of R +. Note that u(x, x ) = (x +) (/e x ) will be another utility representation of the lexicographic preference when the first commodity is integer-indivisible. It is left to the reader to confirm that (x +) (/e x ) is a monotone transform of (x +) /(x +). The above exercise can be extended to a case in which x can be consumed only in increments of /k. That is, x = 0, /k, /k,, n/k,. This k can be taken as large as one wishes, so that /k can be made as small as one wishes. Consider ( k) u( x, x) = ( x + ) k x +. Then, x u( x, ) < x +, k and u(x, x ) increases in x approaching the asymptotic value of x + as x. Once again, k

10 Fall 93/94/95/ Hajime Miyazaki 3 0 = u( 0, 0) < u( 0, x) < u(, ) < u(, ) < u(, ) k k k + < L < u( n, ) < u( n, ) < L. k k Relevance of Despite the logically compelling example of a lexicographic preference, economists have generally dismissed it as curiosum. For example, Malinvaud (97, p. 0) says, Such a preference relation has sometimes been considered; it hardly seems likely to arise in economics, since it assumes that, for the consumer, the good is immeasurably more important than the good. We loose little in the way of realism if we eliminate this and similar cases which do not satisfy [the continuity axiom]. (italics and brackets mine). But, modified forms of the basic lexicographic preference can arise and rather plausible. Consider for example, the case of yellow and red apples. I will not be surprised that if my neighbor compares two bushels of apples first by the total quantity of apples and, only if two baskets contain the same number of apples, she prefers the one with more red apples than yellow apples. Letting the good be a red apple and the good a yellow apple, we can have a modified lexicographic preference given by (x, x ) f (x ~, x ) if either (x + x > x + x ) or ( x + x = x + x and x > x ) holds. Clearly, this consumer s upper contour set is neither open nor closed as in the standard lexicographic preference. Note that this consumer s demand behavior is identical to

11 Fall 93/94/95/ Hajime Miyazaki the consumer with the continuous preference given by the utility function as u(x, x ) = x + x on every linear budget except when the relative prices of yellow and red apples are unitary ( p / p. = ). When p = p, the lexicographic consumer chooses all red apples (x = m/ p. and x = 0 ), but the consumer with u(x, x ) = x + x can chooses any vector on the budget line. The plausibility of such a lexicographic preference is not eclipsed when the consumer s choice includes three types of apples: red, green and yellow. In fact, from any consumer that has a continuous preference, I can create a modified lexicographic preference that behaves in the identical fashion on every linear budget set except when the original consumer s choice is a demand correspondence, rather than a demand function. Let f be an original preference that is continuous in the ~ sense that its upper contour set is closed. Then, induce a modified lexicographic preference L defined as follows: xlx if and only if either ( x f x ) or (x x and x > x ) holds. Letting U be a utility function that represents the continuous preference, we define xlx if and only if either U(x) > U(x ) or ( U(x) = U(x ) and x > x ) holds. For example, by this method, a Cobb-Douglas utility function can be modified into a Cobb-Douglas type lexicographic preference. Expenditure Minimization If a consumer s choice is a demand correspondence, it means that the consumer is indifferent among multiple vectors, but the lexicographic choice on the same demand set will select the one that has the largest first component. A nonconvex preference can induce a demand correspondence. The previous example produced the case of a demand set when the relative price of yellow and red apples was unity.

