c 2008 Je rey A. Miron We have described the constraints that a consumer faces, i.e., discussed the budget constraint.

Size: px
Start display at page:

Download "c 2008 Je rey A. Miron We have described the constraints that a consumer faces, i.e., discussed the budget constraint."

Transcription

1 Lecture 2b: Utility c 2008 Je rey A. Miron Outline: 1. Introduction 2. Utility: A De nition 3. Monotonic Transformations 4. Cardinal Utility 5. Constructing a Utility Function 6. Examples of Utility Functions 7. Preview: Marginal Utility and the MRS 1 Introduction Let s summarize what we have done so far. We have described the constraints that a consumer faces, i.e., discussed the budget constraint. In many cases, this constraint is simple, whether when seen as an equation or as a graph. But in others, it gets messier. We have also discussed a way of modeling what the consumer wants. We referred to this as preferences, and while we have only described them graphically, they can also be described algebraically, sometimes in a simple way, as we will see. So far we have said nothing about what the consumer is going to choose; that is, we have not modeled the problem the consumer faces, although it has obviously been implied, and you may have seen it in other classes. We won t do that for another lecture or so. 1

2 Instead, we are next going to discuss another way of describing the consumer s desires / preferences. This approach is known as the utility function approach; we are going to think about each possible consumption bundle as giving the consumer some amount of utility, whatever that is. We ll de ne it more explicitly shortly. The early versions of consumer theory focused on utility, without any mention of preferences, and assumed we could assign a unique utility number to every choice or bundle. These utility numbers had some sort of independent meaning. In particular, we could add and subtract them across people. Modern economic theory does not proceed in this way. It focuses instead on preferences, and simply assumes that given various bundles, consumers can rank them and make choices. Utility turns out to be, usually, a convenient way to summarize these preferences, but it is the preferences that are fundamental. The reason we are going to move to utility functions from preferences is that they are easier to work with. But it is useful to understand that they are less fundamental than preferences per se. In particular, the utility numbers have no cardinal meaning; they simply indicate that, if one bundle has higher utility number than another, then the one is preferred to the other. The actual value of the utility numbers is irrelevant, sort of like rankings at a golf tournament. Note also that economists use both the preference approach and the utility function approach. We want to understand the relation between them. For many purposes, the two approaches are equivalent. Also, in many settings, it is not obvious why ones needs to have bothered with the preference approach. In a few crucial cases, however, it matters a lot, and that is why we need to do both. We ll see this more explicitly later in the course. 2

3 2 Utility: A De nition A utility function is a way of assigning numbers to every possible consumption bundle in such a way that more-preferred bundles get assigned larger numbers than lesspreferred bundles. That is, X is preferred to Y if and only if the utility of X is larger than the utility of Y. In symbols, (x 1 ; x 2 ) (y 1 ; y 2 ) i u(x 1 ; x 2 ) > u(y 1 ; y 2 ): This relies only on the ordering of the bundles, not the absolute numbers. The function u is referred to as an ordinal utility function. To illustrate, consider a consumer who prefers the bundle X to the bundle Y and the bundle Y to the bundle Z. U 1 U 2 U 3 X Y Z The numbers in each column are the utilities assigned to each bundle by three possible utility functions. The point of this comparison is that each of the three set of numbers is a valid utility function. Since only ranking matters, a unique way to assign utilities does not exist. If one utility function exists that corresponds to a particular set of preferences, then an in nite number exists. Consider u(x 1; x 2 ). Then 2u(x 1 ; x 2 ) gives same ranking. Indeed, any monotonic, increasing transformation gives the same ranking. 3

