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


 Trevor Wilkinson
 2 years ago
 Views:
Transcription
1 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 2 Utility Function U x U x 2 MRS(x, x 2 ) MRS(2,3) (i) U(x, x 2 ) = x x 2 x 2 x x 2 x 3 2 (ii) U(x, x 2 ) = x 3 x 5 2 3x 2 x 5 2 5x 3 x 4 2 3x 2 5x 9 0 (iv) U(x, x 2 ) = 3 ln x + 5 ln x 2 3 x 5 x 2 3x 2 5x 9 0 (b) MRS(2, 3) = 9/0 for utility function U(x, x 2 ) = x 3 x 5 2 has the following interpretation: At bundle (2, 3), to reain indifferent about the change (i.e., reain at the sae utility level), a consuer is willing to give up 9/0 of x 2 for one additional unit of x. (Or, after losing one unit of x, he ust receive 9/0 of a unit of x 2 to be as well off as he was at bundle (2, 3).) So at the point (2, 3), good two is ore valuable since he needs to get less of it than he lost of the other good to reain as satisfied. If of good one is taken away, he would have to receive approxiately ( 9 ) = units of good two 0 to reain indifferent to the change. (c) The two utility functions share the sae MRS functions because U(x, x 2 ) = 3 ln x + 5 ln x 2 is a onotonic transforation of U(x, x 2 ) = x 3 x 5 2. To see this, let f(u) = ln(u) (f(u) is a onotonic function). Then letting u = x 3 x 5 2, we have that f(u) = ln(x 3 x 5 2) = 3 ln x + 5 ln x 2. If one function is a onotonic transforation of another, the two describe the sae preferences since they will they rank bundles in the sae way. (They assign different values to the bundle, but we do not use these cardinal nubers in deterining the utilityaxiizing choices we only care about ordinal coparisons.) Proble 2 (WellBehaved Preferences) (a) Instead of using utility function U(x, x 2 ) = x 3 x 2, we can use a onotonic transforation instead: U(x, x 2 ) = 3 ln x + ln x 2. (To get this, let f(u) = ln(u). Then f(u) = ln(x 3 x 2 ) = 3 ln x + ln x 2. Again, even though these are not the sae utility func
2 tions, they ll give the sae MRS and thus the sae results.) Using U(x, x 2 ) = 3 ln x + ln x 2, we get than MU = U x = 3 x and MU 2 = U MRS(x, x 2 ) = MU MU 2 = ( ) U x ( U x 2 ), we have here that MRS(x, x 2 ) = 3x 2 x 2 = x 2. Since x. This is the MRS for any bundle (x, x 2 ), which is also the slope of the indifference curve passing through that point. (b) Using our answer in (a), we get that MRS(, ) = 3 = 3. This tells us that the slope of the indifference curve passing through the point (, ) is 3: CDs, x 2 MRS = 3 DVDs, x At (, ), good one is locally ore valued since, to copensate for a loss of 3 units of good two (the CDs), Alicia only needs unit of good one (the DVDs) to aintain the initial level of happiness. (c) The two secrets of happiness for wellbehaved preferences are: () x + x 2 = (Since ore is preferred to less, spend all of your incoe.) (2) MRS = (Marginal utility per dollar spent is equalized.) Note: An equivalent way of writing this is MU MU 2 = (using the definition of MRS) or MU. All three ways are exactly the sae. = MU 2 Graphically, we re finding the bundle for which the budget line is tangent to an indifference curve: 2