12 Fall 93/94/95/ Hajime Miyazaki The duality theory of consumption is built on the use of expenditure function. The expenditure function calculates the minimum expenditure necessary for a consumer to attain a given level of preference. That is, e(p, x o ) = min{p x x f x o }. The ~ expenditure function does not generally exist unless the upper contour set A(x o ) = {x x f x o } is closed. The need for this technical ~ requirement is fairly transparent if we express the expenditure function as e(p, x o ) = min{p x x in A(x o )}. In fact, for the case of the original lexicographic preference, there is no expenditure minimizer and the expenditure function is not defined unless x o is on the horizontal axis, i.e., x o = (x o, 0). In contrast, the consumer with u( x, x ) = x has the expenditure function e(p, x o ) = p x o = min {p x u(x) u(x o )} for all x o. Similarly, for the lexicographic preference for yellow and red apples, there is no expenditure minimizer, and the expenditure function is not defined, whenever p > p. Again, for the consumer with u(x, x ) = x + x the expenditure function is well defined for all p and x, namely, e(p, p, x, x ) = (x + x ) Min {p, p }. Since the expenditure function is itself a utility function, called a money-metric utility function, we can heuristically confirm Debreu s Theorem on the existence of utility function. The existence of the expenditure function is tantamount to the existence of a utility function. Because a closed upper contour set guarantees a well- If we redefine the expenditure function as the infimum, rather than the minimum, expenditure, then we have it well-defined. e(p, x o ) = inf{p x x f ~ x o } = p x o for all nonnegative x o. But, the preference still fails to have any utility function representation. In this sense, I consider the infimum version an isolated technical remedy, and prefer using the minimum definition to underscore the economic-theoretic underpinning of the relations among preference, demand and utility function.

13 Fall 93/94/95/ Hajime Miyazaki defined expenditure function, we know that a continuous preference, as it necessarily has closed upper contour sets, can be represented by a utility function. Remark: We can minimize expenditure on the lexicographic preference over yellow and red apples, provided that p p. Even so, the minimizer is of the form x = (x, 0 ) = (m/ p, 0 ), and whenever x o is not on the boundary, the minimizer is strictly preferred to x o : x = (x, 0 ) f x o = (x o, x o ). In that sense, the constraint x f ~ x o is not binding at the minimization solution. We often say casually that the indifference curve of a given consumption vector x f x o is the boundary of the upper contour set {x x f x o }. ~ ~ If the preference is lexicographic, the boundary definition of an indifference curve does not hold. The boundary of the upper contour set, even where the boundary is included in the upper contour set, is not necessarily an indifference curve.

Follow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu

Follow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu 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

More information

Gains from Trade. Christopher P. Chambers and Takashi Hayashi. March 25, 2013. Abstract

Gains from Trade. Christopher P. Chambers and Takashi Hayashi. March 25, 2013. Abstract Gains from Trade Christopher P. Chambers Takashi Hayashi March 25, 2013 Abstract In a market design context, we ask whether there exists a system of transfers regulations whereby gains from trade can always

More information

Lecture 2: Consumer Theory

Lecture 2: Consumer Theory Lecture 2: Consumer Theory Preferences and Utility Utility Maximization (the primal problem) Expenditure Minimization (the dual) First we explore how consumers preferences give rise to a utility fct which

More information

CITY AND REGIONAL PLANNING 7230. Consumer Behavior. Philip A. Viton. March 4, 2015. 1 Introduction 2

CITY AND REGIONAL PLANNING 7230. Consumer Behavior. Philip A. Viton. March 4, 2015. 1 Introduction 2 CITY AND REGIONAL PLANNING 7230 Consumer Behavior Philip A. Viton March 4, 2015 Contents 1 Introduction 2 2 Foundations 2 2.1 Consumption bundles........................ 2 2.2 Preference relations.........................

More information

Notes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand

Notes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such

More information

Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets

Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets Nathaniel Hendren January, 2014 Abstract Both Akerlof (1970) and Rothschild and Stiglitz (1976) show that

More information

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely

More information

Surface bundles over S 1, the Thurston norm, and the Whitehead link

Surface bundles over S 1, the Thurston norm, and the Whitehead link Surface bundles over S 1, the Thurston norm, and the Whitehead link Michael Landry August 16, 2014 The Thurston norm is a powerful tool for studying the ways a 3-manifold can fiber over the circle. In

More information

This chapter is all about cardinality of sets. At first this looks like a

This chapter is all about cardinality of sets. At first this looks like a CHAPTER Cardinality of Sets This chapter is all about cardinality of sets At first this looks like a very simple concept To find the cardinality of a set, just count its elements If A = { a, b, c, d },

More information

FIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.

FIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper. FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is

More information

Imprecise probabilities, bets and functional analytic methods in Łukasiewicz logic.

Imprecise probabilities, bets and functional analytic methods in Łukasiewicz logic. Imprecise probabilities, bets and functional analytic methods in Łukasiewicz logic. Martina Fedel joint work with K.Keimel,F.Montagna,W.Roth Martina Fedel (UNISI) 1 / 32 Goal The goal of this talk is to

More information

Economics 200B Part 1 UCSD Winter 2015 Prof. R. Starr, Mr. John Rehbeck Final Exam 1

Economics 200B Part 1 UCSD Winter 2015 Prof. R. Starr, Mr. John Rehbeck Final Exam 1 Economics 200B Part 1 UCSD Winter 2015 Prof. R. Starr, Mr. John Rehbeck Final Exam 1 Your Name: SUGGESTED ANSWERS Please answer all questions. Each of the six questions marked with a big number counts

More information

INTRODUCTORY SET THEORY

INTRODUCTORY SET THEORY M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

More information

Metric Spaces. Chapter 1

Metric Spaces. Chapter 1 Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence

More information

THE CENTRAL LIMIT THEOREM TORONTO

THE CENTRAL LIMIT THEOREM TORONTO THE CENTRAL LIMIT THEOREM DANIEL RÜDT UNIVERSITY OF TORONTO MARCH, 2010 Contents 1 Introduction 1 2 Mathematical Background 3 3 The Central Limit Theorem 4 4 Examples 4 4.1 Roulette......................................

More information

Investigación Operativa. The uniform rule in the division problem

Investigación Operativa. The uniform rule in the division problem Boletín de Estadística e Investigación Operativa Vol. 27, No. 2, Junio 2011, pp. 102-112 Investigación Operativa The uniform rule in the division problem Gustavo Bergantiños Cid Dept. de Estadística e

More information

Convex Rationing Solutions (Incomplete Version, Do not distribute)

Convex Rationing Solutions (Incomplete Version, Do not distribute) Convex Rationing Solutions (Incomplete Version, Do not distribute) Ruben Juarez rubenj@hawaii.edu January 2013 Abstract This paper introduces a notion of convexity on the rationing problem and characterizes

More information

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5. PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

More information

MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

More information

Name. Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions.

Name. Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions. Name Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions. Question 1. A firm has a production function F (x 1, x 2 ) = ( x 1 + x 2 ) 2. It is a price

More information

Insurance. Michael Peters. December 27, 2013

Insurance. Michael Peters. December 27, 2013 Insurance Michael Peters December 27, 2013 1 Introduction In this chapter, we study a very simple model of insurance using the ideas and concepts developed in the chapter on risk aversion. You may recall

More information

UC Berkeley Haas School of Business Economic Analysis for Business Decisions (EWMBA 201A)

UC Berkeley Haas School of Business Economic Analysis for Business Decisions (EWMBA 201A) UC Berkeley Haas School of Business Economic Analysis for Business Decisions (EWMBA 201A) The economic agent (PR 3.1-3.4) Standard economics vs. behavioral economics Lectures 1-2 Aug. 15, 2009 Prologue

More information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

Economics 2020a / HBS 4010 / HKS API-111 Fall 2011 Practice Problems for Lectures 1 to 11

Economics 2020a / HBS 4010 / HKS API-111 Fall 2011 Practice Problems for Lectures 1 to 11 Economics 2020a / HBS 4010 / HKS API-111 Fall 2011 Practice Problems for Lectures 1 to 11 LECTURE 1: BUDGETS AND REVEALED PREFERENCE 1.1. Quantity Discounts and the Budget Constraint Suppose that a consumer

More information

( ) which must be a vector

( ) which must be a vector MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are

More information

Lecture 11 Uncertainty

Lecture 11 Uncertainty Lecture 11 Uncertainty 1. Contingent Claims and the State-Preference Model 1) Contingent Commodities and Contingent Claims Using the simple two-good model we have developed throughout this course, think

More information

How many numbers there are?