4 3 Monotonic Transformations A monotonically increasing transformation is a function f with the property that it preserves the order of numbers. That is, if f is a monotonically increasing function, and then u 1 > u 2 f(u 1 ) > f(u 2 ) Examples of monotonic functions include the following: Multiplication by a positive number Adding a positive number Taking the ln Exponentiation Raising to an odd power Raising to an even power for functions de ned over non-negative values Step functions, assuming they are increasing Examples of functions that are NOT monotonically increasing include: Sin or cosine Squaring if the function is de ned over negative values Multiplying by a negative Mathematically, f always has a positive derivative, assuming it is di erentiable. More generally, f always has a positive di erence. 4

5 Graph: Monotone and Non-Monotone Functions Monotone: y = x 2 for 0 x y x Non-monotone: y = sin 2x + cos 3x + 5 y x So, a key result we want is the following: 5

6 If u is a utility function and f is monotone, then f(u) is a utility function. Why? 1. Saying u is a utility function corresponding to some preferences means u(x) > u(y ) i X Y 2. If f is monotone, then u(x) > u(y ) i f(u(x)) > f(u(y )) 3. Therefore, f(u(x)) > f(u(y )) i X Y so the new function f(u()) represents the preferences in the same way as the original u. Geometrically, a utility function is a way of labeling the indi erence curves. Every bundle on same curve has the same u. A higher indi erence curve has a higher number (assuming monotonicity of preferences). A monotone transform is just a relabeling that preserves the order of the utility numbers. 4 Cardinal Utility We can, in principle, assign cardinal utilities. The di erence in a consumer s preference for particular bundles could be based on how much she is willing to pay for the bundle, how far she needs to walk to get the bundle, how long she has to wait in line, etc. These are defensible interpretations, but none is compelling. Also, in the context of an individual s behavior, they do not add anything. We can predict an individual s behavior based on the ordinal numbers by themselves. 6

7 Knowing how much more a consumer values one bundle relative to another does not tell us anything useful. We tend to think we want cardinal utilities to do interpersonal comparisons. We will come back to that later in the course, its an important issue. 5 Constructing a Utility Function Now, let s try to make the basic concept clearer. Let s turn to the question of how we might assign ordinal utilities, given that we have some well-de ned preferences. The key question is whether we can always nd a utility function that correctly represents those preferences? In the most general case, no. Say someone prefers A to B to C to A. Then a utility function representing those preferences would have to satisfy u(a) > u(b) > u(c) > u(a), which is impossible. So, we do have to adopt some restrictions, such as transitivity, to assure a utility function exists for a given set of preferences. (This shows why the utility function approach is less general than just using preferences, although it s not that much less general.) Start with the indi erence curve map. We in essence want a way to label the indi erence curves so that higher curves get higher numbers. Here s a way: draw a diagonal line and label each point on the line with its distance from the origin. If preferences are monotonic, the line through the origin must intersect every indi erence curve exactly once. Why? Because by assuming monotonicity, we have ruled out bads and satiation. So, every bundle is getting exactly one label. That s all it takes. There might be other ways to assign utility (assuming monotonicity). But this example shows there is always one way, assuming well-behaved preferences. 7

8 6 Examples of Utility Functions Now let s look at examples. 6.1 Indi erence Curves from Utility Functions Suppose you are given the utility function u(x 1 ; x 2 ) = x 1 x 2. Take as given for the moment that this is a well-de ned utility function. What do the indi erence curves look like? To do this, note that the indi erence curve associated with a particular level of utility is the set of all (x 1 ; x 2 ) such that u(x 1 ; x 2 ) = u, where u is a constant. This is known as a level set. Thus, for each constant, we get a di erent indi erence curve. In this case, implies u(x 1; x 2 ) = x 1 x 2 = u x 2 = u=x 1 8

9 Graph: Indi erence Curves for the Hyperbolic Level Set x 2 = u=x 1 x 2 = 3=x 1 x 2 = 5=x 1 x 2 = 7=x 1 x x1 A di erent example: v(x 1; x 2 ) = x 2 1x 2 2 This is just the square of the previous one. The original one cannot be negative. Thus, over the range of positive x 1 and x 2, the square is a monotonic transformation. Given that it is a monotonic transformation of the original utility function, these must be the same preferences. Therefore, we should have the same indi erence curves, but with di erent labels. For example, 1, 2, 3 corresponds to 1, 4, 9 now. 9

