(b) Can Charlie afford any bundles that give him a utility of 150? (c) Can Charlie afford any bundles that give him a utility of 300?

Size: px
Start display at page:

Download "(b) Can Charlie afford any bundles that give him a utility of 150? (c) Can Charlie afford any bundles that give him a utility of 300?"

Transcription

1 Micro PS2 - Choice, Demand and Consumer Surplus 1. We begin again with Charlie of the apples and bananas. Recall that Charlie s utility function is U(x A, x B ) = x A x B. Suppose that the price of apples is 1, the price of bananas is 2, and Charlie s income is 40. (a) On the graph below, draw Charlie s budget line. (Use the graphs marks and try to make this line accurate putting bananas on the y-axis.) Plot a few points on the indifference curve that gives Charlie a utility of 150 and sketch this curve. Now plot a few points on the indifference curve that gives Charlie a utility of 300 and sketch this curve. (b) Can Charlie afford any bundles that give him a utility of 150? (c) Can Charlie afford any bundles that give him a utility of 300? (d) On your graph, mark a point that Charlie can just afford (ie. on the budget line) and that gives him a higher utility than 150. Label that point A. (e) Neither of the indifference curves that you drew is tangent to Charlie s budget line. Let s try to find one that is. At any point, (x A, x B ), Charlie s marginal rate of substitution is a function of x A and x B. In fact, if you calculate the ratio of marginal utilities for Charlie s utility function, you will find that Charlie s marginal rate of substitution is MRS(x A, x B ) = x B /x A. This is the slope of his indifference curve at any (x A, x B ). The slope of Charlie s budget line is what? (Give an actual number): 1

2 (f) Write an equation that says that the budget line is tangent to an indifference curve at (x A, x B ): There are many solutions to this equation. Each of these solutions corresponds to a point on a different indifference curve. Draw a line that passes through all of these points. (g) The best bundle that Charlie can afford must lie somewhere on the line you just penciled in. It must also lie on his budget line. If the point is outside of his budget line, he can t afford it. If the point lies inside of his budget line, he can afford to do better by buying more of both goods. On your graph, label this best affordable bundle with an E. This happens where x A =? and x B =?. Verify your answer by solving the two simultaneous equations given by his (i) the budget equation and the (ii) tangency condition (MRS = ratio of prices). (HINT: 1st get both equations. 2nd solve one for either x A or x B. 3rd plug this into the other equation and solve for the unknown. 4th use this answer to plug back into the budget equation and solve for the other.) (h) What is Charlie s utility if he consumes the bundle (20, 10)? (i) On the graph above, use a dotted line to draw his indifference curve through (20, 10). Does this indifference curve cross Charlie s budget line, just touch it, or never touch it? 2. Charlie consumes apples (x a ) and bananas (x b ). His budget line is p a x a + p b x b = m. The slope of his indifference curve at a bundle (x a, x b ) is: MU 1(x a,x b ) MU 2 (x a,x b ) = x b/x a. The slope of his budget line is p a /p b. a. Charlies indifference curve will be tangent to his budget line at the point (x a, x b ) if the follow equation is satisfied: b. You have two equations, the budget line and the tangency equation, that must be satisfied by the optimal bundle. Solve these two equations for x a and x b. Charlies demand for apples is x a (p a, p b, m) = His demand for bananas is x b (p a, p b, m) =. c. In general, the demand for both commodities will depend on the price of each and on income. But for Charlie the demand only depends on income and the price of that good. He always spends 2

3 the same fraction of his income on bananas. What is that fraction? 3. Linus has the utility function U(x, y) = x + 3y. (a) On the graph below, draw the indifference curve passing through the point (x, y) = (3, 3). Now sketch the indifference curve connecting bundles that give Linus a utility of 6. (b) On the same graph, use a dotted line to draw Linus s budget line if the price of x is 1 and the price of y is 2 and his income is 8. What bundle does Linus choose in this situation? (c) What bundle would Linus choose if the price of x is 1, the price of y is 4, and his income is 8? 4. (Here you will walk through a unique characteristics of quasi-linear preferences - see pg 62 and 102 for brief discussion) Donald only consumes stamps and Twinkies. His preferences take the form u(s, t) = s + ln(t). Stamps and Twinkies have prices p s and p t, and he has income m. a. Write an expression that says the ratio of Donald s marginal utility for Twinkies to his marginal utility for stamps is equal to the ratio of the price of Twinkies to the price of stamps. (Hint: his marginal utility of Twinkies is 1/t, and his MU of stamps is 1.) b. You can use the equation you found in the last part to show that if he buys both goods, Donald s demand function for Twinkies depends only on the price ratio and not on his income. What is Donald s demand function for Twinkies? (Note: the equation found in (a) is the tangency point 3