How many numbers there are? How many numbers there are? RADEK HONZIK Radek Honzik: Charles University, Department of Logic, Celetná 20, Praha 1, 116 42, Czech Republic radek.honzik@ff.cuni.cz Contents 1 What are numbers 2 1.1 Natural

More information

Persuasion by Cheap Talk - Online Appendix

Persuasion by Cheap Talk - Online Appendix Persuasion by Cheap Talk - Online Appendix By ARCHISHMAN CHAKRABORTY AND RICK HARBAUGH Online appendix to Persuasion by Cheap Talk, American Economic Review Our results in the main text concern the case

More information

Chapter 21: The Discounted Utility Model

Chapter 21: The Discounted Utility Model Chapter 21: The Discounted Utility Model 21.1: Introduction This is an important chapter in that it introduces, and explores the implications of, an empirically relevant utility function representing intertemporal

More information

Duality in General Programs. Ryan Tibshirani Convex Optimization 10-725/36-725

Duality in General Programs. Ryan Tibshirani Convex Optimization 10-725/36-725 Duality in General Programs Ryan Tibshirani Convex Optimization 10-725/36-725 1 Last time: duality in linear programs Given c R n, A R m n, b R m, G R r n, h R r : min x R n c T x max u R m, v R r b T

More information

Estimating the Average Value of a Function

Estimating the Average Value of a Function Estimating the Average Value of a Function Problem: Determine the average value of the function f(x) over the interval [a, b]. Strategy: Choose sample points a = x 0 < x 1 < x 2 < < x n 1 < x n = b and

More information

THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING

THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The Black-Scholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages

More information

24. The Branch and Bound Method

24. The Branch and Bound Method 24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NP-complete. Then one can conclude according to the present state of science that no

More information

Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2

Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2 Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2 Ameya Pitale, Ralf Schmidt 2 Abstract Let µ(n), n > 0, be the sequence of Hecke eigenvalues of a cuspidal Siegel eigenform F of degree

More information

PROBLEM SET 6: POLYNOMIALS

PROBLEM SET 6: POLYNOMIALS PROBLEM SET 6: POLYNOMIALS 1. introduction In this problem set we will consider polynomials with coefficients in K, where K is the real numbers R, the complex numbers C, the rational numbers Q or any other

More information

On the Existence of Nash Equilibrium in General Imperfectly Competitive Insurance Markets with Asymmetric Information

On the Existence of Nash Equilibrium in General Imperfectly Competitive Insurance Markets with Asymmetric Information analysing existence in general insurance environments that go beyond the canonical insurance paradigm. More recently, theoretical and empirical work has attempted to identify selection in insurance markets

More information

Some representability and duality results for convex mixed-integer programs.

Some representability and duality results for convex mixed-integer programs. Some representability and duality results for convex mixed-integer programs. Santanu S. Dey Joint work with Diego Morán and Juan Pablo Vielma December 17, 2012. Introduction About Motivation Mixed integer

More information

Bachelor Degree in Business Administration Academic year 2015/16

Bachelor Degree in Business Administration Academic year 2015/16 University of Catania Department of Economics and Business Bachelor Degree in Business Administration Academic year 2015/16 Mathematics for Social Sciences (1st Year, 1st Semester, 9 Credits) Name of Lecturer:

More information

Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson

Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement

More information

2. Information Economics

2. Information Economics 2. Information Economics In General Equilibrium Theory all agents had full information regarding any variable of interest (prices, commodities, state of nature, cost function, preferences, etc.) In many

More information

Week 7 - Game Theory and Industrial Organisation

Week 7 - Game Theory and Industrial Organisation Week 7 - Game Theory and Industrial Organisation The Cournot and Bertrand models are the two basic templates for models of oligopoly; industry structures with a small number of firms. There are a number

More information

Indifference Curves: An Example (pp. 65-79) 2005 Pearson Education, Inc.