10 x 2 2 = u x 2 1 That is, if with the original labeling, we had, say u 0 = x 1 x 2 and we now have u 1 = x 2 1x 2 2 for the same x 1 and x 2, then squaring the rst one implies that u 2 0 = u 1 So, if u 0 had label of 3, then u 1 has a label of 9. bundles the same way. Thus, this utility function orders This is an example in one direction: given a utility function, determine the indifference curves. 6.2 Utility Functions from Indi erence Curves We also might want to go other way: given the preferences / indi erence curves, nd a corresponding utility function. We can often do it mathematically. Here, we will just gure it out intuitively in some interesting cases Perfect Substitutes In the case of preferences that regard two di erent goods as perfect substitutes, the only thing a consumer cares about is the (possibly weighted) number of units of the two goods. So, consider utility functions of the form u(x 1 ; x 2 ) = ax 1 + bx 2 10

11 This says the consumer would give up 1 unit of x 1 for a=b units of x 2, no matter what the initial levels are. Or, in other words, the consumer would give up b units of x 1 for a units of x 2. The rate of trade-o is a constant, which is what we mean by perfect substitutes. This formula also gives a higher level of utility to bundles with more units. If a = b = 1, this just says that the consumer cares only about the number of goods, not the composition; thus, the goods are perfect substitutes. The slope of the indi erence curves is a=b. We could also use, of course, the square of ax 1 + bx Perfect Complements A utility function that describes perfect complements is u(x 1 ; x 2 ) = minfax 1 ; bx 2 g Why? This is because if you always want to consume these goods in exact proportions, then anything beyond this proportion for one good, but not the other, adds nothing to utility. Take the case where a = b = 1 (e.g. right shoes and left shoes). The utility you get from (2, 1) or (1, 2) or (1, 1) is the same. Likewise for (50, 1), (1, 50), etc. All that matters is the good of which you have the least, i.e., the minimum. If a or b is not one, this becomes mildly tricky. Say we are considering gin and vermouth, which you only consume in martinis in the xed proportion of 5 gin to 1 vermouth. Then the utility function is u(g; v) = minf(1=5)g; vg 11

12 Unless we have 5 times as many units of gin as vermouth, we do not get any extra utility. Any monotonic transformation works as well. For instance, multiplying by 5 to get rid of the fraction: u(g; v) = minfg; 5vg Quasilinear Preferences Imagine that the consumer s indi erence curves are just vertical shifts, as in graph: 12

13 Graph: Quasilinear Preferences x 2 = k v(x 1 ) x 2 = k ln x 1 x x1 We can show that this must mean x 2 = k v(x 1 ) where k is a di erent constant for each indi erence curve. A higher k means a higher indi erence curve. A natural way to label the indi erence curves is with k. Solving for k and setting it equal to utility gives u(x 1 ; x 2 ) = k = v (x 1 ) + x 2 So, the utility function is linear in good 2, but not necessarily in good 1. Hence the name quasilinear. Examples: x :5 1 + x 2 13

14 ln x 1 + x 2 These functions are easy to work with, even if they are not always realistic Cobb-Douglas Preferences Another common utility function is Cobb Douglas: u(x 1 ; x 2 ) = x c 1x d 2 where c and d are positive numbers that describe preferences. The general graphical shape looks like: 14

15 Graph: Cobb-Douglas Indi erence Curves u(x 1 ; x 2 ) = x c 1x d 2 k = x 1=2 1 x 1=2 2 (k = 2; 4; 6 here) x x1 Cobb-Douglas utility functions correspond to convex, monotonic preferences. Once again, monotonic transformations also work: c ln x 1 + d ln x 2 Or, raising x c 1x d 2 to the power 1=(c + d) implies x a 1x 1 2 a where a = c=(c + d) This is useful for future examples. 15

