Mat hs Revision. = 2ax + b,which is. We can see this by dif ferentiating y: dy. zero tomaximize y,at x * = b/2a.
|
|
- Sharleen Hutchinson
- 5 years ago
- Views:
Transcription
1 Mat hs Revision Fr om the document, MBAMat hs Unit 3 Functions & Int eres ts (sic) pages 46/47: Functions Quadratics. Aquadr atic function constains a var iable squared: y = ax 2 + bx + c The (single) maximum or minimum of y alw ays occur s at : x = b 2a We can see this by dif ferentiating y: dy dx zero tomaximize y,at x * = b/2a. = 2ax + b,which is Whet her y is a maximum or minimum depends on the sign of the coefficient a: when a is positive, y is a minimum, when a is negative, y is a maximum, and when a is zero, y is a linear function in x. >
2 Lecture 5a UNSW 2009 Page 2 Let s find the profit-maximizing values in L5. p.5 VonStackelberg: Reaction Function π C = (10 (Q S + Q C )) Q C 3Q C π C = Q 2 C + (7 Q S) Q C + 10 is a quadr atic in Q C, wit h a = 1, b = (7 Q S ),and c =10. π C max at Q * C = b 2a = 7 Q S 2
3 Lecture 5a UNSW 2009 Page 2 Let s find the profit-maximizing values in L5. p.5 VonStackelberg: Reaction Function π C = (10 (Q S + Q C )) Q C 3Q C π C = Q 2 C + (7 Q S) Q C + 10 is a quadr atic in Q C, wit h a = 1, b = (7 Q S ),and c =10. π C max at Q * C = b 2a = 7 Q S 2 p.6 VonStackelberg: Solution π S = 1 2 Q2 S Q S is a quadr atic in Q S, wit h a = ½, b =3½, and c =0. π S max at Q * S = b 2a =
4 Lecture 5a UNSW 2009 Page 2 Let s find the profit-maximizing values in L5. p.5 VonStackelberg: Reaction Function π C = (10 (Q S + Q C )) Q C 3Q C π C = Q 2 C + (7 Q S) Q C + 10 is a quadr atic in Q C, wit h a = 1, b = (7 Q S ),and c =10. π C max at Q * C = b 2a = 7 Q S 2 p.6 VonStackelberg: Solution π S = 1 2 Q2 S Q S is a quadr atic in Q S, wit h a = ½, b =3½, and c =0. π S max at Q * S = b 2a = p.8 Monopolis t: Solution π M = P M Q M 3Q M = Q 2 M + 7Q M is a quadr atic in Q M, wit h a = 1, b =7,and c =0. π M is max at Q * M =7/2.
5 Lecture 5a UNSW 2009 Page 3 p.11 Cour not: Reaction Function π S = (10 Q C Q S ) Q S 3Q S π S = Q 2 S + (7 Q C)Q S is a quadr atic in Q S, wit h a = 1, b = 7 Q C,and c =0. π S is a max at Q * S = 7 Q C 2.
6 Lecture 5a UNSW 2009 Page 3 p.11 Cour not: Reaction Function π S = (10 Q C Q S ) Q S 3Q S π S = Q 2 S + (7 Q C)Q S is a quadr atic in Q S, wit h a = 1, b = 7 Q C,and c =0. π S is a max at Q * S = 7 Q C 2. p.15 Imper fect Bertrand: Solution π p = P 2 p + ( P d ) P p 144 3P d is a quadr atic in P p, wit h a = 1, b = P d,and c = P d π p is max at P * p = P d
7 Lecture 5a UNSW 2009 Page 3 p.11 Cour not: Reaction Function π S = (10 Q C Q S ) Q S 3Q S π S = Q 2 S + (7 Q C)Q S is a quadr atic in Q S, wit h a = 1, b = 7 Q C,and c =0. π S is a max at Q * S = 7 Q C 2. p.15 Imper fect Bertrand: Solution π p = P 2 p + ( P d ) P p 144 3P d is a quadr atic in P p, wit h a = 1, b = P d,and c = P d π p is max at P * p = P d p.26 Monopolis tic Car tel: Solution π M = (10 Q M ) Q M 1 Q M = Q 2 M + 9Q M is a quadr atic in Q M, wit h a = 1, b =9,and c =0. π M is max at Q * M = 9/2 =4½.
8 Lecture 5a UNSW 2009 Page 4 p.27 Cour not: Reaction Function π 2 = (10 y 2 y e 1 ) y 2 1 y 2 π 2 = y (9 y e 1 ) y 2 is a quadr atic in y 2, wit h a = 1, b = 9 y e 1,and c =0. π 2 is max at y * 2 = 1 2 (9 y e 1 ).
9 Lecture 5a UNSW 2009 Page 4 p.27 Cour not: Reaction Function π 2 = (10 y 2 y e 1 ) y 2 1 y 2 π 2 = y (9 y e 1 ) y 2 is a quadr atic in y 2, wit h a = 1, b = 9 y e 1,and c =0. π 2 is max at y * 2 = 1 2 (9 y e 1 ). p.28 VonStackelberg: Solution π 1 = (10 y 2 y 1 ) y 1 1 y 1 π 1 = ( (9 y 1) y 1 ) y 1 y 1 π 1 = 10y y y 2 1 y 2 1 y 1 π 1 = y y 2 1 is a quadr atic in y 1, wit h a = ½,b =4½, and c =0. π 1 is max at y * 1 =4½.
