Economic Principles Solutions to Problem Set 1


 Moris York
 2 years ago
 Views:
Transcription
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 < ) strict quasiconcavit of u ). For x; 2 R n +;suppose < x and 6 x. Strict convexit of < implies that + (1 ) x x for all 2 (0; 1): Since u represents <, this means that u( + (1 Hence, u is strictl quasiconcave. ) x) > u(x) minfu(x); u()g: Onl if part: ( strict quasiconcavit of u ) strict convexit of < ). Suppose z < x; < x; with z 6 : We have to show that + (1 ) z x for all 2 (0; 1): Without loss of generalit, suppose z < ; i.e., u (z) u () : Strict quasiconcavit of u implies Hence, < is strictl convex. u( + (1 )z) > u () u (x) : Question 2. To prove that the two functions have the same indi erence curves, pick an arbitrar bundle ( ; x 2 ). For utilit function u(x) p x 2 the bundles indi erent to ( ; x 2 ) satisf the equalit p x 2 p 1 2 which b appling logarithms to both sides is equivalent to log + log x 2 log 1 + log 2 ; but this is the same indi erence condition we obtain if we use the utilit function v(x) log + log x 2. We conclude that the two utilit functions give the same set of bundles indi erent to x. Since x was arbitrar, the two utilit functions have the same indi erence curves. Since u 0 was chosen arbitraril, u and v have the same indi erence curves. For u; MRS MU 1 MU 2 p x2 2 p p x1 2 p x 2 x 2
2 For v; MRS MU x 2 x 2 u and v have the same indi erence curves and the same MRS because each utilit function is a strictl increasing transformation of the other. Speci call, v(:) 2 ln(u(:)): Question 3. Graph an indi erence curve, and compute the M RS and the Marshallian demand functions for the following utilit functions: a) Perfect substitutes: u ( ; x 2 ) + x 2 ; where > 0; > 0; x2 x1 Perfect Substitutes (slope ). MRS MU MU x2 It is possible to solve the problem graphicall. Here we do a little more algebraic solution. The problem we want to solve is max x1 ;x 2 + x 2 s.t. + x 2 0; x 2 0 The strateg we can use is to solve for x 2 in the budget constraint and substitute it in the objective function, turning it in a maximization problem in one variable. We need some care though, since we need to remember the nonnegativit constraints. From the budget constraint we can write: x 2 2
3 However we need to remember that x 2 0, that is must be contained in the interval [0; ]. So our original maximization problem is equivalent to the following: max ( ) + s.t. 2 [0; _ ] This is a ver eas maximization problem, since we are maximazing a straight line over an interval. The slope of the straight line is ( ): Therefore, if the slope is strictl positive (negative), the straight line is strictl increasing (decreasing), so the point of maximum is at the right (left) endpoint of the interval. If the objective function is constant (the slope is zero), the consumer will be indi erent tamong all the values in the interval. We can summarize these observations in our Marshallian demand function: (p; ) x 2 (p; ) 8 < : 8 < : 0 if < [0; ] if if > 0 if > (p; ) if if < b) The ke to solve this problem is the following observation. Suppose the solution satis es the budget constraint and has 6 x 2 Without loss of generalit suppose > x 2. The utilit of this bundle is x 2. This utilit can be increased if we decrease the consumption of b a small amount " > 0, use the mone we are saving ( ") to bu a little more of x 2 ( ") : The new bundle ( ",x 2 + ") still satis es the budget constraint with equalit, and we can choose " small enough so that it satis es the inequalit ( ") > (x 2 + "). The utilit of our new bundle is therefore (x 2 + ") > x 2 contradicting the hpothesis that our original bundle ( ; x 2 ) was optimal. We conclude that the solution to the maximization problem with goods that are perfect complements must satisf: a x 2. We can then solve the following sistem: a x 2 + x 2 Hence, the Marshallian demand functions are: ( ; ; ) x 2 ( ; ; ) + + An indi erence curve is shown in red in the following graph: 3