16 7 Preview: Marginal Utility and the Marginal Rate of Substitution We have de ned and described utility functions, looked at examples, and related them to indi erence curves We have not mentioned marginal utility or the marginal rate of substitution; the order in which I am doing things is a bit di erent than the text. We will get to that when we examine the actual choice problem later. 16

Utility. M. Utku Ünver Micro Theory. M. Utku Ünver Micro Theory Utility 1 / 15

Utility M. Utku Ünver Micro Theory M. Utku Ünver Micro Theory Utility 1 / 15 Utility Function The preferences are the fundamental description useful for analyzing choice and utility is simply a way of

Common sense, and the model that we have used, suggest that an increase in p means a decrease in demand, but this is not the only possibility.

Lecture 6: Income and Substitution E ects c 2009 Je rey A. Miron Outline 1. Introduction 2. The Substitution E ect 3. The Income E ect 4. The Sign of the Substitution E ect 5. The Total Change in Demand

Chapter 4 Online Appendix: The Mathematics of Utility Functions

Chapter 4 Online Appendix: The Mathematics of Utility Functions We saw in the text that utility functions and indifference curves are different ways to represent a consumer s preferences. Calculus can

Advanced Microeconomics Ordinal preference theory Harald Wiese University of Leipzig Harald Wiese (University of Leipzig) Advanced Microeconomics 1 / 68 Part A. Basic decision and preference theory 1 Decisions

Economic Principles Solutions to Problem Set 1

Economic Principles Solutions to Problem Set 1 Question 1. Let < be represented b u : R n +! R. Prove that u (x) is strictl quasiconcave if and onl if < is strictl convex. If part: ( strict convexit of

Representation of functions as power series

Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions

Deriving Demand Functions - Examples 1

Deriving Demand Functions - Examples 1 What follows are some examples of different preference relations and their respective demand functions. In all the following examples, assume we have two goods x

Our development of economic theory has two main parts, consumers and producers. We will start with the consumers.

Lecture 1: Budget Constraints c 2008 Je rey A. Miron Outline 1. Introduction 2. Two Goods are Often Enough 3. Properties of the Budget Set 4. How the Budget Line Changes 5. The Numeraire 6. Taxes, Subsidies,

Economics 326: Duality and the Slutsky Decomposition. Ethan Kaplan

Economics 326: Duality and the Slutsky Decomposition Ethan Kaplan September 19, 2011 Outline 1. Convexity and Declining MRS 2. Duality and Hicksian Demand 3. Slutsky Decomposition 4. Net and Gross Substitutes

36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?

36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this

4.1 Ordinal versus cardinal utility

Microeconomics I. Antonio Zabalza. Universit of Valencia 1 Micro I. Lesson 4. Utilit In the previous lesson we have developed a method to rank consistentl all bundles in the (,) space and we have introduced

Price Elasticity of Supply; Consumer Preferences

1 Price Elasticity of Supply 1 14.01 Principles of Microeconomics, Fall 2007 Chia-Hui Chen September 12, 2007 Lecture 4 Price Elasticity of Supply; Consumer Preferences Outline 1. Chap 2: Elasticity -

Demand. Lecture 3. August 2015. Reading: Perlo Chapter 4 1 / 58

Demand Lecture 3 Reading: Perlo Chapter 4 August 2015 1 / 58 Introduction We saw the demand curve in chapter 2. We learned about consumer decision making in chapter 3. Now we bridge the gap between the

The fundamental question in economics is 2. Consumer Preferences

A Theory of Consumer Behavior Preliminaries 1. Introduction The fundamental question in economics is 2. Consumer Preferences Given limited resources, how are goods and service allocated? 1 3. Indifference

Preferences. M. Utku Ünver Micro Theory. Boston College. M. Utku Ünver Micro Theory (BC) Preferences 1 / 20