10 Lecture 5a UNSW 2009 Page 5 The Economics of Profit-Maximizing The profit-maximizing firmwill continue increasing its rat e of output until the revenue associated withselling another unit of output is equal toorless than the costofproducing that output, assuming rising costs.
11 Lecture 5a UNSW 2009 Page 5 The Economics of Profit-Maximizing The profit-maximizing firmwill continue increasing its rat e of output until the revenue associated withselling another unit of output is equal toorless than the costofproducing that output, assuming rising costs. These two measures arecalled the Marginal Revenue and the Marginal Cost, respectivel y.
12 Lecture 5a UNSW 2009 Page 5 The Economics of Profit-Maximizing The profit-maximizing firmwill continue increasing its rat e of output until the revenue associated withselling another unit of output is equal toorless than the costofproducing that output, assuming rising costs. These two measures arecalled the Marginal Revenue and the Marginal Cost, respectivel y. The difference between Marginal Revenue and Marginal Cost is the Marginal Profit associated withthe lastunit of output produced and sold.
13 Lecture 5a UNSW 2009 Page 5 The Economics of Profit-Maximizing The profit-maximizing firmwill continue increasing its rat e of output until the revenue associated withselling another unit of output is equal toorless than the costofproducing that output, assuming rising costs. These two measures arecalled the Marginal Revenue and the Marginal Cost, respectivel y. The difference between Marginal Revenue and Marginal Cost is the Marginal Profit associated withthe lastunit of output produced and sold. In algebr a: M π = MR MC, where all three are functions of the level ofoutput Q (amongs t ot her things, such as the demand curve, orthe going market price P).
14 Lecture 5a UNSW 2009 Page 6 With rising costs, profit is maximized when Marginal Profit is no longer positive: when Tot alprofit no longer rises with the rat e of production.
15 Lecture 5a UNSW 2009 Page 6 With rising costs, profit is maximized when Marginal Profit is no longer positive: when Tot alprofit no longer rises with the rat e of production. Or, when Mπ = MR MC = 0.
16 Lecture 5a UNSW 2009 Page 6 With rising costs, profit is maximized when Marginal Profit is no longer positive: when Tot alprofit no longer rises with the rat e of production. Or, when Mπ = MR MC = 0. If we have functions for MR and MC in ter ms of output Q, then we can deter mine theprofit-maximizing levelofoutput Q *.
17 Lecture 5a UNSW 2009 Page 6 With rising costs, profit is maximized when Marginal Profit is no longer positive: when Tot alprofit no longer rises with the rat e of production. Or, when Mπ = MR MC = 0. If we have functions for MR and MC in ter ms of output Q, then we can deter mine theprofit-maximizing levelofoutput Q *. Marginal Revenue and Costare jus t theder ivatives of Tot al Revenue and Tot al Cos t wit h respect tooutput Q.
18 Lecture 5a UNSW 2009 Page 6 With rising costs, profit is maximized when Marginal Profit is no longer positive: when Tot alprofit no longer rises with the rat e of production. Or, when Mπ = MR MC = 0. If we have functions for MR and MC in ter ms of output Q, then we can deter mine theprofit-maximizing levelofoutput Q *. Marginal Revenue and Costare jus t theder ivatives of Tot al Revenue and Tot al Cos t wit h respect tooutput Q. TR is just P Q. Wit h alinear demand curve (say P = 10 Q), and a firmwit h some market power (which means can set its own price, subject todemand), MR can be calculat ed: TR = P Q = (10 Q) Q = 10Q Q 2.
19 Lecture 5a UNSW 2009 Page 7 Dif ferentiating with respect tooutput Q: = MR[Q] = 10 2Q dtr dq
20 Lecture 5a UNSW 2009 Page 7 Dif ferentiating with respect tooutput Q: = MR[Q] = 10 2Q dtr dq Let s say that it costs a constant amount (say, $3) toproduce and sell a unit of output. Then TC = 3Q,and MC =3.
21 Lecture 5a UNSW 2009 Page 7 Dif ferentiating with respect tooutput Q: = MR[Q] = 10 2Q dtr dq Let s say that it costs a constant amount (say, $3) toproduce and sell a unit of output. Then TC = 3Q,and MC =3. For this firm, choosing Q * to equat e MR and MC will result in the higes t profit : sol ve: 10 2Q = 3 to get Q * =3½.
22 Lecture 5a UNSW 2009 Page 7 Dif ferentiating with respect tooutput Q: = MR[Q] = 10 2Q dtr dq Let s say that it costs a constant amount (say, $3) toproduce and sell a unit of output. Then TC = 3Q,and MC =3. For this firm, choosing Q * to equat e MR and MC will result in the higes t profit : sol ve: 10 2Q = 3 to get Q * =3½. Fr om the demand function, with Q * =3½, the price P will be $6.50: the higher the output, the lowerthe price tosell all units.
23 Lecture 5a UNSW 2009 Page 8 Profit-Maximizing, Graphicall y We can plot the MC = $3/unit line and the demand line P = 10 Q. Wecan identify the Monopolist sprice & quantity (P M,Q M )and the price-t aker sprice & quantity (P C, Q C ).
24 Lecture 5a UNSW 2009 Page 8 Profit-Maximizing, Graphicall y We can plot the MC = $3/unit line and the demand line P = 10 Q. Wecan identify the Monopolist sprice & quantity (P M,Q M )and the price-t aker sprice & quantity (P C, Q C ). Monopolis ts s TR = P M Q M = (10 Q M ) Q M = 10Q M Q 2 M
25 Lecture 5a UNSW 2009 Page 8 Profit-Maximizing, Graphicall y We can plot the MC = $3/unit line and the demand line P = 10 Q. Wecan identify the Monopolist sprice & quantity (P M,Q M )and the price-t aker sprice & quantity (P C, Q C ). Monopolis ts s TR = P M Q M = (10 Q M ) Q M = 10Q M Q 2 M Dif ferentiating TR, weget MR = dtr = 10 2Q dq M M,which is exactl y twice as steep as the demand line, as shown below.