3 CDs, x 2 = = 20 DVDs, x Given that = 40, = 20, and = 800, we can rewrite these two equations as () 40x + 20x 2 = 800 (2) 3x 2 x = = x 2 = 2 3 x (d) To find Alicia s optial bundle, we just use the two equations above to solve for our two unknowns, which are x and x 2. (So there s no econoics here, only Algebra.) You can just take x 2 = 2 3 x fro equation (2) and plug it into () to get 40x + 20( 2 3 x ) = 800 = x = 5. Plug x = 5 into either equation to find that x 2 = 0. Alicia s optial bundle, given these prices and her incoe, is (5, 0), which is interior (she s consuing nonzero aounts of both). This is shown in the figure above. Proble 3 (Perfect Copleents) (a) The indifference curves passing through (5, ), (0, 0), and (5, 4) are shown below. The vertices all fall along the dotted line along which x 2 = 5x. (It shows the cobinations for which Trevor consues five ties as any strawberries (x 2 ) as he does units of ilk (x )). The MRS at each of these points (without using any forulas and only looking at the graph) is zero: MRS(5, ) = MRS(0, 0) = MRS(5, 4) = 0. Strawberries, x 2 x 2 =5x (0, 0) (5, 4) (5, ) Milk, x 3
4 (b) His preferences can be represented by the utility function U(x, x 2 ) = in{5x, x 2 }. In general, if preferences are perfect copleents where a of x ust be consued for every b of x 2, the utility function can be expressed as U(x, x 2 ) = in{ a x, b x 2}, and the line along which all of the vertices of those Lshaped indifference curves lie is a x = b x 2. So using this forula directly U(x, x 2 ) = in{x, 5 x 2} but, ultiplying everything through by 5 (which would be a onotonic transforation!) we get U(x, x 2 ) = in{5x, x 2 }. To find the level of utility associated with the indifference curves passing through (5, ), (0, 0), and (5, 4), we use this utility function to find that: U(5, ) = in{5 5, } = in{25, } = U(0, 0) = in{5 0, 0} = in{50, 0} = 0 U(5, 4) = in{5 5, 4} = in{75, 4} = 4 Strawberries, x 2 x 2 =5x (0, 0) u = 0 (5, ) (5, 4) u =4 u = Milk, x Notice that if you would have used utility function U(x, x 2 ) = in{x, 5 x 2}, you would get: U(5, ) = in{5, 5 } = in{5, 5 } = 5 U(0, 0) = in{0, 0} = in{0, 2} = 2 5 U(5, 4) = in{5, 5 4} = in{5, 4 5 } = 4 5 Either way, we see that (0, 0) is the ost preferred (i.e., gives the highest utility aong the three), followed by (5, 4) and then (5, ). (c) Multiplying our utility function by ten and adding two is equivalent to taking a onotonic transforation f(u) = 0u + 2. If we take our utility U(x, x 2 ) = in{5x, x 2 }, we get U trans (x, x 2 ) = 0 in{5x, x 2 } + 2. Then U(5, ) = 0 in{25, } + 2 = = 2 U(0, 0) = 0 in{50, 0} + 2 = = 02 U(5, 4) = 0 in{75, 4} + 2 = = 42 4
5 (New) Strawberries, x 2 x 2 =2x Again, the indifference curves do not ove and the preference ranking aong the bundles is preserved, we just have the above levels of utility attached to each of the indifference curves. (d) Letting =, =, and = 00, the two secrets of happiness for perfect copleents are () x + x 2 = = x + x 2 = 00 (Trevor spends all of his incoe.) (2) x 2 = 5x (He consues only to optial proportions along the dotted line along which x 2 = 5x.) Note: These types of preferences are not well behaved like Cobb Douglas preferences are, so we use a different second secret of happiness for these preferences. We can no longer use MRS = since the MRS of the indifference curve is not defined at the kink. To find the deand for both ilk (x ) and strawberries (x 2 ) we solve the equations in () and (2): Plug x 2 = 5x into equation () for x 2, so x + (5x ) = 00 = x = 00/6. Plug this into either equation to solve for x 2 and get x 2 = 500/6. This is interior since Trevor is consuing nonzero aounts of both goods (i.e., x > 0 and x 2 > 0). (e) With larger strawberries, the new optial proportion of ilk (x ) and strawberries (x 2 ) is two strawberries for every unit of ilk, or x 2 = 2x. Our indifference curves are the sae shape as they were before, but now the vertices of these Lshaped indifference curves (which will be the optial bundles) lie along x 2 = 2x, as seen below. Milk, x These new preferences can be represented by utility function U(x, x 2 ) = in{2x, x 2 }. 5