4 needed for an interior solution) c. Notice that for this special utility function, if Donald buys both goods, then the total amount of money that he spends on Twinkies has the peculiar property that it depends on only one of the three variables (m, p t, p s ), show this. (Hint: what is the amount spent on Twinkies?) d. Since there are only two goods, any money that is not spent on Twinkies must be spent on stamps. Use the budget equation and Donald s demand function for Twinkies to find an expression for the number of stamps he will buy. e. The expression you just wrote down is negative if m < p s. Surely it makes no sense for him to be demanding negative amounts of postage stamps. If m < p s, Donald s demand for postage stamps will be 0. What would his demand for Twinkies be? f. Donald s wife complains that whenever Donald gets an extra dollar, he always spends it all on stamps. Is she right? (Assume that m > p s.) g. Suppose that the price of Twinkies is $2 and the price of stamps is $1. Draw Donald s Engel curve for Twinkies and stamps - be sure to mark which is which. (Hint: First draw the Engel curves for incomes greater than $1, then draw them for incomes less than $1.) 5. Bob likes beef, his demand for kilograms of beef is: D(p) = 100 p, where p is the price of a kilo of beef. a. If the price of beef is 50, how much will he consume? 4

5 b. How much gross consumer s surplus does he get from his consumption? c. How much money does he spend on beef? d. What is his net consumer s surplus? 6. In the graph below, you see a representation of Sarah Gamp s indifference curves between cucumbers and other goods. Suppose that the reference price of cucumbers and the reference price of other goods are both 1. (a) What is the minimum amount of money that Sarah would need in order to purchase a bundle that is indifferent to A? (b) What is the minimum amount of money that Sarah would need in order to purchase a bundle that is indifferent to B? (c) Suppose that the reference price for cucumbers is 2 and the reference price for other goods is 1. How much money does she need in order to purchase a bundle that is indifferent to bundle A? (d) What is the minimum amount of money that Sarah would need to purchase a bundle that is indifferent to B using these new prices? 5

6 (e) No matter what prices Sarah faces, the amount of money she needs to purchase a bundle indifferent to A must be (higher or lower?) than the amount she needs to purchase a bundle indifferent to B. 6

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 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

More information

Chapter 4 NAME. Utility

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

More information

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

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

More information

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

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

More information

CHAPTER 4 Consumer Choice

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

More information

The fundamental question in economics is 2. Consumer Preferences

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

More information

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

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).

More information

Elements of a graph. Click on the links below to jump directly to the relevant section

Elements of a graph. Click on the links below to jump directly to the relevant section Click on the links below to jump directly to the relevant section Elements of a graph Linear equations and their graphs What is slope? Slope and y-intercept in the equation of a line Comparing lines on

More information

Chapter 4 Online Appendix: The Mathematics of Utility Functions

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

More information

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

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

More information

Deriving Demand Functions - Examples 1

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

More information

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

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

More information

Chapter 3 Consumer Behavior

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

More information

What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b.

What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b. PRIMARY CONTENT MODULE Algebra - Linear Equations & Inequalities T-37/H-37 What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of

More information

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

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

More information

Table of Contents MICRO ECONOMICS

Table of Contents MICRO ECONOMICS economicsentrance.weebly.com Basic Exercises Micro Economics AKG 09 Table of Contents MICRO ECONOMICS Budget Constraint... 4 Practice problems... 4 Answers... 4 Supply and Demand... 7 Practice Problems...