Indifference Curves: An Example (pp. 65-79) 2005 Pearson Education, Inc. Indifference Curves: An Example (pp. 65-79) Market Basket A B D E G H Units of Food 20 10 40 30 10 10 Units of Clothing 30 50 20 40 20 40 Chapter 3 1 Indifference Curves: An Example (pp. 65-79) Graph the

More information

Working Paper Series

Working Paper Series RGEA Universidade de Vigo http://webs.uvigo.es/rgea Working Paper Series A Market Game Approach to Differential Information Economies Guadalupe Fugarolas, Carlos Hervés-Beloso, Emma Moreno- García and

More information

Lecture 8 The Subjective Theory of Betting on Theories

Lecture 8 The Subjective Theory of Betting on Theories Lecture 8 The Subjective Theory of Betting on Theories Patrick Maher Philosophy 517 Spring 2007 Introduction The subjective theory of probability holds that the laws of probability are laws that rational

More information

Graphs without proper subgraphs of minimum degree 3 and short cycles

Graphs without proper subgraphs of minimum degree 3 and short cycles Graphs without proper subgraphs of minimum degree 3 and short cycles Lothar Narins, Alexey Pokrovskiy, Tibor Szabó Department of Mathematics, Freie Universität, Berlin, Germany. August 22, 2014 Abstract

More information

Mathematics for Econometrics, Fourth Edition

Mathematics for Econometrics, Fourth Edition Mathematics for Econometrics, Fourth Edition Phoebus J. Dhrymes 1 July 2012 1 c Phoebus J. Dhrymes, 2012. Preliminary material; not to be cited or disseminated without the author s permission. 2 Contents

More information

Classification - Examples

Classification - Examples Lecture 2 Scheduling 1 Classification - Examples 1 r j C max given: n jobs with processing times p 1,...,p n and release dates r 1,...,r n jobs have to be scheduled without preemption on one machine taking

More information

3. INNER PRODUCT SPACES

3. INNER PRODUCT SPACES . INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

More information

CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC. 1. Introduction

CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC. 1. Introduction Acta Math. Univ. Comenianae Vol. LXXXI, 1 (2012), pp. 71 77 71 CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC E. BALLICO Abstract. Here we study (in positive characteristic) integral curves

More information

Module1. x 1000. y 800.

Module1. x 1000. y 800. Module1 1 Welcome to the first module of the course. It is indeed an exciting event to share with you the subject that has lot to offer both from theoretical side and practical aspects. To begin with,

More information

Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

More information

Mathematics Review for MS Finance Students

Mathematics Review for MS Finance Students Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,

More information

Bargaining Solutions in a Social Network

Bargaining Solutions in a Social Network Bargaining Solutions in a Social Network Tanmoy Chakraborty and Michael Kearns Department of Computer and Information Science University of Pennsylvania Abstract. We study the concept of bargaining solutions,

More information

OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN. E.V. Grigorieva. E.N. Khailov

OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN. E.V. Grigorieva. E.N. Khailov DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Supplement Volume 2005 pp. 345 354 OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN E.V. Grigorieva Department of Mathematics

More information

ECON 305 Tutorial 7 (Week 9)

ECON 305 Tutorial 7 (Week 9) H. K. Chen (SFU) ECON 305 Tutorial 7 (Week 9) July 2,3, 2014 1 / 24 ECON 305 Tutorial 7 (Week 9) Questions for today: Ch.9 Problems 15, 7, 11, 12 MC113 Tutorial slides will be posted Thursday after 10:30am,

More information

MOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu

MOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing

More information

Understanding Basic Calculus

Understanding Basic Calculus Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other

More information

ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION. E. I. Pancheva, A. Gacovska-Barandovska

ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION. E. I. Pancheva, A. Gacovska-Barandovska Pliska Stud. Math. Bulgar. 22 (2015), STUDIA MATHEMATICA BULGARICA ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION E. I. Pancheva, A. Gacovska-Barandovska Smirnov (1949) derived four