Preferences M. Utku Ünver Micro Theory Boston College M. Utku Ünver Micro Theory (BC) Preferences 1 / 20 Preference Relations Given any two consumption bundles x = (x 1, x 2 ) and y = (y 1, y 2 ), the

Chapter 27: Taxation. 27.1: Introduction. 27.2: The Two Prices with a Tax. 27.2: The Pre-Tax Position

Chapter 27: Taxation 27.1: Introduction We consider the effect of taxation on some good on the market for that good. We ask the questions: who pays the tax? what effect does it have on the equilibrium

CHAPTER 4 Consumer Choice

CHAPTER 4 Consumer Choice CHAPTER OUTLINE 4.1 Preferences Properties of Consumer Preferences Preference Maps 4.2 Utility Utility Function Ordinal Preference Utility and Indifference Curves Utility and

5.1 Radical Notation and Rational Exponents

Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots

Notes on Orthogonal and Symmetric Matrices MENU, Winter 2013

Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,

Elasticity. I. What is Elasticity?

Elasticity I. What is Elasticity? The purpose of this section is to develop some general rules about elasticity, which may them be applied to the four different specific types of elasticity discussed in

Chapter 4 NAME. Utility

Chapter 4 Utility NAME Introduction. In the previous chapter, you learned about preferences and indifference curves. Here we study another way of describing preferences, the utility function. A utility

1 Present and Future Value

Lecture 8: Asset Markets c 2009 Je rey A. Miron Outline:. Present and Future Value 2. Bonds 3. Taxes 4. Applications Present and Future Value In the discussion of the two-period model with borrowing and

x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1

Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs

Risk Aversion. Expected value as a criterion for making decisions makes sense provided that C H A P T E R 2. 2.1 Risk Attitude

C H A P T E R 2 Risk Aversion Expected value as a criterion for making decisions makes sense provided that the stakes at risk in the decision are small enough to \play the long run averages." The range

Linear Programming Notes V Problem Transformations

Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material

ANSWER KEY 3 UTILITY FUNCTIONS, THE CONSUMER S PROBLEM, DEMAND CURVES

ANSWER KEY 3 UTILITY FUNCTIONS, THE CONSUMER S PROBLEM, DEMAND CURVES ECON 210 (1) Perfect Substitutes. Suppose that Jack s utility is entirely based on number of hours spent camping (c) and skiing (s).

Review of Fundamental Mathematics

Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.

Chapter 9. Systems of Linear Equations

Chapter 9. Systems of Linear Equations 9.1. Solve Systems of Linear Equations by Graphing KYOTE Standards: CR 21; CA 13 In this section we discuss how to solve systems of two linear equations in two variables

Constrained Optimisation

CHAPTER 9 Constrained Optimisation Rational economic agents are assumed to make choices that maximise their utility or profit But their choices are usually constrained for example the consumer s choice

Problem Set 2: Solutions ECON 301: Intermediate Microeconomics Prof. Marek Weretka. Problem 1 (Marginal Rate of Substitution)

Proble Set 2: Solutions ECON 30: Interediate Microeconoics Prof. Marek Weretka Proble (Marginal Rate of Substitution) (a) For the third colun, recall that by definition MRS(x, x 2 ) = ( ) U x ( U ). x

Economics 2020a / HBS 4010 / HKS API-111 FALL 2010 Solutions to Practice Problems for Lectures 1 to 4

Economics 00a / HBS 4010 / HKS API-111 FALL 010 Solutions to Practice Problems for Lectures 1 to 4 1.1. Quantity Discounts and the Budget Constraint (a) The only distinction between the budget line with

REVIEW OF MICROECONOMICS

ECO 352 Spring 2010 Precepts Weeks 1, 2 Feb. 1, 8 REVIEW OF MICROECONOMICS Concepts to be reviewed Budget constraint: graphical and algebraic representation Preferences, indifference curves. Utility function