More information

CHAPTER 3 CONSUMER BEHAVIOR

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

More information

ECN 221 Chapter 5 practice problems This is not due for a grade

ECN 221 Chapter 5 practice problems This is not due for a grade ECN 221 Chapter 5 practice problems This is not due for a grade 1. Assume the price of pizza is $2.00 and the price of Beer is $1.00 and that at your current levels of consumption, the Marginal Utility

More information

A Utility Maximization Example

A Utility Maximization Example A Utilit Maximization Example Charlie Gibbons Universit of California, Berkele September 17, 2007 Since we couldn t finish the utilit maximization problem in section, here it is solved from the beginning.

More information

1.3 LINEAR EQUATIONS IN TWO VARIABLES. Copyright Cengage Learning. All rights reserved.

1.3 LINEAR EQUATIONS IN TWO VARIABLES. Copyright Cengage Learning. All rights reserved. 1.3 LINEAR EQUATIONS IN TWO VARIABLES Copyright Cengage Learning. All rights reserved. What You Should Learn Use slope to graph linear equations in two variables. Find the slope of a line given two points

More information

Review of Fundamental Mathematics

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

More information

AK 4 SLUTSKY COMPENSATION

AK 4 SLUTSKY COMPENSATION AK 4 SLUTSKY COMPENSATION ECON 210 A. JOSEPH GUSE (1) (a) First calculate the demand at the original price p b = 2 b(p b,m) = 1000 20 5p b b 0 = b(2) = 40 In general m c = m+(p 1 b p0 b )b 0. If the price

More information

Econ 100A: Intermediate Microeconomics Notes on Consumer Theory

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

More information

Consumers face constraints on their choices because they have limited incomes.

Consumers face constraints on their choices because they have limited incomes. Consumer Choice: the Demand Side of the Market Consumers face constraints on their choices because they have limited incomes. Wealthy and poor individuals have limited budgets relative to their desires.

More information

3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

More information

Lab 17: Consumer and Producer Surplus

Lab 17: Consumer and Producer Surplus Lab 17: Consumer and Producer Surplus Who benefits from rent controls? Who loses with price controls? How do taxes and subsidies affect the economy? Some of these questions can be analyzed using the concepts

More information

Gains From Trade Consumer Surplus Quantifying Welfare Effects Producer Surplus Welfare in Equilibrium. Consumer Surplus and Welfare Measurement

Gains From Trade Consumer Surplus Quantifying Welfare Effects Producer Surplus Welfare in Equilibrium. Consumer Surplus and Welfare Measurement Consumer Surplus and Welfare Measurement Questions Q: How can we... Find a monetary measure of a consumer s utility/happiness? Evaluate a consumer s willingness to pay for a unit of a good? Evaluate whether

More information

DEMAND FORECASTING. Demand. Law of Demand. Definition of Law of Demand

DEMAND FORECASTING. Demand. Law of Demand. Definition of Law of Demand DEMAND FORECASTING http://www.tutorialspoint.com/managerial_economics/demand_forecasting.htm Copyright tutorialspoint.com Demand Demand is a widely used term, and in common is considered synonymous with

More information

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

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

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

Microeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS

Microeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS DUSP 11.203 Frank Levy Microeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS These notes have three purposes: 1) To explain why some simple calculus formulae are useful in understanding

More information

Price Elasticity of Supply; Consumer Preferences

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 -

More information

Theory of Demand. ECON 212 Lecture 7. Tianyi Wang. Winter 2013. Queen s Univerisity. Tianyi Wang (Queen s Univerisity) Lecture 7 Winter 2013 1 / 46

Theory of Demand. ECON 212 Lecture 7. Tianyi Wang. Winter 2013. Queen s Univerisity. Tianyi Wang (Queen s Univerisity) Lecture 7 Winter 2013 1 / 46 Theory of Demand ECON 212 Lecture 7 Tianyi Wang Queen s Univerisity Winter 2013 Tianyi Wang (Queen s Univerisity) Lecture 7 Winter 2013 1 / 46 Intro Note: Quiz 1 can be picked up at Distribution Center.