More information

Discrete Optimization

Discrete Optimization Discrete Optimization [Chen, Batson, Dang: Applied integer Programming] Chapter 3 and 4.1-4.3 by Johan Högdahl and Victoria Svedberg Seminar 2, 2015-03-31 Todays presentation Chapter 3 Transforms using

More information

CONSUMER PREFERENCES THE THEORY OF THE CONSUMER

CONSUMER PREFERENCES THE THEORY OF THE CONSUMER CONSUMER PREFERENCES The underlying foundation of demand, therefore, is a model of how consumers behave. The individual consumer has a set of preferences and values whose determination are outside the

More information

The Trip Scheduling Problem

The Trip Scheduling Problem The Trip Scheduling Problem Claudia Archetti Department of Quantitative Methods, University of Brescia Contrada Santa Chiara 50, 25122 Brescia, Italy Martin Savelsbergh School of Industrial and Systems

More information

BX in ( u, v) basis in two ways. On the one hand, AN = u+

BX in ( u, v) basis in two ways. On the one hand, AN = u+ 1. Let f(x) = 1 x +1. Find f (6) () (the value of the sixth derivative of the function f(x) at zero). Answer: 7. We expand the given function into a Taylor series at the point x = : f(x) = 1 x + x 4 x

More information

Lecture 6: Logistic Regression

Lecture 6: Logistic Regression Lecture 6: CS 194-10, Fall 2011 Laurent El Ghaoui EECS Department UC Berkeley September 13, 2011 Outline Outline Classification task Data : X = [x 1,..., x m]: a n m matrix of data points in R n. y { 1,

More information

ANALYTIC HIERARCHY PROCESS (AHP) TUTORIAL

ANALYTIC HIERARCHY PROCESS (AHP) TUTORIAL Kardi Teknomo ANALYTIC HIERARCHY PROCESS (AHP) TUTORIAL Revoledu.com Table of Contents Analytic Hierarchy Process (AHP) Tutorial... 1 Multi Criteria Decision Making... 1 Cross Tabulation... 2 Evaluation

More information

CHAPTER 7 GENERAL PROOF SYSTEMS

CHAPTER 7 GENERAL PROOF SYSTEMS CHAPTER 7 GENERAL PROOF SYSTEMS 1 Introduction Proof systems are built to prove statements. They can be thought as an inference machine with special statements, called provable statements, or sometimes

More information

4. Expanding dynamical systems

4. Expanding dynamical systems 4.1. Metric definition. 4. Expanding dynamical systems Definition 4.1. Let X be a compact metric space. A map f : X X is said to be expanding if there exist ɛ > 0 and L > 1 such that d(f(x), f(y)) Ld(x,

More information

SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by

SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples

More information

An optimal transportation problem with import/export taxes on the boundary

An optimal transportation problem with import/export taxes on the boundary An optimal transportation problem with import/export taxes on the boundary Julián Toledo Workshop International sur les Mathématiques et l Environnement Essaouira, November 2012..................... Joint

More information

Price Discrimination: Part 2. Sotiris Georganas

Price Discrimination: Part 2. Sotiris Georganas Price Discrimination: Part 2 Sotiris Georganas 1 More pricing techniques We will look at some further pricing techniques... 1. Non-linear pricing (2nd degree price discrimination) 2. Bundling 2 Non-linear

More information

Degree Hypergroupoids Associated with Hypergraphs

Degree Hypergroupoids Associated with Hypergraphs Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated

More information

Modeling Insurance Markets

Modeling Insurance Markets Modeling Insurance Markets Nathaniel Hendren Harvard April, 2015 Nathaniel Hendren (Harvard) Insurance April, 2015 1 / 29 Modeling Competition Insurance Markets is Tough There is no well-agreed upon model