26 Lecture 5a UNSW 2009 Page 8 Profit-Maximizing, Graphicall y We can plot the MC = $3/unit line and the demand line P = 10 Q. Wecan identify the Monopolist sprice & quantity (P M,Q M )and the price-t aker sprice & quantity (P C, Q C ). Monopolis ts s TR = P M Q M = (10 Q M ) Q M = 10Q M Q 2 M Dif ferentiating TR, weget MR = dtr = 10 2Q dq M M,which is exactl y twice as steep as the demand line, as shown below. Fr om above, profit-maximizing monopolistic output Q * M occur s when MR = MC,but here MC =$3. So Q * M from 10 2Q * M = 3,or Q * M =
27 Lecture 5a UNSW 2009 Page 8 Profit-Maximizing, Graphicall y We can plot the MC = $3/unit line and the demand line P = 10 Q. Wecan identify the Monopolist sprice & quantity (P M,Q M )and the price-t aker sprice & quantity (P C, Q C ). Monopolis ts s TR = P M Q M = (10 Q M ) Q M = 10Q M Q 2 M Dif ferentiating TR, weget MR = dtr = 10 2Q dq M M,which is exactl y twice as steep as the demand line, as shown below. Fr om above, profit-maximizing monopolistic output Q * M occur s when MR = MC,but here MC =$3. So Q * M from 10 2Q * M = 3,or Q * M = The diagram shows that the output Q * M =3½occur s where the red downwards-sloping MR line cuts the MC = $3 line, and reading up to the demand line gives P * M =$6.50.
28 Lecture 5a UNSW 2009 Page 9 Pr ice P $/unit Demand: P = 10 Q MC = AC = Quantity Q
29 Lecture 5a UNSW 2009 Page 9 Pr ice P $/unit MR Demand: P = 10 Q Monopol y (Q M =3.5, P M =$6.50) MC = AC = Quantity Q
30 Lecture 5a UNSW 2009 Page 9 Pr ice P $/unit MR Demand: P = 10 Q Monopol y (Q M =3.5, P M =$6.50) Pr ice-taking MC = AC (Q C=7, P C =$3) = Quantity Q
31 Lecture 5a UNSW 2009 Page 10 If the firm isbehaving as a price-t aker,then its Marginal Revenue is just the going price P. So it chooses its output whereits Marginal Revenue = $3, the Marginal Cost. The Marginal Cost =$3/unit is in effect the market Supply cur ve. The market quantity is Q C = 7,at P C =$3/unit, where Demand = the horizont al green Supplyline, as shown above. <
Lecture 5a UNSW 2009 Page 1. Mat hs Revision
Lecture 5a UNSW 2009 Page 1 Mat hs Revision Fr om the document, MBAMat hs Unit 3 Functions & Int eres ts (sic) pages 46/47: Functions Quadratics. Aquadr atic function constains a var iable squared: y =
More informationSecond Order Derivatives. Background to Topic 6 Maximisation and Minimisation
Second Order Derivatives Course Manual Background to Topic 6 Maximisation and Minimisation Jacques (4 th Edition): Chapter 4.6 & 4.7 Y Y=a+bX a X Y= f (X) = a + bx First Derivative dy/dx = f = b constant
More informationy = F (x) = x n + c dy/dx = F`(x) = f(x) = n x n-1 Given the derivative f(x), what is F(x)? (Integral, Anti-derivative or the Primitive function).
Integration Course Manual Indefinite Integration 7.-7. Definite Integration 7.-7.4 Jacques ( rd Edition) Indefinite Integration 6. Definite Integration 6. y F (x) x n + c dy/dx F`(x) f(x) n x n- Given
More informationStudy Unit 2 : Linear functions Chapter 2 : Sections and 2.6
1 Study Unit 2 : Linear functions Chapter 2 : Sections 2.1 2.4 and 2.6 1. Function Humans = relationships Function = mathematical form of a relationship Temperature and number of ice cream sold Independent
More informationThe Monopolist. The Pure Monopolist with symmetric D matrix
University of California, Davis Department of Agricultural and Resource Economics ARE 252 Optimization with Economic Applications Lecture Notes 5 Quirino Paris The Monopolist.................................................................
More informationDCDM BUSINESS SCHOOL FACULTY OF MANAGEMENT ECONOMIC TECHNIQUES 102 LECTURE 4 DIFFERENTIATION
DCDM BUSINESS SCHOOL FACULTY OF MANAGEMENT ECONOMIC TECHNIUES 1 LECTURE 4 DIFFERENTIATION 1 Differentiation Managers are often concerned with the way that a variable changes over time Prices, for example,
More informationAnswer Key: Problem Set 1
Answer Key: Problem Set 1 Econ 409 018 Fall Question 1 a The profit function (revenue minus total cost) is π(q) = P (q)q cq The first order condition with respect to (henceforth wrt) q is P (q )q + P (q
More informationEssential Mathematics for Economics and Business, 4 th Edition CHAPTER 6 : WHAT IS THE DIFFERENTIATION.