More information

PPA 723, Fall 2006 Professor John McPeak

PPA 723, Fall 2006 Professor John McPeak Quiz One PPA 723, Fall 2006 Professor John McPeak Name: The total quiz is worth 20 points. Each question is worth 2 points, and each sub question is worth an equal share of the two points. 1) The demand

More information

Practice Problem Set 2 (ANSWERS)

Practice Problem Set 2 (ANSWERS) Economics 370 Professor H.J. Schuetze Practice Problem Set 2 (ANSWERS) 1. See the figure below, where the initial budget constraint is given by ACE. After the new legislation is passed, the budget constraint

More information

Where are we? To do today: finish the derivation of the demand curve using indifference curves. Go on then to chapter Production and Cost

Where are we? To do today: finish the derivation of the demand curve using indifference curves. Go on then to chapter Production and Cost Where are we? To do today: finish the derivation of the demand curve using indifference curves Go on then to chapter Production and Cost Utility and indifference curves The point is to find where on the

More information

Utility Maximization

Utility Maximization Utility Maimization Given the consumer's income, M, and prices, p and p y, the consumer's problem is to choose the a ordable bundle that maimizes her utility. The feasible set (budget set): total ependiture

More information

Economics 100A. Final Exam

Economics 100A. Final Exam Name form number 1 Economics 100A Final Exam Fill in the bubbles on your scantron with your id number (starting from the left side of the box), your name, and the form type. Students who do this successfully

More information

Economics 301 Problem Set 4 5 October 2007

Economics 301 Problem Set 4 5 October 2007 Economics 301 Name Problem Set 4 5 October 2007 Budget Lines and Indifference Curves and the Consumer Optimum 1. Parvez, a pharmacology student, has allocated $120 per month to spend on paperback novels

More information

Math 1526 Consumer and Producer Surplus

Math 1526 Consumer and Producer Surplus Math 156 Consumer and Producer Surplus Scenario: In the grocery store, I find that two-liter sodas are on sale for 89. This is good news for me, because I was prepared to pay $1.9 for them. The store manager

More information

REVIEW OF MICROECONOMICS

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

More information

Constrained Optimisation

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

More information

COMMERCE MENTORSHIP PROGRAM COMM295: MANAGERIAL ECONOMICS FINAL EXAM REVIEW SOLUTION KEY

COMMERCE MENTORSHIP PROGRAM COMM295: MANAGERIAL ECONOMICS FINAL EXAM REVIEW SOLUTION KEY COMMERCE MENTORSHIP PROGRAM COMM295: MANAGERIAL ECONOMICS FINAL EXAM REVIEW SOLUTION KEY WR1 Sam-I-Am is a local restaurant chain located in Vancouver. It is considering different pricing strategies for

More information

Average rate of change of y = f(x) with respect to x as x changes from a to a + h:

Average rate of change of y = f(x) with respect to x as x changes from a to a + h: L15-1 Lecture 15: Section 3.4 Definition of the Derivative Recall the following from Lecture 14: For function y = f(x), the average rate of change of y with respect to x as x changes from a to b (on [a,

More information

The Point-Slope Form

The Point-Slope Form 7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope

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

Prot Maximization and Cost Minimization

Prot Maximization and Cost Minimization Simon Fraser University Prof. Karaivanov Department of Economics Econ 0 COST MINIMIZATION Prot Maximization and Cost Minimization Remember that the rm's problem is maximizing prots by choosing the optimal

More information

Test 1 10 October 2008. 1. Assume that tea and lemons are complements and that coffee and tea are substitutes.

Test 1 10 October 2008. 1. Assume that tea and lemons are complements and that coffee and tea are substitutes. Eco 301 Name Test 1 10 October 2008 100 points. Please write all answers in ink. Please use pencil and a straight edge to draw graphs. Allocate your time efficiently. 1. Assume that tea and lemons are

More information

Chapter 4 Individual and Market Demand

Chapter 4 Individual and Market Demand Chapter 4 Individual and Market Demand Questions for Review 1. Explain the difference between each of the following terms: a. a price consumption curve and a demand curve The price consumption curve (PCC)

More information

GRAPHING IN POLAR COORDINATES SYMMETRY

GRAPHING IN POLAR COORDINATES SYMMETRY GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Also remember that there are three types of symmetry - y-axis,

More information

Tastes and Indifference Curves

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

More information

Consumer Theory. The consumer s problem

Consumer Theory. The consumer s problem Consumer Theory The consumer s problem 1 The Marginal Rate of Substitution (MRS) We define the MRS(x,y) as the absolute value of the slope of the line tangent to the indifference curve at point point (x,y).