Lecture 1: Systems of Linear Equations

MTH Elementary Matrix Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Lecture 1 Systems of Linear Equations ² Systems of two linear equations with two variables

6.4 Normal Distribution

Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under

Economics 121b: Intermediate Microeconomics Problem Set 2 1/20/10

Dirk Bergemann Department of Economics Yale University s by Olga Timoshenko Economics 121b: Intermediate Microeconomics Problem Set 2 1/20/10 This problem set is due on Wednesday, 1/27/10. Preliminary

Chapter 3 Consumer Behavior

Chapter 3 Consumer Behavior Read Pindyck and Rubinfeld (2013), Chapter 3 Microeconomics, 8 h Edition by R.S. Pindyck and D.L. Rubinfeld Adapted by Chairat Aemkulwat for Econ I: 2900111 1/29/2015 CHAPTER

The Method of Partial Fractions Math 121 Calculus II Spring 2015

Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization

Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization 2.1. Introduction Suppose that an economic relationship can be described by a real-valued

Introductory Notes on Demand Theory

Introductory Notes on Demand Theory (The Theory of Consumer Behavior, or Consumer Choice) This brief introduction to demand theory is a preview of the first part of Econ 501A, but it also serves as a prototype

EQUATIONS and INEQUALITIES

EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line

Examples on Monopoly and Third Degree Price Discrimination

1 Examples on Monopoly and Third Degree Price Discrimination This hand out contains two different parts. In the first, there are examples concerning the profit maximizing strategy for a firm with market

Domain of a Composition

Domain of a Composition Definition Given the function f and g, the composition of f with g is a function defined as (f g)() f(g()). The domain of f g is the set of all real numbers in the domain of g such

Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is

1. Briefly explain what an indifference curve is and how it can be graphically derived.

Chapter 2: Consumer Choice Short Answer Questions 1. Briefly explain what an indifference curve is and how it can be graphically derived. Answer: An indifference curve shows the set of consumption bundles

Multi-variable Calculus and Optimization

Multi-variable Calculus and Optimization Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 1 / 51 EC2040 Topic 3 - Multi-variable Calculus

1.7 Graphs of Functions

64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most

Correlation key concepts:

CORRELATION Correlation key concepts: Types of correlation Methods of studying correlation a) Scatter diagram b) Karl pearson s coefficient of correlation c) Spearman s Rank correlation coefficient d)

Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving

Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words

Partial Derivatives. @x f (x; y) = @ x f (x; y) @x x2 y + @ @x y2 and then we evaluate the derivative as if y is a constant.

Partial Derivatives Partial Derivatives Just as derivatives can be used to eplore the properties of functions of 1 variable, so also derivatives can be used to eplore functions of 2 variables. In this

The Mathematics 11 Competency Test Percent Increase or Decrease

The Mathematics 11 Competency Test Percent Increase or Decrease The language of percent is frequently used to indicate the relative degree to which some quantity changes. So, we often speak of percent

CALCULATIONS & STATISTICS

CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 1-5 scale to 0-100 scores When you look at your report, you will notice that the scores are reported on a 0-100 scale, even though respondents

Determine If An Equation Represents a Function

Question : What is a linear function? The term linear function consists of two parts: linear and function. To understand what these terms mean together, we must first understand what a function is. The

Module 3: Correlation and Covariance

Using Statistical Data to Make Decisions Module 3: Correlation and Covariance Tom Ilvento Dr. Mugdim Pašiƒ University of Delaware Sarajevo Graduate School of Business O ften our interest in data analysis

To give it a definition, an implicit function of x and y is simply any relationship that takes the form:

2 Implicit function theorems and applications 21 Implicit functions The implicit function theorem is one of the most useful single tools you ll meet this year After a while, it will be second nature to

More Quadratic Equations Math 99 N1 Chapter 8 1 Quadratic Equations We won t discuss quadratic inequalities. Quadratic equations are equations where the unknown appears raised to second power, and, possibly

Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

Managerial Economics Prof. Trupti Mishra S.J.M. School of Management Indian Institute of Technology, Bombay. Lecture - 13 Consumer Behaviour (Contd )

(Refer Slide Time: 00:28) Managerial Economics Prof. Trupti Mishra S.J.M. School of Management Indian Institute of Technology, Bombay Lecture - 13 Consumer Behaviour (Contd ) We will continue our discussion

Section 2.4: Equations of Lines and Planes

Section.4: Equations of Lines and Planes An equation of three variable F (x, y, z) 0 is called an equation of a surface S if For instance, (x 1, y 1, z 1 ) S if and only if F (x 1, y 1, z 1 ) 0. x + y

CURVE FITTING LEAST SQUARES APPROXIMATION

CURVE FITTING LEAST SQUARES APPROXIMATION Data analysis and curve fitting: Imagine that we are studying a physical system involving two quantities: x and y Also suppose that we expect a linear relationship

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

1.6 The Order of Operations

1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative

Chapter 25: Exchange in Insurance Markets

Chapter 25: Exchange in Insurance Markets 25.1: Introduction In this chapter we use the techniques that we have been developing in the previous 2 chapters to discuss the trade of risk. Insurance markets

Problem Set #5-Key. Economics 305-Intermediate Microeconomic Theory

Problem Set #5-Key Sonoma State University Economics 305-Intermediate Microeconomic Theory Dr Cuellar (1) Suppose that you are paying your for your own education and that your college tuition is \$200 per

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

Solving Rational Equations

Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,

Tastes and Indifference Curves

C H A P T E R 4 Tastes and Indifference Curves This chapter begins a -chapter treatment of tastes or what we also call preferences. In the first of these chapters, we simply investigate the basic logic

This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.

Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course

Partial Fractions Decomposition

Partial Fractions Decomposition Dr. Philippe B. Laval Kennesaw State University August 6, 008 Abstract This handout describes partial fractions decomposition and how it can be used when integrating rational

Year 9 set 1 Mathematics notes, to accompany the 9H book.

Part 1: Year 9 set 1 Mathematics notes, to accompany the 9H book. equations 1. (p.1), 1.6 (p. 44), 4.6 (p.196) sequences 3. (p.115) Pupils use the Elmwood Press Essential Maths book by David Raymer (9H

3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

Linear Programming Notes VII Sensitivity Analysis

Linear Programming Notes VII Sensitivity Analysis 1 Introduction When you use a mathematical model to describe reality you must make approximations. The world is more complicated than the kinds of optimization

Solving Systems of Two Equations Algebraically

8 MODULE 3. EQUATIONS 3b Solving Systems of Two Equations Algebraically Solving Systems by Substitution In this section we introduce an algebraic technique for solving systems of two equations in two unknowns

9.4. The Scalar Product. Introduction. Prerequisites. Learning Style. Learning Outcomes

The Scalar Product 9.4 Introduction There are two kinds of multiplication involving vectors. The first is known as the scalar product or dot product. This is so-called because when the scalar product of

Graphing Linear Equations in Two Variables

Math 123 Section 3.2 - Graphing Linear Equations Using Intercepts - Page 1 Graphing Linear Equations in Two Variables I. Graphing Lines A. The graph of a line is just the set of solution points of the

Solutions of Linear Equations in One Variable

2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools

Microeconomic Theory: Basic Math Concepts

Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts

1 Error in Euler s Method

1 Error in Euler s Method Experience with Euler s 1 method raises some interesting questions about numerical approximations for the solutions of differential equations. 1. What determines the amount of

1 Review of Least Squares Solutions to Overdetermined Systems

cs4: introduction to numerical analysis /9/0 Lecture 7: Rectangular Systems and Numerical Integration Instructor: Professor Amos Ron Scribes: Mark Cowlishaw, Nathanael Fillmore Review of Least Squares