More information

Mechanisms for Fair Attribution

Mechanisms for Fair Attribution Mechanisms for Fair Attribution Eric Balkanski Yaron Singer Abstract We propose a new framework for optimization under fairness constraints. The problems we consider model procurement where the goal is

More information

17. If a good is normal, then the Engel curve A. Slopes upward B. Slopes downward C. Is vertical D. Is horizontal

17. If a good is normal, then the Engel curve A. Slopes upward B. Slopes downward C. Is vertical D. Is horizontal Sample Exam 1 1. Suppose that when the price of hot dogs is $2 per package, there is a demand for 10,000 bags of hot dog buns. When the price of hot dogs is $3 per package, the demand for hot dog buns

More information

Working Paper Secure implementation in economies with indivisible objects and money

Working Paper Secure implementation in economies with indivisible objects and money econstor www.econstor.eu Der Open-Access-Publikationsserver der ZBW Leibniz-Informationszentrum Wirtschaft The Open Access Publication Server of the ZBW Leibniz Information Centre for Economics Fujinaka,

More information

Linear Codes. Chapter 3. 3.1 Basics

Linear Codes. Chapter 3. 3.1 Basics Chapter 3 Linear Codes In order to define codes that we can encode and decode efficiently, we add more structure to the codespace. We shall be mainly interested in linear codes. A linear code of length

More information

Computational Finance Options

Computational Finance Options 1 Options 1 1 Options Computational Finance Options An option gives the holder of the option the right, but not the obligation to do something. Conversely, if you sell an option, you may be obliged to

More information

The number of generalized balanced lines

The number of generalized balanced lines The number of generalized balanced lines David Orden Pedro Ramos Gelasio Salazar Abstract Let S be a set of r red points and b = r + 2δ blue points in general position in the plane, with δ 0. A line l

More information

The Banach-Tarski Paradox

The Banach-Tarski Paradox University of Oslo MAT2 Project The Banach-Tarski Paradox Author: Fredrik Meyer Supervisor: Nadia S. Larsen Abstract In its weak form, the Banach-Tarski paradox states that for any ball in R, it is possible

More information

DEGREES OF ORDERS ON TORSION-FREE ABELIAN GROUPS

DEGREES OF ORDERS ON TORSION-FREE ABELIAN GROUPS DEGREES OF ORDERS ON TORSION-FREE ABELIAN GROUPS ASHER M. KACH, KAREN LANGE, AND REED SOLOMON Abstract. We construct two computable presentations of computable torsion-free abelian groups, one of isomorphism

More information

The Optimal Growth Problem

The Optimal Growth Problem LECTURE NOTES ON ECONOMIC GROWTH The Optimal Growth Problem Todd Keister Centro de Investigación Económica, ITAM keister@itam.mx January 2005 These notes provide an introduction to the study of optimal

More information

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear

More information

Working Paper Series

Working Paper Series RGEA Universidade de Vigo http://webs.uvigo.es/rgea Working Paper Series General Equilibrium with Private State Verification João Correia-da-Silva and Carlos Hervés-Beloso 5-08 Facultade de Ciencias Económicas

More information

Cycles in a Graph Whose Lengths Differ by One or Two

Cycles in a Graph Whose Lengths Differ by One or Two Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDE-BERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS

More information

Error Bound for Classes of Polynomial Systems and its Applications: A Variational Analysis Approach

Error Bound for Classes of Polynomial Systems and its Applications: A Variational Analysis Approach Outline Error Bound for Classes of Polynomial Systems and its Applications: A Variational Analysis Approach The University of New South Wales SPOM 2013 Joint work with V. Jeyakumar, B.S. Mordukhovich and

More information

An example of a computable

An example of a computable An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept

More information

Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)

Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m) Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.

More information

SOLVING POLYNOMIAL EQUATIONS

SOLVING POLYNOMIAL EQUATIONS C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra

More information

Section 3.7. Rolle s Theorem and the Mean Value Theorem. Difference Equations to Differential Equations

Section 3.7. Rolle s Theorem and the Mean Value Theorem. Difference Equations to Differential Equations Difference Equations to Differential Equations Section.7 Rolle s Theorem and the Mean Value Theorem The two theorems which are at the heart of this section draw connections between the instantaneous rate

More information

If n is odd, then 3n + 7 is even.