Essential Mathematics for Economics and Business, 4 th Edition CHAPTER 6 : WHAT IS THE DIFFERENTIATION. John Wiley and Sons 13 Slopes/rates of change Recall linear functions For linear functions slope
More informationAdvanced Microeconomics
Advanced Microeconomics Leonardo Felli EC441: Room D.106, Z.332, D.109 Lecture 8 bis: 24 November 2004 Monopoly Consider now the pricing behavior of a profit maximizing monopolist: a firm that is the only
More informationEconS Oligopoly - Part 2
EconS 305 - Oligopoly - Part 2 Eric Dunaway Washington State University eric.dunaway@wsu.edu November 29, 2015 Eric Dunaway (WSU) EconS 305 - Lecture 32 November 29, 2015 1 / 28 Introduction Last time,
More informationEC611--Managerial Economics
EC611--Managerial Economics Optimization Techniques and New Management Tools Dr. Savvas C Savvides, European University Cyprus Models and Data Model a framework based on simplifying assumptions it helps
More informationOligopoly Notes. Simona Montagnana
Oligopoly Notes Simona Montagnana Question 1. Write down a homogeneous good duopoly model of quantity competition. Using your model, explain the following: (a) the reaction function of the Stackelberg
More informationBusiness Mathematics. Lecture Note #11 Chapter 6-(2)
Business Mathematics Lecture Note #11 Chapter 6-(2) 1 Applications of Differentiation Total functions 1) TR (Total Revenue) function TR = P Q where P = unit price, Q = quantity Note that P(price) is given
More informationSECTION 5.1: Polynomials
1 SECTION 5.1: Polynomials Functions Definitions: Function, Independent Variable, Dependent Variable, Domain, and Range A function is a rule that assigns to each input value x exactly output value y =
More informationChapter 6: Sections 6.1, 6.2.1, Chapter 8: Section 8.1, 8.2 and 8.5. In Business world the study of change important
Study Unit 5 : Calculus Chapter 6: Sections 6., 6.., 6.3. Chapter 8: Section 8., 8. and 8.5 In Business world the study of change important Example: change in the sales of a company; change in the value
More informationQuadratic function and equations Quadratic function/equations, supply, demand, market equilibrium
Exercises 8 Quadratic function and equations Quadratic function/equations, supply, demand, market equilibrium Objectives - know and understand the relation between a quadratic function and a quadratic
More informationIndustrial Organization, Fall 2011: Midterm Exam Solutions and Comments Date: Wednesday October
Industrial Organization, Fall 2011: Midterm Exam Solutions and Comments Date: Wednesday October 23 2011 1 Scores The exam was long. I know this. Final grades will definitely be curved. Here is a rough
More informationOptimization Techniques
Optimization Techniques Methods for maximizing or minimizing an objective function Examples Consumers maximize utility by purchasing an optimal combination of goods Firms maximize profit by producing and
More informationFirms and returns to scale -1- Firms and returns to scale
Firms and returns to scale -1- Firms and returns to scale. Increasing returns to scale and monopoly pricing 2. Constant returns to scale 19 C. The CRS economy 25 D. pplication to trade 47 E. Decreasing
More informationLecture Notes for Chapter 9
Lecture Notes for Chapter 9 Kevin Wainwright April 26, 2014 1 Optimization of One Variable 1.1 Critical Points A critical point occurs whenever the firest derivative of a function is equal to zero. ie.
More informationChapter 4 Differentiation
Chapter 4 Differentiation 08 Section 4. The derivative of a function Practice Problems (a) (b) (c) 3 8 3 ( ) 4 3 5 4 ( ) 5 3 3 0 0 49 ( ) 50 Using a calculator, the values of the cube function, correct
More informationEXAMINATION #4 ANSWER KEY. I. Multiple choice (1)a. (2)e. (3)b. (4)b. (5)d. (6)c. (7)b. (8)b. (9)c. (10)b. (11)b.
William M. Boal Version A EXAMINATION #4 ANSWER KEY I. Multiple choice (1)a. ()e. (3)b. (4)b. (5)d. (6)c. (7)b. (8)b. (9)c. (10)b. (11)b. II. Short answer (1) a. 4 units of food b. 1/4 units of clothing
More informationMathematical Foundations -1- Constrained Optimization. Constrained Optimization. An intuitive approach 2. First Order Conditions (FOC) 7
Mathematical Foundations -- Constrained Optimization Constrained Optimization An intuitive approach First Order Conditions (FOC) 7 Constraint qualifications 9 Formal statement of the FOC for a maximum
More informationREVIEW OF MATHEMATICAL CONCEPTS
REVIEW OF MATHEMATICAL CONCEPTS Variables, functions and slopes: A Variable is any entity that can take different values such as: price, output, revenue, cost, etc. In economics we try to 1. Identify the
More informationLesson 6: Switching Between Forms of Quadratic Equations Unit 5 Quadratic Functions
(A) Lesson Context BIG PICTURE of this UNIT: CONTEXT of this LESSON: How do we analyze and then work with a data set that shows both increase and decrease What is a parabola and what key features do they
More informationAdvanced Microeconomic Analysis, Lecture 6
Advanced Microeconomic Analysis, Lecture 6 Prof. Ronaldo CARPIO April 10, 017 Administrative Stuff Homework # is due at the end of class. I will post the solutions on the website later today. The midterm
More informationAnswer Key: Problem Set 3