6 Proble 4 (Perfect Substitutes) (a) When two goods are perfect substitutes, we know the indifference curves are linear and downwardsloping, in this case having a constant slope of. The indifference curves passing through points (3, 2) and (3, 3) are shown below: Jonagold, x (3, 3) (3, 2) 3 Red Delicious, x (b) Soe utility functions that could represent these functions: (i) U(x, x 2 ) = x + x 2 (ii) U(x, x 2 ) = ln(x + x 2 ) (iii) U(x, x 2 ) = (x + x 2 ) 2 (iv) U(x, x 2 ) = 8 x + 8 x 2 (v) U(x, x 2 ) = 6x + 6x 2 Each of these five utility functions represents a onotonic transforation of any of the others; they all represent the sae underlying perfectsubstitute preferences over Red Delicious (x ) and Jonagold (x 2 ) apples and give indifference curves having the sae MRS = (you can verify this). (c) MRS(x, x 2 ) = for any (x, x 2 ). This eans that starting fro any (x, x 2 ) bundle, Kate is always willing to give up one Red Delicious (x ) apple to get one additional Jonagold (x 2 ) (or vice versa). This eans, as noted above, that the indifferences curves will be linear everywhere. (d) Letting = 2, =, and = 00 we can first turn to the coodity space to see how to go about funding the optial bundle. We know that the first secret of happiness (spending all of one s incoe) will always hold, so the optial choice is along the budget line. But the budget line is not tangent to any indifference curve here: The budget line has a slope of = 2 and we just deterined that MRS(x, x 2 ) = everywhere, so 6
7 MRS anywhere. Jonagold, x 2 = 00 (0, 00) = 50 Red Delicious, x We can see that at the indifference curve furthest out fro the origin (the one with highest utility) but still touching the budget line, Kate is consuing only Jonagolds (x 2 ) and no Red Delicious (x ): x = 0, x 2 = = 00 = 00. This is not an interior solution (since x = 0) but rather is a corner solution. As a general rule, when MRS < (the indifference curves are ore flat than the budget line), the consuer chooses to consue only x 2. Here, MRS = < = 2. This akes sense intuitively here: If the two types of apples are : perfect substitutes for Kate, she should just consue the one that is cheaper. (e) Now we have that MRS = > = and Kate consues only the cheaper apple, the 2 Red Delicious (x ), with x = = 00 = 00, x 2 = 0. Jonagold, x 2 = 50 (00, 0) = 00 Red Delicious, x (f) With this price change, MRS = everywhere, so all of the cobinations along the budget line lie along the sae indifference curve, and thus any bundle is equally as good as 7
8 any other bundle on the budget line. Kate can choose any (x, x 2 ) cobination such that x + x 2 = or x + x 2 = 00: Jonagold, x 2 = 00 all optial bundles along here = 00 Red Delicious, x 8
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 informationDemand  Basic Review and Examples 1
Deand  Basic Review and Exaples The Basic Deand For What exactly is a deand function? In Varian s words, a consuer s deand functions give the optial aounts of each of the goods as a function of the prices
More informationChapter 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(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?