4 x2 x1 Perfect complements MRS 1 when x 2 > ; MRS 0 when x 2 < ; and MRS is not well de ned when x 2 : Question 4. (JR 1.21). We have noted that u (x) is invariant to positive monotonic transformation. One common transformation is the logarithmic transform, ln (u (x)) : Take the logarithmic transform of the CobbDouglas utilit function; then using that as the utilit function, derive the Marshallian demand functions and verif that the are identical to those derived in class. CobbDouglas utilit function: Taking the logarithmic transformation, u( ; x 2 ) x 1 x 2 v( ; x 2 ) ln(u( ; x 2 )) ln + ln x 2 To nd the Marshallian demand functions, we solve the problem: The Lagrangian for this problem is: The F.O.C. are: max ln + ln x 2 s.t. + x 2 L ln + ln x 2 + ( x 2 ) x 2 + x 2 4
5 Taking the ratio of the rst two equations gives: x 2 Together with the budget constraint, we can solve for the optimal choice of and x 2. ( ) ) Substitute this into the budget constraint, we get x Hence, the Marshallian demand functions are the same as those we derived in class. Question 5. (JR 1.27). A consumer of two goods faces positive prices and has a positive income. Her utilit function is u ( ; x 2 ) max fa ; ax 2 g + min f ; x 2 g ; where 0 < a < 1: Derive the Marshallian demand functions. We can express the utilit function in the following wa: u( ; x 2 ) maxfa ; ax 2 g + minf ; x 2 g ax1 + x 2 if x 2 + ax 2 if x 2 Graphicall, an indi erence curve looks like this: x2 Indi erence Curve. Let s solve the maximization problem. We can distinguish several cases, depending on the relationship between the price ratio (i.e. the slope of the budget constraint) and the MRS (i.e. the slope of indi erence curves). 5 x1
6 < a < 1 a : In this case the budget constraint is atter than both the MRS above and below the 45degree line. So the consumer will bu onl good 1. (p; ) x 2 (p; ) 0 a < 1 a. In this case the consumer is indi erent among all the bundles on the budget set and below the 45degree line. (p; ) 2 [ + ; ] x 2 (p; ) (p; ) a < < 1 a. In this case the slope of the budget constraint is between the two MRS and the maximal point is at the kink, that is where x 2 B solving keeping in mind that the solution must satisf the budget constraint we get (; ; ) + x 2 (; ; ) + a < 1 a. In this case the consumer is indi erent between all the bundles that are on the budget line and above the 45degree line. (p; ) 2 [0; + ] x 2 (p; ) (p; ) a < 1 a < In this case the budget constraint is steeper than both MRS, therefore the consumer will consume onl good 2. (p; ) 0 x 2 (p; ) Question 6 Consider the following monotonic transformation of the u(:): v(:) (u (:)) 2 + 2x 2 + 3x 3 : The three goods are perfect substitutes for each other and the consumer will choose the good that gives the highest MU i p i. 6
7 The Marshallian demand functions are as follows: if < minf 2 ; p 3 if 2 < minf; p 3 if p 3 < minfp 3 1; if g; x 3 1(p; ) ; x 2 (p; ) 0; x 3 (p; ) 0; g; x 3 1(p; ) 0; x 2 (p; ) ; x 3 (p; ) 0; g; x 2 1(p; ) 0; x 2 (p; ) 0; x 3 (p; ) p 3 ; < p 3 ; x 2 3 3(p; ) 0; (p; ) 0, x 2 (p; ) and (p; ) + x 2 (p; ) ; if p 3 < ; x 3 2 2(p; ) 0; (p; ) 0, x 3 (p; ) and (p; ) + p 3 x 3 (p; ) ; if p 3 < 3 1; (p; ) 0; x 2 (p; ) 0, x 3 (p; ) and x 2 (p; ) + p 3 x 3 (p; ) ; if p 3 ; x 2 3 1(p; ) 0; x 2 (p; ) 0, x 3 (p; ) and (p; ) + x 2 (p; ) + p 3 x 3 (p; ) : When 2, 3, p 3 5, we have < minfp 2 1 ; p 3 g and therefore x 3 1(p; ) 0; x 2 (p; ) ; x 3 3(p; ) 0. To achieve utilit level 6, she needs income such that r ) 54 7
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 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 informationEconomics 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
More informationAdvanced Microeconomics
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
More informationDeriving MRS from Utility Function, Budget Constraints, and Interior Solution of Optimization
Utilit Function, Deriving MRS. Principles of Microeconomics, Fall ChiaHui Chen September, Lecture Deriving MRS from Utilit Function, Budget Constraints, and Interior Solution of Optimization Outline.