Answer Key: Problem Set Econ 409 018 Fall Question 1 a This is a standard monopoly problem; using MR = a 4Q, let MR = MC and solve: Q M = a c 4, P M = a + c, πm = (a c) 8 The Lerner index is then L M P
More informationTopic 6: Optimization I. Maximisation and Minimisation Jacques (4th Edition): Chapter 4.6 & 4.7
Topic 6: Optimization I Maximisation and Minimisation Jacques (4th Edition): Chapter 4.6 & 4.7 1 For a straight line Y=a+bX Y= f (X) = a + bx First Derivative dy/dx = f = b constant slope b Second Derivative
More informationFirms and returns to scale -1- John Riley
Firms and returns to scale -1- John Riley Firms and returns to scale. Increasing returns to scale and monopoly pricing 2. Natural monopoly 1 C. Constant returns to scale 21 D. The CRS economy 26 E. pplication
More informationTrade policy III: Export subsidies
The Vienna Institute for International Economic Studies - wiiw June 25, 2015 Overview Overview 1 1 Under perfect competition lead to welfare loss 2 Effects depending on market structures 1 Subsidies to
More informationECONOMICS 207 SPRING 2008 PROBLEM SET 13
ECONOMICS 207 SPRING 2008 PROBLEM SET 13 Problem 1. The cost function for a firm is a rule or mapping that tells the minimum total cost of production of any output level produced by the firm for a fixed
More informationREVIEW OF MATHEMATICAL CONCEPTS
REVIEW OF MATHEMATICAL CONCEPTS 1 Variables, functions and slopes A variable is any entity that can take different values such as: price, output, revenue, cost, etc. In economics we try to 1. Identify
More informationx ax 1 2 bx2 a bx =0 x = a b. Hence, a consumer s willingness-to-pay as a function of liters on sale, 1 2 a 2 2b, if l> a. (1)
Answers to Exam Economics 201b First Half 1. (a) Observe, first, that no consumer ever wishes to consume more than 3/2 liters (i.e., 1.5 liters). To see this, observe that, even if the beverage were free,
More informationEconomics 203: Intermediate Microeconomics. Calculus Review. A function f, is a rule assigning a value y for each value x.
Economics 203: Intermediate Microeconomics Calculus Review Functions, Graphs and Coordinates Econ 203 Calculus Review p. 1 Functions: A function f, is a rule assigning a value y for each value x. The following
More informationb. /(x) = -x^-;r^ r(x)= -SX"" f'(^^^ ^ -^x"- "3.^ ^ Mth 241 Test 2 Extra Practice: ARC and IRC Name:
Mth 241 Test 2 Extra Practice: ARC and IRC Name: This is not a practice exam. This is a supplement to the activities v\/e do in class and the homework. 1. Use the graph to answer the following questions.
More informationSymmetric Lagrange function
University of California, Davis Department of Agricultural and Resource Economics ARE 5 Optimization with Economic Applications Lecture Notes 7 Quirino Paris Symmetry.......................................................................page
More informationLecture 6: Sections 2.2 and 2.3 Polynomial Functions, Quadratic Models
L6-1 Lecture 6: Sections 2.2 and 2.3 Polynomial Functions, Quadratic Models Polynomial Functions Def. A polynomial function of degree n is a function of the form f(x) = a n x n + a n 1 x n 1 +... + a 1
More informationClassic Oligopoly Models: Bertrand and Cournot
Classic Oligopoly Models: Bertrand and Cournot Class Note: There are supplemental readings, including Werden (008) Unilateral Competitive Effects of Horizontal Mergers I: Basic Concepts and Models, that
More informationMATH 104 THE SOLUTIONS OF THE ASSIGNMENT
MTH 4 THE SOLUTIONS OF THE SSIGNMENT Question9. (Page 75) Solve X = if = 8 and = 4 and write a system. X =, = 8 4 = *+ *4= = 8*+ 4*= For finding the system, we use ( ) = = 6= 5, 8 /5 /5 = = 5 8 8/5 /5
More informationGS/ECON 5010 Answers to Assignment 3 W2005
GS/ECON 500 Answers to Assignment 3 W005 Q. What are the market price, and aggregate quantity sold, in long run equilibrium in a perfectly competitive market f which the demand function has the equation
More informationMathematics for Economics ECON MA/MSSc in Economics-2017/2018. Dr. W. M. Semasinghe Senior Lecturer Department of Economics
Mathematics for Economics ECON 53035 MA/MSSc in Economics-2017/2018 Dr. W. M. Semasinghe Senior Lecturer Department of Economics MATHEMATICS AND STATISTICS LERNING OUTCOMES: By the end of this course unit
More informationDifferentiation. 1. What is a Derivative? CHAPTER 5
CHAPTER 5 Differentiation Differentiation is a technique that enables us to find out how a function changes when its argument changes It is an essential tool in economics If you have done A-level maths,
More informationTopic 3: Application of Differential Calculus in
Mathematics 2 for Business Schools Topic 3: Application of Differential Calculus in Economics Building Competence. Crossing Borders. Spring Semester 2017 Learning objectives After finishing this section
More informationUnit 3: Producer Theory
Unit 3: Producer Theory Prof. Antonio Rangel December 13, 2013 1 Model of the firm 1.1 Key properties of the model Key assumption: firms maximize profits subject to Technological constraints: natural limits
More information2. Which of the following is the ECONOMISTS inverse of the function y = 9/x 2 (i.e. find x as a function of y, x = f(y))
Anwers for Review Quiz #1. Material Covered. Klein 1, 2; Schaums 1, 2 1. Solve the following system of equations for x, y and z: x + y = 2 2x + 2y + z = 5 7x + y + z = 9 Answers: x = 1, y = 1, z = 1. 2.