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
More informationANSWER KEY 1 BUDGETS
ANSWER KEY 1 BUDGETS W & L INTERMEDIATE MICROECONOMICS PROFESSOR A. JOSEPH GUSE (1) Draw the budget set for the following paraeters. =, p beer = 1, p pizza =. The units for the two goods are pints and
More informationEconomics 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 informationc 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 informationFrom preferences to numbers Cardinal v ordinal Examples MRS. Utility. Intermediate Micro. Lecture 4. Chapter 4 of Varian
Utility Intermediate Micro Lecture 4 Chapter 4 of Varian Preferences and decisionmaking 1. Last lecture: Ranking consumption bundles by preference/indifference 2. Today: Assigning values (numbers) to
More information1 Deriving demand function
Econoics II: Micro Fall 2009 Eercise session 2 VŠE Deriving deand function Assue that consuer s utility function is of CobbDouglass for: U (; y) = y () To solve the consuer s otiisation roble it is necessary
More information1. 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 informationREVIEW 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 informationANSWER 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 informationUtility. 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 informationDemand. 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 informationProblem Set #5Key. Economics 305Intermediate Microeconomic Theory
Problem Set #5Key Sonoma State University Economics 305Intermediate Microeconomic Theory Dr Cuellar (1) Suppose that you are paying your for your own education and that your college tuition is $200 per
More informationCalculating the Return on Investment (ROI) for DMSMS Management. The Problem with Cost Avoidance
Calculating the Return on nvestent () for DMSMS Manageent Peter Sandborn CALCE, Departent of Mechanical Engineering (31) 453167 sandborn@calce.ud.edu www.ene.ud.edu/escml/obsolescence.ht October 28, 21
More informationChapter 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 informationDeriving 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 informationCV and EV Examples 1
CV and EV Examples 1 Before we do any examples, let s make sure we have CV and EV straight. They re similar, but not the exact same thing, and which is which can get confusing. Both deal with changes
More informationUNIT 2: CONSUMER EQUILIBRIUM AND DEMAND
KEY CONCEPTS UNIT 2: CONSUMER EQUILIBRIUM AND DEMAND 1. UTILITY A) MARGINAL UTILITY B) LAW OF DIMINISHING MARGINAL UTILITY 2. CONDITIONS OF CONSUMER S EQUILIBRIUM 3. INDIFFERENCE CURVE ANALYSIS 4. THE
More informationIV Approximation of Rational Functions 1. IV.C Bounding (Rational) Functions on Intervals... 4
Contents IV Approxiation of Rational Functions 1 IV.A Constant Approxiation..................................... 1 IV.B Linear Approxiation....................................... 3 IV.C Bounding (Rational)
More informationEcon 101: Principles of Microeconomics Fall 2012
Problem 1: Use the following graph to answer the questions. a. From the graph, which good has the price change? Did the price go down or up? What is the fraction of the new price relative to the original
More informationIndifference Curves and the Marginal Rate of Substitution
Introduction Introduction to Microeconomics Indifference Curves and the Marginal Rate of Substitution In microeconomics we study the decisions and allocative outcomes of firms, consumers, households and
More informationLecture CT2: Utility Function
Econ Urban Economics Lecture CT: Utility Function Instructor: Hiroki Watanabe Fall Watanabe Econ CT Utility Function / Introduction Utility Function Indifference Curves Examples Trinity Now We Know Watanabe
More informationEconomics 2020a / HBS 4010 / HKS API111 FALL 2010 Solutions to Practice Problems for Lectures 1 to 4
Economics 00a / HBS 4010 / HKS API111 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 information1 Homogenous and Homothetic Functions
1 Homogenous and Homothetic Functions Reading: [Simon], Chapter 20, p. 483504. 1.1 Homogenous Functions Definition 1 A real valued function f(x 1,..., x n ) is homogenous of degree k if for all t > 0
More informationEconomics 326 (Utility, Marginal Utility, MRS, Substitutes and Complements ) Ethan Kaplan
Economics 326 (Utility, Marginal Utility, MRS, Substitutes and Complements ) Ethan Kaplan September 10, 2012 1 Utility From last lecture: a utility function U (x; y) is said to represent preferences if
More informationCHAPTER 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 informationA 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 information5.7 Chebyshev Multisection Matching Transformer
/9/ 5_7 Chebyshev Multisection Matching Transforers / 5.7 Chebyshev Multisection Matching Transforer Reading Assignent: pp. 555 We can also build a ultisection atching network such that Γ f is a Chebyshev
More informationConstruction Economics & Finance. Module 3 Lecture1
Depreciation: Construction Econoics & Finance Module 3 Lecture It represents the reduction in arket value of an asset due to age, wear and tear and obsolescence. The physical deterioration of the asset
More informationAnalysis of the purchase option of computers
Analysis of the of coputers N. Ahituv and I. Borovits Faculty of Manageent, The Leon Recanati Graduate School of Business Adinistration, TelAviv University, University Capus, RaatAviv, TelAviv, Israel
More informationCONSUMER 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 information3. Solve the equation containing only one variable for that variable.