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 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 informationEconomics 326: Marshallian Demand and Comparative Statics. Ethan Kaplan
Economics 326: Marshallian Demand and Comparative Statics Ethan Kaplan September 17, 2012 Outline 1. Utility Maximization: General Formulation 2. Marshallian Demand 3. Homogeneity of Degree Zero of Marshallian
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 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 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 informationEconomics 401. Sample questions 1
Economics 40 Sample questions. Do each of the following choice structures satisfy WARP? (a) X = {a, b, c},b = {B, B 2, B 3 }, B = {a, b}, B 2 = {b, c}, B 3 = {c, a}, C (B ) = {a}, C (B 2 ) = {b}, C (B
More informationSolving x < a. Section 4.4 Absolute Value Inequalities 391
Section 4.4 Absolute Value Inequalities 391 4.4 Absolute Value Inequalities In the last section, we solved absolute value equations. In this section, we turn our attention to inequalities involving absolute
More informationThe Graph of a Linear Equation
4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that
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 informationConstrained Optimization: The Method of Lagrange Multipliers:
Constrained Optimization: The Method of Lagrange Multipliers: Suppose the equation p(x,) x 60x 7 00 models profit when x represents the number of handmade chairs and is the number of handmade rockers produced
More informationChapter 3: Section 32 Graphing Linear Inequalities
Chapter : Section Graphing Linear Inequalities D. S. Malik Creighton Universit, Omaha, NE D. S. Malik Creighton Universit, Omaha, NE Chapter () : Section Graphing Linear Inequalities / 9 Geometric Approach
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 informationEcon 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 informationName. Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions.
Name Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions. Question 1. A firm has a production function F (x 1, x 2 ) = ( x 1 + x 2 ) 2. It is a price
More informationSolve the linear programming problem graphically: Minimize w 4. subject to. on the vertical axis.
Do a similar example with checks along the wa to insure student can find each corner point, fill out the table, and pick the optimal value. Example 3 Solve the Linear Programming Problem Graphicall Solve
More information1 Maximizing pro ts when marginal costs are increasing
BEE12 Basic Mathematical Economics Week 1, Lecture Tuesda 12.1. Pro t maimization 1 Maimizing pro ts when marginal costs are increasing We consider in this section a rm in a perfectl competitive market
More informationLecture 4: Equality Constrained Optimization. Tianxi Wang
Lecture 4: Equality Constrained Optimization Tianxi Wang wangt@essex.ac.uk 2.1 Lagrange Multiplier Technique (a) Classical Programming max f(x 1, x 2,..., x n ) objective function where x 1, x 2,..., x
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 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 informationMathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions
Natalia Lazzati Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions Note 5 is based on Madden (1986, Ch. 1, 2, 4 and 7) and Simon and Blume (1994, Ch. 13 and 21). Concave functions
More informationConstrained 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 informationChapter 2 TWOPERIOD INTERTEMPORAL DECISIONS
Chapter 2 TWOPERIOD INTERTEMPORAL DECISIONS The decisions on consumption and savings are at the heart of modern macroeconomics. This decision is about the tradeo between current consumption and future
More informationLecture 2: Consumer Theory
Lecture 2: Consumer Theory Preferences and Utility Utility Maximization (the primal problem) Expenditure Minimization (the dual) First we explore how consumers preferences give rise to a utility fct which
More information36 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
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 informationReadings. D Chapter 1. Lecture 2: Constrained Optimization. Cecilia Fieler. Example: Input Demand Functions. Consumer Problem
Economics 245 January 17, 2012 : Example Readings D Chapter 1 : Example The FOCs are max p ( x 1 + x 2 ) w 1 x 1 w 2 x 2. x 1,x 2 0 p 2 x i w i = 0 for i = 1, 2. These are two equations in two unknowns,
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 informationThe Real Business Cycle Model
The Real Business Cycle Model Ester Faia Goethe University Frankfurt Nov 2015 Ester Faia (Goethe University Frankfurt) RBC Nov 2015 1 / 27 Introduction The RBC model explains the comovements in the uctuations
More informationUtility 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 informationIntroduction. Agents have preferences over the two goods which are determined by a utility function. Speci cally, type 1 agents utility is given by
Introduction General equilibrium analysis looks at how multiple markets come into equilibrium simultaneously. With many markets, equilibrium analysis must take explicit account of the fact that changes