More informationLecture 1: Introduction to IO Tom Holden
Lecture 1: Introduction to IO Tom Holden http://io.tholden.org/ Email: t.holden@surrey.ac.uk Standard office hours: Thursday, 12-1PM + 3-4PM, 29AD00 However, during term: CLASSES will be run in the first
More informationMATH150-E01 Test #2 Summer 2016 Show all work. Name 1. Find an equation in slope-intercept form for the line through (4, 2) and (1, 3).
1. Find an equation in slope-intercept form for the line through (4, 2) and (1, 3). 2. Let the supply and demand functions for sugar be given by p = S(q) = 1.4q 0.6 and p = D(q) = 2q + 3.2 where p is the
More informationOligopoly Theory. This might be revision in parts, but (if so) it is good stu to be reminded of...
This might be revision in parts, but (if so) it is good stu to be reminded of... John Asker Econ 170 Industrial Organization January 23, 2017 1 / 1 We will cover the following topics: with Sequential Moves
More informationDCDM BUSINESS SCHOOL FACULTY OF MANAGEMENT ECONOMIC TECHNIQUES 102 LECTURE 3 NON-LINEAR FUNCTIONS
DCDM BUSINESS SCHOOL FACULTY OF MANAGEMENT ECONOMIC TECHNIQUES 10 LECTURE NON-LINEAR FUNCTIONS 0. Preliminaries The following functions will be discussed briefly first: Quadratic functions and their solutions
More informationOligopoly. Firm s Profit Maximization Firm i s profit maximization problem: Static oligopoly model with n firms producing homogenous product.
Oligopoly Static oligopoly model with n firms producing homogenous product. Firm s Profit Maximization Firm i s profit maximization problem: Max qi P(Q)q i C i (q i ) P(Q): inverse demand curve: p = P(Q)
More informationOligopoly. Oligopoly. Xiang Sun. Wuhan University. March 23 April 6, /149
Oligopoly Xiang Sun Wuhan University March 23 April 6, 2016 1/149 Outline 1 Introduction 2 Game theory 3 Oligopoly models 4 Cournot competition Two symmetric firms Two asymmetric firms Many symmetric firms
More informationFunctions. A function is a rule that gives exactly one output number to each input number.
Functions A function is a rule that gives exactly one output number to each input number. Why it is important to us? The set of all input numbers to which the rule applies is called the domain of the function.
More informationOctober 16, 2018 Notes on Cournot. 1. Teaching Cournot Equilibrium
October 1, 2018 Notes on Cournot 1. Teaching Cournot Equilibrium Typically Cournot equilibrium is taught with identical zero or constant-mc cost functions for the two firms, because that is simpler. I
More informationMathematical Foundations II
Mathematical Foundations 2-1- Mathematical Foundations II A. Level and superlevel sets 2 B. Convex sets and concave functions 4 C. Parameter changes: Envelope Theorem I 17 D. Envelope Theorem II 41 48
More informationH-Pre-Calculus Targets Chapter I can write quadratic functions in standard form and use the results to sketch graphs of the function.
H-Pre-Calculus Targets Chapter Section. Sketch and analyze graphs of quadratic functions.. I can write quadratic functions in standard form and use the results to sketch graphs of the function. Identify
More informationz = f (x; y) f (x ; y ) f (x; y) f (x; y )
BEEM0 Optimization Techiniques for Economists Lecture Week 4 Dieter Balkenborg Departments of Economics University of Exeter Since the fabric of the universe is most perfect, and is the work of a most
More informationStudy Guide - Part 2
Math 116 Spring 2015 Study Guide - Part 2 1. Which of the following describes the derivative function f (x) of a quadratic function f(x)? (A) Cubic (B) Quadratic (C) Linear (D) Constant 2. Find the derivative
More information1 + x 1/2. b) For what values of k is g a quasi-concave function? For what values of k is g a concave function? Explain your answers.
Questions and Answers from Econ 0A Final: Fall 008 I have gone to some trouble to explain the answers to all of these questions, because I think that there is much to be learned b working through them
More information32. Use a graphing utility to find the equation of the line of best fit. Write the equation of the line rounded to two decimal places, if necessary.