Question : How do you solve a system of linear equations? There are two basic strategies for solving a system of two linear equations and two variables. In each strategy, one of the variables is eliminated
More informationSection 7.1 Solving Linear Systems by Graphing. System of Linear Equations: Two or more equations in the same variables, also called a.
Algebra 1 Chapter 7 Notes Name Section 7.1 Solving Linear Systems by Graphing System of Linear Equations: Two or more equations in the same variables, also called a. Solution of a System of Linear Equations:
More informationPlane Trusses. Section 7: Flexibility Method  Trusses. A plane truss is defined as a twodimensional
lane Trusses A plane truss is defined as a twodiensional fraework of straight prisatic ebers connected at their ends by frictionless hinged joints, and subjected to loads and reactions that act only at
More informationA 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 informationEconomics 101 Fall 2013 Answers to Homework 5 Due Tuesday, November 19, 2013
Economics 101 Fall 2013 Answers to Homework 5 Due Tuesday, November 19, 2013 Directions: The homework will be collected in a box before the lecture. Please place your name, TA name and section number on
More informationMathematical Economics: Lecture 15
Mathematical Economics: Lecture 15 Yu Ren WISE, Xiamen University November 19, 2012 Outline 1 Chapter 20: Homogeneous and Homothetic Functions New Section Chapter 20: Homogeneous and Homothetic Functions
More information3 The Utility Maximization Problem
3 The Utility Mxiiztion Proble We hve now discussed how to describe preferences in ters of utility functions nd how to forulte siple budget sets. The rtionl choice ssuption, tht consuers pick the best
More informationUC 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 3.13.4) Standard economics vs. behavioral economics Lectures 12 Aug. 15, 2009 Prologue
More informationExample: Suppose that we deposit $1000 in a bank account offering 3% interest, compounded monthly. How will our money grow?
Finance 111 Finance We have to work with oney every day. While balancing your checkbook or calculating your onthly expenditures on espresso requires only arithetic, when we start saving, planning for retireent,
More informationSemiinvariants IMOTC 2013 Simple semiinvariants
Seiinvariants IMOTC 2013 Siple seiinvariants These are soe notes (written by Tejaswi Navilarekallu) used at the International Matheatical Olypiad Training Cap (IMOTC) 2013 held in Mubai during AprilMay,
More informationAnalyzing Functions Intervals of Increase & Decrease Lesson 76
(A) Lesson Objectives a. Understand what is meant by the terms increasing/decreasing as it relates to functions b. Use graphic and algebraic methods to determine intervals of increase/decrease c. Apply
More informationservings of fries, income is exhausted and MU per dollar spent is the same for both goods, so this is the equilibrium.