More information

Lesson 19: Equations for Tangent Lines to Circles

Lesson 19: Equations for Tangent Lines to Circles Student Outcomes Given a circle, students find the equations of two lines tangent to the circle with specified slopes. Given a circle and a point outside the circle, students find the equation of the line

More information

PPF's of Germany and France

PPF's of Germany and France Economics 165 Winter 2 Problem Set #1 Problem 1: Let Germany and France have respective labor forces of 8 and 6. Suppose both countries produce wine and cares according to the following unit labor requirements:

More information

COST THEORY. I What costs matter? A Opportunity Costs

COST THEORY. I What costs matter? A Opportunity Costs COST THEORY Cost theory is related to production theory, they are often used together. However, the question is how much to produce, as opposed to which inputs to use. That is, assume that we use production

More information

Introductory Notes on Demand Theory

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

More information

Graphing Linear Equations

Graphing Linear Equations Graphing Linear Equations I. Graphing Linear Equations a. The graphs of first degree (linear) equations will always be straight lines. b. Graphs of lines can have Positive Slope Negative Slope Zero slope

More information

Fractions Practice: Answers

Fractions Practice: Answers Click on the links below to jump directly to the relevant section Fractions Practice: Answers Percents Practice: Answers Ratios Practice: Answers Proportions Practice: Answers Graphing Practice: Answers

More information

c. Given your answer in part (b), what do you anticipate will happen in this market in the long-run?

c. Given your answer in part (b), what do you anticipate will happen in this market in the long-run? Perfect Competition Questions Question 1 Suppose there is a perfectly competitive industry where all the firms are identical with identical cost curves. Furthermore, suppose that a representative firm

More information

Principles of Economics: Micro: Exam #2: Chapters 1-10 Page 1 of 9

Principles of Economics: Micro: Exam #2: Chapters 1-10 Page 1 of 9 Principles of Economics: Micro: Exam #2: Chapters 1-10 Page 1 of 9 print name on the line above as your signature INSTRUCTIONS: 1. This Exam #2 must be completed within the allocated time (i.e., between

More information

Linear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)}

Linear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)} Linear Equations Domain and Range Domain refers to the set of possible values of the x-component of a point in the form (x,y). Range refers to the set of possible values of the y-component of a point in

More information

Slutsky Equation. M. Utku Ünver Micro Theory. Boston College. M. Utku Ünver Micro Theory (BC) Slutsky Equation 1 / 15

Slutsky Equation. M. Utku Ünver Micro Theory. Boston College. M. Utku Ünver Micro Theory (BC) Slutsky Equation 1 / 15 Slutsky Equation M. Utku Ünver Micro Theory Boston College M. Utku Ünver Micro Theory (BC) Slutsky Equation 1 / 15 Effects of a Price Change: What happens when the price of a commodity decreases? 1 The

More information

Section 1.1 Linear Equations: Slope and Equations of Lines

Section 1.1 Linear Equations: Slope and Equations of Lines Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

More information

LINEAR INEQUALITIES. Mathematics is the art of saying many things in many different ways. MAXWELL

LINEAR INEQUALITIES. Mathematics is the art of saying many things in many different ways. MAXWELL Chapter 6 LINEAR INEQUALITIES 6.1 Introduction Mathematics is the art of saying many things in many different ways. MAXWELL In earlier classes, we have studied equations in one variable and two variables

More information

The Walrasian Model and Walrasian Equilibrium

The Walrasian Model and Walrasian Equilibrium The Walrasian Model and Walrasian Equilibrium 1.1 There are only two goods in the economy and there is no way to produce either good. There are n individuals, indexed by i = 1,..., n. Individual i owns

More information

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

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

More information

MICROECONOMICS AND POLICY ANALYSIS - U8213 Professor Rajeev H. Dehejia Class Notes - Spring 2001

MICROECONOMICS AND POLICY ANALYSIS - U8213 Professor Rajeev H. Dehejia Class Notes - Spring 2001 MICROECONOMICS AND POLICY ANALYSIS - U8213 Professor Rajeev H. Dehejia Class Notes - Spring 2001 General Equilibrium and welfare with production Wednesday, January 24 th and Monday, January 29 th Reading:

More information

Chapter 4 Consumption, Saving, and Investment

Chapter 4 Consumption, Saving, and Investment Chapter 4 Consumption, Saving, and Investment Multiple Choice Questions 1. Desired national saving equals (a) Y C d G. (b) C d + I d + G. (c) I d + G. (d) Y I d G. 2. With no inflation and a nominal interest

More information

In following this handout, sketch appropriate graphs in the space provided.