Sample Midterm Solutions

Sample Midterm Solutions Instructions: Please answer both questions. You should show your working and calculations for each applicable problem. Correct answers without working will get you relatively few

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

Linear Programming. Solving LP Models Using MS Excel, 18

SUPPLEMENT TO CHAPTER SIX Linear Programming SUPPLEMENT OUTLINE Introduction, 2 Linear Programming Models, 2 Model Formulation, 4 Graphical Linear Programming, 5 Outline of Graphical Procedure, 5 Plotting

Session 7 Bivariate Data and Analysis

Session 7 Bivariate Data and Analysis Key Terms for This Session Previously Introduced mean standard deviation New in This Session association bivariate analysis contingency table co-variation least squares

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

or, put slightly differently, the profit maximizing condition is for marginal revenue to equal marginal cost:

Chapter 9 Lecture Notes 1 Economics 35: Intermediate Microeconomics Notes and Sample Questions Chapter 9: Profit Maximization Profit Maximization The basic assumption here is that firms are profit maximizing.

G. GRAPHING FUNCTIONS

G. GRAPHING FUNCTIONS To get a quick insight int o how the graph of a function looks, it is very helpful to know how certain simple operations on the graph are related to the way the function epression

No Solution Equations Let s look at the following equation: 2 +3=2 +7

5.4 Solving Equations with Infinite or No Solutions So far we have looked at equations where there is exactly one solution. It is possible to have more than solution in other types of equations that are

On Lexicographic (Dictionary) Preference

MICROECONOMICS LECTURE SUPPLEMENTS Hajime Miyazaki File Name: lexico95.usc/lexico99.dok DEPARTMENT OF ECONOMICS OHIO STATE UNIVERSITY Fall 993/994/995 Miyazaki.@osu.edu On Lexicographic (Dictionary) Preference

CHAPTER 3 CONSUMER BEHAVIOR

CHAPTER 3 CONSUMER BEHAVIOR EXERCISES 2. Draw the indifference curves for the following individuals preferences for two goods: hamburgers and beer. a. Al likes beer but hates hamburgers. He always prefers

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

Chapter 3. Distribution Problems. 3.1 The idea of a distribution. 3.1.1 The twenty-fold way

Chapter 3 Distribution Problems 3.1 The idea of a distribution Many of the problems we solved in Chapter 1 may be thought of as problems of distributing objects (such as pieces of fruit or ping-pong balls)

Econ 100A: Intermediate Microeconomics Notes on Consumer Theory

Econ 100A: Interediate Microeconoics Notes on Consuer Theory Linh Bun Winter 2012 (UCSC 1. Consuer Theory Utility Functions 1.1. Types of Utility Functions The following are soe of the type of the utility

MATHEMATICS FOR ENGINEERING BASIC ALGEBRA

MATHEMATICS FOR ENGINEERING BASIC ALGEBRA TUTORIAL 3 EQUATIONS This is the one of a series of basic tutorials in mathematics aimed at beginners or anyone wanting to refresh themselves on fundamentals.

-2- Reason: This is harder. I'll give an argument in an Addendum to this handout.

LINES Slope The slope of a nonvertical line in a coordinate plane is defined as follows: Let P 1 (x 1, y 1 ) and P 2 (x 2, y 2 ) be any two points on the line. Then slope of the line = y 2 y 1 change in

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.

8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent

ALGEBRA. sequence, term, nth term, consecutive, rule, relationship, generate, predict, continue increase, decrease finite, infinite

ALGEBRA Pupils should be taught to: Generate and describe sequences As outcomes, Year 7 pupils should, for example: Use, read and write, spelling correctly: sequence, term, nth term, consecutive, rule,

Equations, Lenses and Fractions

46 Equations, Lenses and Fractions The study of lenses offers a good real world example of a relation with fractions we just can t avoid! Different uses of a simple lens that you may be familiar with are