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 information2.6. The Circle. Introduction. Prerequisites. Learning Outcomes
The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures and has been around a long time! In this brief Section we discuss the basic coordinate geometr of a circle  in particular
More informationEcon306 Intermediate Microeconomics Fall 2007 Midterm exam Solutions
Econ306 Intermediate Microeconomics Fall 007 Midterm exam Solutions Question ( points) Perry lives on avocado and beans. The price of avocados is $0, the price of beans is $, and his income is $0. (i)
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 informationPractice with Proofs
Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using
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 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 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 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 informationReview 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 decisionmaking tools
More informationCommon 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 information1. Then f has a relative maximum at x = c if f(c) f(x) for all values of x in some
Section 3.1: First Derivative Test Definition. Let f be a function with domain D. 1. Then f has a relative maximum at x = c if f(c) f(x) for all values of x in some open interval containing c. The number
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 information2.6. The Circle. Introduction. Prerequisites. Learning Outcomes
The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures. In this brief Section we discuss the basic coordinate geometr of a circle  in particular the basic equation representing
More informationRepresentation 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
More informationLESSON EIII.E EXPONENTS AND LOGARITHMS
LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential
More informationReasoning with Equations and Inequalities
Instruction Goal: To provide opportunities for students to develop concepts and skills related to solving sstems of linear inequalities, including realworld problems through graphing two and three variables
More informationReview for Calculus Rational Functions, Logarithms & Exponentials
Definition and Domain of Rational Functions A rational function is defined as the quotient of two polynomial functions. F(x) = P(x) / Q(x) The domain of F is the set of all real numbers except those for
More informationPreferences. 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 information2.4 Inequalities with Absolute Value and Quadratic Functions
08 Linear and Quadratic Functions. Inequalities with Absolute Value and Quadratic Functions In this section, not onl do we develop techniques for solving various classes of inequalities analticall, we
More information8.7 Systems of NonLinear Equations and Inequalities
8.7 Sstems of NonLinear Equations and Inequalities 67 8.7 Sstems of NonLinear Equations and Inequalities In this section, we stud sstems of nonlinear equations and inequalities. Unlike the sstems of
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 informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
More informationFirst Welfare Theorem
First Welfare Theorem Econ 2100 Fall 2015 Lecture 17, November 2 Outline 1 First Welfare Theorem 2 Preliminaries to Second Welfare Theorem Last Class Definitions A feasible allocation (x, y) is Pareto
More informationExponential and Logarithmic Functions
Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,
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 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 informationUsing the data above Height range: 6 1to 74 inches Weight range: 95 to 205
Plotting When plotting data, ou will normall be using two numbers, one for the coordinate, another for the coordinate. In some cases, like the first assignment, ou ma have onl one value. There, the second
More informationECONOMICS 101 WINTER 2004 TA LECTURE 5: GENERAL EQUILIBRIUM
ECONOMICS 0 WINTER 2004 TA LECTURE 5: GENERAL EQUILIBRIUM KENNETH R. AHERN. Production Possibility Frontier We now move from partial equilibrium, where we examined only one market at a time, to general
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 informationExamples 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
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 informationPreferences. Preferences. Budget Constraint and Line 9/10/2014
9/0/04 references Chapter 4 At this point, we know a lot about preferences and their representation with utilit functions references tell us how a consumer ranks a given bundle compared to another bundle,
More information4.4 Logarithmic Functions
SECTION 4.4 Logarithmic Functions 87 4.4 Logarithmic Functions PREPARING FOR THIS SECTION Before getting started, review the following: Solving Inequalities (Appendi, Section A.8, pp. 04 05) Polnomial
More informationWalrasian Demand. u(x) where B(p, w) = {x R n + : p x w}.