Problem Set # 5 Unless told otherwise, assume that individuals think that more of any good is better (that is, marginal utility is positive). Also assume that indifference curves have their normal shape,
More information4.3 The Graph of a Rational Function
4.3 The Graph of a Rational Function Section 4.3 Notes Page EXAMPLE: Find the intercepts, asyptotes, and graph of + y =. 9 First we will find the intercept by setting the top equal to zero: + = 0 so =
More informationLecture L9  Linear Impulse and Momentum. Collisions
J. Peraire, S. Widnall 16.07 Dynaics Fall 009 Version.0 Lecture L9  Linear Ipulse and Moentu. Collisions In this lecture, we will consider the equations that result fro integrating Newton s second law,
More informationThe 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 informationConsumer 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 informationElectric Forces between Charged Plates
CP.1 Goals of this lab Electric Forces between Charged Plates Overview deterine the force between charged parallel plates easure the perittivity of the vacuu (ε 0 ) In this experient you will easure the
More information( C) CLASS 10. TEMPERATURE AND ATOMS
CLASS 10. EMPERAURE AND AOMS 10.1. INRODUCION Boyle s understanding of the pressurevolue relationship for gases occurred in the late 1600 s. he relationships between volue and teperature, and between
More informationEconomic 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
More informationChapter 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 informationSlutsky 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 informationAdvanced Microeconomics (ES30025)
Advanced Microeconoics (ES3005) Advanced Microeconoics (ES3005) Matheatics Review : The Lagrange Multiplier Outline: I. Introduction II. Duality Theory: Co Douglas Exaple III. Final Coents I. Introduction
More informationA Gas Law And Absolute Zero Lab 11
HB 040605 A Gas Law And Absolute Zero Lab 11 1 A Gas Law And Absolute Zero Lab 11 Equipent safety goggles, SWS, gas bulb with pressure gauge, 10 C to +110 C theroeter, 100 C to +50 C theroeter. Caution
More information5.1. Systems of Linear Equations. Linear Systems Substitution Method Elimination Method Special Systems
5.1 Systems of Linear Equations Linear Systems Substitution Method Elimination Method Special Systems 5.11 Linear Systems The possible graphs of a linear system in two unknowns are as follows. 1. The
More informationUse of extrapolation to forecast the working capital in the mechanical engineering companies
ECONTECHMOD. AN INTERNATIONAL QUARTERLY JOURNAL 2014. Vol. 1. No. 1. 23 28 Use of extrapolation to forecast the working capital in the echanical engineering copanies A. Cherep, Y. Shvets Departent of finance
More information2.1 Systems of Linear Equations
. Systems of Linear Equations Question : What is a system of linear equations? Question : Where do systems of equations come from? In Chapter, we looked at several applications of linear functions. One
More informationSOME APPLICATIONS OF FORECASTING Prof. Thomas B. Fomby Department of Economics Southern Methodist University May 2008
SOME APPLCATONS OF FORECASTNG Prof. Thoas B. Foby Departent of Econoics Southern Methodist University May 8 To deonstrate the usefulness of forecasting ethods this note discusses four applications of forecasting
More informationa. Meaning: The amount (as a percentage of total) that quantity demanded changes as price changes. b. Factors that make demand more price elastic
Things to know about elasticity. 1. Price elasticity of demand a. Meaning: The amount (as a percentage of total) that quantity demanded changes as price changes. b. Factors that make demand more price
More information2. Efficiency and Perfect Competition
General Equilibrium Theory 1 Overview 1. General Equilibrium Analysis I Partial Equilibrium Bias 2. Efficiency and Perfect Competition 3. General Equilibrium Analysis II The Efficiency if Competition The
More informationSAMPLING METHODS LEARNING OBJECTIVES
6 SAMPLING METHODS 6 Using Statistics 66 2 Nonprobability Sapling and Bias 66 Stratified Rando Sapling 62 6 4 Cluster Sapling 64 6 5 Systeatic Sapling 69 6 6 Nonresponse 62 6 7 Suary and Review of
More informationLesson 13: Voltage in a Uniform Field
Lesson 13: Voltage in a Unifor Field Most of the tie if we are doing experients with electric fields, we use parallel plates to ensure that the field is unifor (the sae everywhere). This carries over to
More informationFixedIncome Securities and Interest Rates
Chapter 2 FixedIncoe Securities and Interest Rates We now begin a systeatic study of fixedincoe securities and interest rates. The literal definition of a fixedincoe security is a financial instruent
More informationProcedure In each case, draw and extend the given series to the fifth generation, then complete the following tasks:
Math IV Nonlinear Algebra 1.2 Growth & Decay Investigation 1.2 B: Nonlinear Growth Introduction The previous investigation introduced you to a pattern of nonlinear growth, as found in the areas of a series
More informationWork, Energy, Conservation of Energy
This test covers Work, echanical energy, kinetic energy, potential energy (gravitational and elastic), Hooke s Law, Conservation of Energy, heat energy, conservative and nonconservative forces, with soe
More informationThe Velocities of Gas Molecules
he Velocities of Gas Molecules by Flick Colean Departent of Cheistry Wellesley College Wellesley MA 8 Copyright Flick Colean 996 All rights reserved You are welcoe to use this docuent in your own classes
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n1 x n1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of
More informationDefinition 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
More informationChapter 7. Costs. C = FC + VC Marginal cost MC = C/ q Note that FC will not change, so marginal cost also means marginal variable cost.