If n is odd, then 3n + 7 is even. Proof: Proof: We suppose... that 3n + 7 is even. that 3n + 7 is even. Since n is odd, there exists an integer k so that n = 2k + 1. that 3n + 7 is even. Since n is odd, there exists an integer k so that

More information

Chapter 7. Sealed-bid Auctions

Chapter 7. Sealed-bid Auctions Chapter 7 Sealed-bid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)

More information

it is easy to see that α = a

it is easy to see that α = a 21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

More information

Lecture 4: BK inequality 27th August and 6th September, 2007

Lecture 4: BK inequality 27th August and 6th September, 2007 CSL866: Percolation and Random Graphs IIT Delhi Amitabha Bagchi Scribe: Arindam Pal Lecture 4: BK inequality 27th August and 6th September, 2007 4. Preliminaries The FKG inequality allows us to lower bound

More information

An Innocent Investigation

An Innocent Investigation An Innocent Investigation D. Joyce, Clark University January 2006 The beginning. Have you ever wondered why every number is either even or odd? I don t mean to ask if you ever wondered whether every number

More information

Ultraproducts and Applications I

Ultraproducts and Applications I Ultraproducts and Applications I Brent Cody Virginia Commonwealth University September 2, 2013 Outline Background of the Hyperreals Filters and Ultrafilters Construction of the Hyperreals The Transfer

More information

[Refer Slide Time: 05:10]

[Refer Slide Time: 05:10] Principles of Programming Languages Prof: S. Arun Kumar Department of Computer Science and Engineering Indian Institute of Technology Delhi Lecture no 7 Lecture Title: Syntactic Classes Welcome to lecture

More information

IMPLEMENTING ARROW-DEBREU EQUILIBRIA BY TRADING INFINITELY-LIVED SECURITIES

IMPLEMENTING ARROW-DEBREU EQUILIBRIA BY TRADING INFINITELY-LIVED SECURITIES IMPLEMENTING ARROW-DEBREU EQUILIBRIA BY TRADING INFINITELY-LIVED SECURITIES Kevin X.D. Huang and Jan Werner DECEMBER 2002 RWP 02-08 Research Division Federal Reserve Bank of Kansas City Kevin X.D. Huang

More information

Equilibrium: Illustrations

Equilibrium: Illustrations Draft chapter from An introduction to game theory by Martin J. Osborne. Version: 2002/7/23. Martin.Osborne@utoronto.ca http://www.economics.utoronto.ca/osborne Copyright 1995 2002 by Martin J. Osborne.

More information

Pest control may make the pest population explode

Pest control may make the pest population explode Pest control may make the pest population explode Hirokazu Ninomiya and Hans F. Weinberger Zeitschrift für angewandte Mathematik un Physik 54 (2003), pp. 869-873. To Larry Payne on his eightieth birthday

More information

Dedekind s forgotten axiom and why we should teach it (and why we shouldn t teach mathematical induction in our calculus classes)

Dedekind s forgotten axiom and why we should teach it (and why we shouldn t teach mathematical induction in our calculus classes) Dedekind s forgotten axiom and why we should teach it (and why we shouldn t teach mathematical induction in our calculus classes) by Jim Propp (UMass Lowell) March 14, 2010 1 / 29 Completeness Three common

More information

GENERIC COMPUTABILITY, TURING DEGREES, AND ASYMPTOTIC DENSITY

GENERIC COMPUTABILITY, TURING DEGREES, AND ASYMPTOTIC DENSITY GENERIC COMPUTABILITY, TURING DEGREES, AND ASYMPTOTIC DENSITY CARL G. JOCKUSCH, JR. AND PAUL E. SCHUPP Abstract. Generic decidability has been extensively studied in group theory, and we now study it in

More information