Pre-Calculus A Final Review Part 2 Calculator Name 31. The price p and the quantity x sold of a certain product obey the demand equation: p = x + 80 where r = xp. What is the revenue to the nearest dollar
More information5.1 - Polynomials. Ex: Let k(x) = x 2 +2x+1. Find (and completely simplify) the following: (a) k(1) (b) k( 2) (c) k(a)
c Kathryn Bollinger, March 15, 2017 1 5.1 - Polynomials Def: A function is a rule (process) that assigns to each element in the domain (the set of independent variables, x) ONE AND ONLY ONE element in
More informationEcon Slides from Lecture 10
Econ 205 Sobel Econ 205 - Slides from Lecture 10 Joel Sobel September 2, 2010 Example Find the tangent plane to {x x 1 x 2 x 2 3 = 6} R3 at x = (2, 5, 2). If you let f (x) = x 1 x 2 x3 2, then this is
More informationMath 1325 Final Exam Review. (Set it up, but do not simplify) lim
. Given f( ), find Math 5 Final Eam Review f h f. h0 h a. If f ( ) 5 (Set it up, but do not simplify) If c. If f ( ) 5 f (Simplify) ( ) 7 f (Set it up, but do not simplify) ( ) 7 (Simplify) d. If f. Given
More informationMath 142 Week-in-Review #11 (Final Exam Review: All previous sections as well as sections 6.6 and 6.7)
Math 142 Week-in-Review #11 (Final Exam Review: All previous sections as well as sections 6.6 and 6.7) Note: This review is intended to highlight the topics covered on the Final Exam (with emphasis on
More informationOnline Supplementary Appendix B
Online Supplementary Appendix B Uniqueness of the Solution of Lemma and the Properties of λ ( K) We prove the uniqueness y the following steps: () (A8) uniquely determines q as a function of λ () (A) uniquely
More informationUnit 6. Firm behaviour and market structure: perfect competition
Uni 6. Firm behaviour and marke srucure: erfec comeiion In accordance wih he APT rogramme objecives of he lecure are o hel You o: deermine shor-run and long-run equilibrium, boh for he rofi-maximizing
More informationDurable goods monopolist
Durable goods monopolist Coase conjecture: A monopolist selling durable good has no monopoly power. Reason: A P 1 P 2 B MC MC D MR Q 1 Q 2 C Q Although Q 1 is optimal output of the monopolist, it faces
More information3. Partial Equilibrium under Imperfect Competition Competitive Equilibrium
3. Imperfect Competition 3. Partial Equilirium under Imperfect Competition Competitive Equilirium Partial equilirium studies the existence of equilirium in the market of a given commodity and analyzes
More informationTotal 100
Math 112 Final Exam June 3, 2017 Name Student ID # Section HONOR STATEMENT I affirm that my work upholds the highest standards of honesty and academic integrity at the University of Washington, and that
More informationT R(y(L)) w L = 0. L so we can write this as p ] (b) Recall that with the perfectly competitive firm, demand for labor was such that
Problem Set 11: Solutions ECO 301: Intermediate Microeconomics Prof. Marek Weretka Problem 1 (Monopoly and the Labor Market) (a) We find the optimal demand for labor for a monopoly firm (in the goods market
More informationChapter 6: Sections 6.1, 6.2.1, Chapter 8: Section 8.1, 8.2 and 8.5. In Business world the study of change important
Study Unit 5 : Calculus Chapter 6: Sections 6., 6.., 6.. Chapter 8: Section 8., 8. and 8.5 In Business world the study of change important Eample: change in the sales of a company; change in the value
More informationIntroduction to Industrial Organization Professor: Caixia Shen Fall 2014 Lecture Note 12 Price discrimination (ch 10)-continue
Introduction to Industrial Organization Professor: Caixia Shen Fall 2014 Lecture Note 12 Price discrimination (ch 10)-continue 2 nd degree price discrimination We have discussed that firms charged different
More information4. Partial Equilibrium under Imperfect Competition
4. Partial Equilibrium under Imperfect Competition Partial equilibrium studies the existence of equilibrium in the market of a given commodity and analyzes its properties. Prices in other markets as well
More informationLecture #11: Introduction to the New Empirical Industrial Organization (NEIO) -
Lecture #11: Introduction to the New Empirical Industrial Organization (NEIO) - What is the old empirical IO? The old empirical IO refers to studies that tried to draw inferences about the relationship
More informationPractice Questions for Math 131 Exam # 1
Practice Questions for Math 131 Exam # 1 1) A company produces a product for which the variable cost per unit is $3.50 and fixed cost 1) is $20,000 per year. Next year, the company wants the total cost
More informationBertrand Model of Price Competition. Advanced Microeconomic Theory 1
Bertrand Model of Price Competition Advanced Microeconomic Theory 1 ҧ Bertrand Model of Price Competition Consider: An industry with two firms, 1 and 2, selling a homogeneous product Firms face market
More informationMicroeconomic Theory -1- Introduction
Microeconomic Theory -- Introduction. Introduction. Profit maximizing firm with monopoly power 6 3. General results on maximizing with two variables 8 4. Model of a private ownership economy 5. Consumer
More informationStatic Models of Oligopoly
Static Models of Oligopoly Cournot and Bertrand Models Mateusz Szetela 1 1 Collegium of Economic Analysis Warsaw School of Economics 3 March 2016 Outline 1 Introduction Game Theory and Oligopolies 2 The
More informationANSWERS, Homework Problems, Fall 2014: Lectures Now You Try It, Supplemental problems in written homework, Even Answers. 24x + 72 (x 2 6x + 4) 4
ANSWERS, Homework Problems, Fall 014: Lectures 19 35 Now You Try It, Supplemental problems in written homework, Even Answers Lecture 19 1. d [ 4 ] dx x 6x + 4) 3 = 4x + 7 x 6x + 4) 4. a) P 0) = 800 b)
More information9. The Monopsonist. π = c x [g s + s Gs] (9.1) subject to. Ax s (9.2) x 0, s 0. The economic interpretation of this model proceeds as in previous
9. The Monopsonist An economic agent who is the sole buer of all the available suppl of an input is called a monopsonist. We will discusss two categories of monopsonists: the pure monopsonist and the perfectl
More informationPart I: Exercise of Monopoly Power. Chapter 1: Monopoly. Two assumptions: A1. Quality of goods is known by consumers; A2. No price discrimination.