In following this handout, sketch appropriate graphs in the space provided. Dr. McGahagan Graphs and microeconomics You will see a remarkable number of graphs on the blackboard and in the text in this course. You will see a fair number on examinations as well, and many exam questions,

More information

Temperature Scales. The metric system that we are now using includes a unit that is specific for the representation of measured temperatures.

Temperature Scales. The metric system that we are now using includes a unit that is specific for the representation of measured temperatures. Temperature Scales INTRODUCTION The metric system that we are now using includes a unit that is specific for the representation of measured temperatures. The unit of temperature in the metric system is

More information

Intermediate Microeconomics (22014)

Intermediate Microeconomics (22014) Intermediate Microeconomics (22014) I. Consumer Instructor: Marc Teignier-Baqué First Semester, 2011 Outline Part I. Consumer 1. umer 1.1 Budget Constraints 1.2 Preferences 1.3 Utility Function 1.4 1.5

More information

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 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

More information

Profit and Revenue Maximization

Profit and Revenue Maximization WSG7 7/7/03 4:36 PM Page 95 7 Profit and Revenue Maximization OVERVIEW The purpose of this chapter is to develop a general framework for finding optimal solutions to managerial decision-making problems.

More information

3. George W. Bush is the current U.S. President. This is an example of a: A. Normative statement B. Positive statement

3. George W. Bush is the current U.S. President. This is an example of a: A. Normative statement B. Positive statement Econ 3144 Fall 2006 Test 1 Dr. Rupp Name Sign Pledge I have neither given nor received aid on this exam Multiple Choice Questions (3 points each) 1. What you give up to obtain an item is called your A.

More information

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

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.

More information

Solving Quadratic Equations

Solving Quadratic Equations 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

More information

Chapter 6: Pure Exchange

Chapter 6: Pure Exchange Chapter 6: Pure Exchange Pure Exchange Pareto-Efficient Allocation Competitive Price System Equitable Endowments Fair Social Welfare Allocation Outline and Conceptual Inquiries There are Gains from Trade

More information

The Graphical Method: An Example

The Graphical Method: An Example The Graphical Method: An Example Consider the following linear program: Maximize 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2 0, where, for ease of reference,

More information

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.

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

More information

Systems of Linear Equations: Two Variables

Systems of Linear Equations: Two Variables OpenStax-CNX module: m49420 1 Systems of Linear Equations: Two Variables OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 In this section,

More information

F.IF.7b: Graph Root, Piecewise, Step, & Absolute Value Functions

F.IF.7b: Graph Root, Piecewise, Step, & Absolute Value Functions F.IF.7b: Graph Root, Piecewise, Step, & Absolute Value Functions F.IF.7b: Graph Root, Piecewise, Step, & Absolute Value Functions Analyze functions using different representations. 7. Graph functions expressed

More information

PLOTTING DATA AND INTERPRETING GRAPHS

PLOTTING DATA AND INTERPRETING GRAPHS PLOTTING DATA AND INTERPRETING GRAPHS Fundamentals of Graphing One of the most important sets of skills in science and mathematics is the ability to construct graphs and to interpret the information they

More information

Week 1: Functions and Equations

Week 1: Functions and Equations Week 1: Functions and Equations Goals: Review functions Introduce modeling using linear and quadratic functions Solving equations and systems Suggested Textbook Readings: Chapter 2: 2.1-2.2, and Chapter

More information

A Detailed Price Discrimination Example

A Detailed Price Discrimination Example A Detailed Price Discrimination Example Suppose that there are two different types of customers for a monopolist s product. Customers of type 1 have demand curves as follows. These demand curves include

More information

Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20

Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20 Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding

More information

LINEAR EQUATIONS IN TWO VARIABLES

LINEAR EQUATIONS IN TWO VARIABLES 66 MATHEMATICS CHAPTER 4 LINEAR EQUATIONS IN TWO VARIABLES The principal use of the Analytic Art is to bring Mathematical Problems to Equations and to exhibit those Equations in the most simple terms that

More information

Teaching and Learning Guide 3: Linear Equations Further Topics

Teaching and Learning Guide 3: Linear Equations Further Topics Guide 3: Linear Equations Further Topics Table of Contents Section 1: Introduction to the guide...3 Section : Solving simultaneous equations using graphs...4 1. The concept of solving simultaneous equations

More information

The Cost of Production

The Cost of Production The Cost of Production 1. Opportunity Costs 2. Economic Costs versus Accounting Costs 3. All Sorts of Different Kinds of Costs 4. Cost in the Short Run 5. Cost in the Long Run 6. Cost Minimization 7. The

More information

EQUATIONS and INEQUALITIES

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

More information

CHAPTER 1 Linear Equations

CHAPTER 1 Linear Equations CHAPTER 1 Linear Equations 1.1. Lines The rectangular coordinate system is also called the Cartesian plane. It is formed by two real number lines, the horizontal axis or x-axis, and the vertical axis or

More information

CHAPTER 5 MARGINAL UTILITY AND CONSUMER CHOICE

CHAPTER 5 MARGINAL UTILITY AND CONSUMER CHOICE CHAPTER 5 MARGINAL UTILITY AND CONSUMER CHOICE Chapter in a Nutshell In Chapter 3, we studied the law of demand, noting that when price falls, quantity demanded increases. But why? It seemed obvious, didn't

More information

Mathematics (Project Maths Phase 3)

Mathematics (Project Maths Phase 3) 2014. M329 Coimisiún na Scrúduithe Stáit State Examinations Commission Leaving Certificate Examination 2014 Mathematics (Project Maths Phase 3) Paper 1 Higher Level Friday 6 June Afternoon 2:00 4:30 300

More information

Algebra Bridge Project Cell Phone Plans

Algebra Bridge Project Cell Phone Plans Algebra Bridge Project Cell Phone Plans Name Teacher Part I: Two Cell Phone Plans You are in the market for a new cell phone, and you have narrowed your search to two different cell phone companies --

More information

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

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

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving

More information

Notes on indifference curve analysis of the choice between leisure and labor, and the deadweight loss of taxation. Jon Bakija

Notes on indifference curve analysis of the choice between leisure and labor, and the deadweight loss of taxation. Jon Bakija Notes on indifference curve analysis of the choice between leisure and labor, and the deadweight loss of taxation Jon Bakija This example shows how to use a budget constraint and indifference curve diagram

More information

Understanding the Slutsky Decomposition: Substitution & Income Effect

Understanding the Slutsky Decomposition: Substitution & Income Effect Understanding the Slutsky Decomposition: Substitution & Income Effect age 1 lacement of the Final Bundle when p : Substitute or Complement Goods? egion A egion B egion C BC 2 S When p, BC rotates inwards

More information

Numerical Solution of Differential Equations

Numerical Solution of Differential Equations Numerical Solution of Differential Equations Dr. Alvaro Islas Applications of Calculus I Spring 2008 We live in a world in constant change We live in a world in constant change We live in a world in constant

More information

U = x 1 2. 1 x 1 4. 2 x 1 4. What are the equilibrium relative prices of the three goods? traders has members who are best off?

U = x 1 2. 1 x 1 4. 2 x 1 4. What are the equilibrium relative prices of the three goods? traders has members who are best off? Chapter 7 General Equilibrium Exercise 7. Suppose there are 00 traders in a market all of whom behave as price takers. Suppose there are three goods and the traders own initially the following quantities:

More information

Solution: The optimal position for an investor with a coefficient of risk aversion A = 5 in the risky asset is y*:

Solution: The optimal position for an investor with a coefficient of risk aversion A = 5 in the risky asset is y*: Problem 1. Consider a risky asset. Suppose the expected rate of return on the risky asset is 15%, the standard deviation of the asset return is 22%, and the risk-free rate is 6%. What is your optimal position

More information