Chapter 7. Costs Shortrun costs Longrun costs Lowering costs in the longrun 0. Economic cost and accounting cost Opportunity cost : the highest value of other alternative activities forgone. To determine
More informationEconomics Homework 3 Fall 2006 Stacy DickertConlin
Economics 0  Homework Fall 00 Stacy DickertConlin nswer Key. ndy collects baseball and football cards. The following graph shows a few of his indifference curves. The price of a pack of baseball cards
More informationA Gas Law And Absolute Zero
A Gas Law And Absolute Zero Equipent safety goggles, DataStudio, gas bulb with pressure gauge, 10 C to +110 C theroeter, 100 C to +50 C theroeter. Caution This experient deals with aterials that are very
More informationImperfect Competition: Monopoly
Ierfect Coetition: Monooly New Toic: Monooly Q: What is a onooly? A onooly is a fir that faces a downward sloing deand, and has a choice about what rice to charge an increase in rice doesn t send ost or
More informationHW 2. Q v. kt Step 1: Calculate N using one of two equivalent methods. Problem 4.2. a. To Find:
HW 2 Proble 4.2 a. To Find: Nuber of vacancies per cubic eter at a given teperature. b. Given: T 850 degrees C 1123 K Q v 1.08 ev/ato Density of Fe ( ρ ) 7.65 g/cc Fe toic weight of iron ( c. ssuptions:
More informationEconomics 165 Winter 2002 Problem Set #2
Economics 165 Winter 2002 Problem Set #2 Problem 1: Consider the monopolistic competition model. Say we are looking at sailboat producers. Each producer has fixed costs of 10 million and marginal costs
More informationDesign of Model Reference Self Tuning Mechanism for PID like Fuzzy Controller
Research Article International Journal of Current Engineering and Technology EISSN 77 46, PISSN 347 56 4 INPRESSCO, All Rights Reserved Available at http://inpressco.co/category/ijcet Design of Model Reference
More informationSection 1. Inequalities 54 32 1 0 1 2 3 4 5
Worksheet 2.4 Introduction to Inequalities Section 1 Inequalities The sign < stands for less than. It was introduced so that we could write in shorthand things like 3 is less than 5. This becomes 3 < 5.
More informationECO364  International Trade
ECO364  International Trade Chapter 2  Ricardo Christian Dippel University of Toronto Summer 2009 Christian Dippel (University of Toronto) ECO364  International Trade Summer 2009 1 / 73 : The Ricardian
More informationMath 018 Review Sheet v.3
Math 018 Review Sheet v.3 Tyrone Crisp Spring 007 1.1  Slopes and Equations of Lines Slopes: Find slopes of lines using the slope formula m y y 1 x x 1. Positive slope the line slopes up to the right.
More informationNotes 10: An Equation Based Model of the Macroeconomy
Notes 10: An Equation Based Model of the Macroeconomy In this note, I am going to provide a simple mathematical framework for 8 of the 9 major curves in our class (excluding only the labor supply curve).