Walrasian Demand Econ 2100 Fall 2015 Lecture 5, September 16 Outline 1 Walrasian Demand 2 Properties of Walrasian Demand 3 An Optimization Recipe 4 First and Second Order Conditions Definition Walrasian
More informationMTH4100 Calculus I. Lecture notes for Week 8. Thomas Calculus, Sections 4.1 to 4.4. Rainer Klages
MTH4100 Calculus I Lecture notes for Week 8 Thomas Calculus, Sections 4.1 to 4.4 Rainer Klages School of Mathematical Sciences Queen Mary University of London Autumn 2009 Theorem 1 (First Derivative Theorem
More information1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
More informationHomework # 3 Solutions
Homework # 3 Solutions February, 200 Solution (2.3.5). Noting that and ( + 3 x) x 8 = + 3 x) by Equation (2.3.) x 8 x 8 = + 3 8 by Equations (2.3.7) and (2.3.0) =3 x 8 6x2 + x 3 ) = 2 + 6x 2 + x 3 x 8
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 informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b
More informationAlternative proof for claim in [1]
Alternative proof for claim in [1] Ritesh Kolte and Ayfer Özgür Aydin The problem addressed in [1] is described in Section 1 and the solution is given in Section. In the proof in [1], it seems that obtaining
More information8 The Problem of Social Choice
8 The Problem of Social Choice One can argue that social choice theor asks the most fundamental question in economics. That, is, consider an econom. Then, provided that we are economists and insists on
More informationUCLA. Department of Economics Ph. D. Preliminary Exam MicroEconomic Theory
UCLA Department of Economics Ph. D. Preliminary Exam MicroEconomic Theory (SPRING 2011) Instructions: You have 4 hours for the exam Answer any 5 out of the 6 questions. All questions are weighted equally.
More informationTastes 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 informationOur 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,
More informationSection 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
More informationThe CobbDouglas Production Function
171 10 The CobbDouglas Production Function This chapter describes in detail the most famous of all production functions used to represent production processes both in and out of agriculture. First used
More informationPrice Theory Lecture 3: Theory of the Consumer
Price Theor Lecture 3: Theor of the Consumer I. Introduction The purpose of this section is to delve deeper into the roots of the demand curve, to see eactl how it results from people s tastes, income,
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 informationGraphing Linear Inequalities in Two Variables
5.4 Graphing Linear Inequalities in Two Variables 5.4 OBJECTIVES 1. Graph linear inequalities in two variables 2. Graph a region defined b linear inequalities What does the solution set look like when
More information4.6 Null Space, Column Space, Row Space
NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear
More informationMultivariable Calculus and Optimization
Multivariable Calculus and Optimization Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Multivariable Calculus and Optimization 1 / 51 EC2040 Topic 3  Multivariable Calculus
More informationSolution of the System of Linear Equations: any ordered pair in a system that makes all equations true.
Definitions: Sstem of Linear Equations: or more linear equations Sstem of Linear Inequalities: or more linear inequalities Solution of the Sstem of Linear Equations: an ordered pair in a sstem that makes
More informationU = 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 informationLinear Inequalities, Systems, and Linear Programming
8.8 Linear Inequalities, Sstems, and Linear Programming 481 8.8 Linear Inequalities, Sstems, and Linear Programming Linear Inequalities in Two Variables Linear inequalities with one variable were graphed
More informationGeneral Equilibrium Theory: Examples
General Equilibrium Theory: Examples 3 examples of GE: pure exchange (Edgeworth box) 1 producer  1 consumer several producers and an example illustrating the limits of the partial equilibrium approach
More informationNeoclassical Theory of Consumption
Neoclassical Theory of Consumption Economics II: Microeconomics VŠE Praha September 2009 Micro (VŠE) Consumption theory 09/09 1 / 24 Economics II: Microeconomics Consumers: Firms: People. Households. Applications.
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 informationProblem Set I: Preferences, W.A.R.P., consumer choice
Problem Set I: Preferences, W.A.R.P., consumer choice Paolo Crosetto paolo.crosetto@unimi.it Exercises solved in class on 18th January 2009 Recap:,, Definition 1. The strict preference relation is x y
More informationChoice under Uncertainty
Choice under Uncertainty Part 1: Expected Utility Function, Attitudes towards Risk, Demand for Insurance Slide 1 Choice under Uncertainty We ll analyze the underlying assumptions of expected utility theory
More informationMicroeconomic 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
More informationFINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether or not the relationship shown in the table is a function. 1) 
More informationMA107 Precalculus Algebra Exam 2 Review Solutions
MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write
More information4.1 PiecewiseDefined Functions
Section 4.1 PiecewiseDefined Functions 335 4.1 PiecewiseDefined Functions In preparation for the definition of the absolute value function, it is etremel important to have a good grasp of the concept
More information