Part I: Exercise of Monopoly Power Chapter 1: Monopoly Two assumptions: A1. Quality of goods is known by consumers; A2. No price discrimination. Best known monopoly distortion: p>mc DWL (section 1). Other
More informationEconomics 401 Sample questions 2
Economics 401 Sample questions 1. What does it mean to say that preferences fit the Gorman polar form? Do quasilinear preferences fit the Gorman form? Do aggregate demands based on the Gorman form have
More information1 Functions and Graphs
1 Functions and Graphs 1.1 Functions Cartesian Coordinate System A Cartesian or rectangular coordinate system is formed by the intersection of a horizontal real number line, usually called the x axis,
More informationHans Zenger University of Munich. Abstract
The Optimal Regulation of Product Quality under Monopoly Hans Zenger University of Munich Abstract This paper characterizes the optimal quality regulation of a monopolist when quality is observable. In
More informationSchool of Business. Blank Page
Maxima and Minima 9 This unit is designed to introduce the learners to the basic concepts associated with Optimization. The readers will learn about different types of functions that are closely related
More informationECO 2901 EMPIRICAL INDUSTRIAL ORGANIZATION
ECO 2901 EMPIRICAL INDUSTRIAL ORGANIZATION Lecture 7 & 8: Models of Competition in Prices & Quantities Victor Aguirregabiria (University of Toronto) Toronto. Winter 2018 Victor Aguirregabiria () Empirical
More informationSTUDY MATERIALS. (The content of the study material is the same as that of Chapter I of Mathematics for Economic Analysis II of 2011 Admn.
STUDY MATERIALS MATHEMATICAL TOOLS FOR ECONOMICS III (The content of the study material is the same as that of Chapter I of Mathematics for Economic Analysis II of 2011 Admn.) & MATHEMATICAL TOOLS FOR
More informationFor logistic growth, we have
Dr. Lozada Econ. 5250 For logistic growth, we have. Harvest H depends on effort E and on X. With harvesting, Ẋ = F (X) H. Suppose that for a fixed level of effort E, H depends linearly on X: If E increases,
More informationMath 110 Final Exam General Review. Edward Yu
Math 110 Final Exam General Review Edward Yu Da Game Plan Solving Limits Regular limits Indeterminate Form Approach Infinities One sided limits/discontinuity Derivatives Power Rule Product/Quotient Rule
More informationOverview. Producer Theory. Consumer Theory. Exchange
Overview Consumer Producer Exchange Edgeworth Box All Possible Exchange Points Contract Curve Overview Consumer Producer Exchange (Multiplicity) Walrasian Equilibrium Walrasian Equilibrium Requirements:
More informationMA 151, Applied Calculus, Fall 2017, Final Exam Preview Identify each of the graphs below as one of the following functions:
MA 5, Applie Calculus, Fall 207, Final Exam Preview Basic Functions. Ientify each of the graphs below as one of the following functions: x 3 4x 2 + x x 2 e x x 3 ln(x) (/2) x 2 x 4 + 4x 2 + 0 5x + 0 5x
More informationAre innocuous Minimum Quality Standards really innocuous?
Are innocuous Minimum Quality Standards really innocuous? Paolo G. Garella University of Bologna 14 July 004 Abstract The present note shows that innocuous Minimum Quality Standards, namely standards that
More informationUniversidad Carlos III de Madrid May Microeconomics Grade
Universidad Carlos III de Madrid May 017 Microeconomics Name: Group: 1 3 5 Grade You have hours and 5 minutes to answer all the questions. 1. Multiple Choice Questions. (Mark your choice with an x. You
More informationWorkbook. Michael Sampson. An Introduction to. Mathematical Economics Part 1 Q 2 Q 1. Loglinear Publishing
Workbook An Introduction to Mathematical Economics Part U Q Loglinear Publishing Q Michael Sampson Copyright 00 Michael Sampson. Loglinear Publications: http://www.loglinear.com Email: mail@loglinear.com.
More informationSCHOOL OF DISTANCE EDUCATION
SCHOOL OF DISTANCE EDUCATION CCSS UG PROGRAMME MATHEMATICS (OPEN COURSE) (For students not having Mathematics as Core Course) MM5D03: MATHEMATICS FOR SOCIAL SCIENCES FIFTH SEMESTER STUDY NOTES Prepared
More informationExam A. Exam 3. (e) Two critical points; one is a local maximum, the other a local minimum.
1.(6 pts) The function f(x) = x 3 2x 2 has: Exam A Exam 3 (a) Two critical points; one is a local minimum, the other is neither a local maximum nor a local minimum. (b) Two critical points; one is a local
More informationLecture 4: Optimization. Maximizing a function of a single variable
Lecture 4: Optimization Maximizing or Minimizing a Function of a Single Variable Maximizing or Minimizing a Function of Many Variables Constrained Optimization Maximizing a function of a single variable
More informationEconS Vertical Integration
EconS 425 - Vertical Integration Eric Dunaway Washington State University eric.dunaway@wsu.edu Industrial Organization Eric Dunaway (WSU) EconS 425 Industrial Organization 1 / 26 Introduction Let s continue
More informationMathematics 2 for Business Schools Topic 7: Application of Integration to Economics. Building Competence. Crossing Borders.
Mathematics 2 for Business Schools Topic 7: Application of Integration to Economics Building Competence. Crossing Borders. Spring Semester 2017 Learning objectives After finishing this section you should
More informationEcon 110: Introduction to Economic Theory. 8th Class 2/7/11
Econ 110: Introduction to Economic Theory 8th Class 2/7/11 go over problem answers from last time; no new problems today given you have your problem set to work on; we'll do some problems for these concepts
More informationBusiness Mathematics. Lecture Note #13 Chapter 7-(1)
1 Business Mathematics Lecture Note #13 Chapter 7-(1) Applications of Partial Differentiation 1. Differentials and Incremental Changes 2. Production functions: Cobb-Douglas production function, MP L, MP
More information