More informationFigure 4.1 Average Hours Worked per Person in the United States
The Supply of Labor Figure 4.1 Average Hours Worked per Person in the United States 1 Table 4.1 Change in Hours Worked by Age: 1950 2000 4.1: Preferences 4.2: The Constraints 4.3: Optimal Choice I: Determination
More informationPart 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
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationChoices. Preferences. Indifference Curves. Preference Relations. ECON 370: Microeconomic Theory Summer 2004 Rice University Stanley Gilbert
Choices Preferences ECON 370: Microeconomic Theor Summer 2004 Rice Universit Stanle Gilbert The theor of consumer preferences is based fundamentall on choices The steak dinner or the salad bar Major in
More informationDifferent Types of Tastes
Chapter 5 Different Types of Tastes In Chapter 4 we demonstrated how tastes can be represented by maps of indifference curves and how 5 basic assumptions about tastes result in particular features of these
More informationChapter 2 Section 4: Equations of Lines. 4.* Find the equation of the line with slope 4 3, and passing through the point (0,2).
Chapter Section : Equations of Lines Answers to Problems For problems , put our answers into slope intercept form..* Find the equation of the line with slope, and passing through the point (,0).. Find
More informationCooperative Caching for Adaptive Bit Rate Streaming in Content Delivery Networks
Cooperative Caching for Adaptive Bit Rate Streaing in Content Delivery Networs Phuong Luu Vo Departent of Coputer Science and Engineering, International University  VNUHCM, Vietna vtlphuong@hciu.edu.vn
More informationThe United States was in the midst of a
A Prier on the Mortgage Market and Mortgage Finance Daniel J. McDonald and Daniel L. Thornton This article is a prier on ortgage finance. It discusses the basics of the ortgage arket and ortgage finance.
More informationChulalongkorn University: BBA International Program, Faculty of Commerce and Accountancy. Solution to Selected Questions: CHAPTER 3 CONSUMER BEHAVIOR
Chulalongkorn University: BBA International Program, Faculty of Commerce and Accountancy 2900111 (Section 1) Chairat Aemkulwat Economics I: Microeconomics Spring 2015 Solution to Selected Questions: CHAPTER
More informationManagerial 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 informationLesson 44: Acceleration, Velocity, and Period in SHM
Lesson 44: Acceleration, Velocity, and Period in SHM Since there is a restoring force acting on objects in SHM it akes sense that the object will accelerate. In Physics 20 you are only required to explain
More informationAlgebra Chapter 6 Notes Systems of Equations and Inequalities. Lesson 6.1 Solve Linear Systems by Graphing System of linear equations:
Algebra Chapter 6 Notes Systems of Equations and Inequalities Lesson 6.1 Solve Linear Systems by Graphing System of linear equations: Solution of a system of linear equations: Consistent independent system:
More informationEvaluating Inventory Management Performance: a Preliminary DeskSimulation Study Based on IOC Model
Evaluating Inventory Manageent Perforance: a Preliinary DeskSiulation Study Based on IOC Model Flora Bernardel, Roberto Panizzolo, and Davide Martinazzo Abstract The focus of this study is on preliinary
More informationNotes 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 informationELECTRIC ENERGY ECONOMICS
ELECTRIC ENERGY ECONOMICS The cost to the consuer for electric energy can be roughly apportioned according to Generation 60% Transission 0% Distribution 20% Adinistrative/Profit 0% Econoics drives the
More informationPrice Elasticity of Supply; Consumer Preferences
1 Price Elasticity of Supply 1 14.01 Principles of Microeconomics, Fall 2007 ChiaHui Chen September 12, 2007 Lecture 4 Price Elasticity of Supply; Consumer Preferences Outline 1. Chap 2: Elasticity 
More information4.1 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
More informationIntroductory 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 informationGeneral Equilibrium. The Producers
General Equilibrium In this section we will combine production possibilities frontiers and community indifference curves in order to create a model of the economy as a whole. This brings together producers,
More information