Exercise 2: Equivalence of the first two definitions for a differentiable function. is a convex combination of
|
|
- Laurence Baker
- 6 years ago
- Views:
Transcription
1 March 07 Mathematical Foundations John Riley Module Marginal analysis and single variable calculus 6 Eercises Eercise : Alternative definitions of a concave function (a) For and that 0, and conve combination t (), elain why the second definition imlies f f t f t t f t 0 0 ( ) ( ( )) ( ( )) ( )) ( ( )) Multilying both sides by t, it follows that 0 0 ( t) f ( ) ( t) f ( ( t)) ( t) f ( ( t)) ( t) ( t)) f ( ( t)) (b) Use a very similar argument to show that tf tf t t f t t t f t ( ) ( ( )) ( ( )) ( )) ( ( )) Add the two inequalities and then use the fact that ( t) ( t) t 0 to show that the second definition imlies the first Eercise : Equivalence of the first two definitions for a differentiable function (a) Define y( t) f ( ( t)) where t () is a conve combination of 0 and Use the first definition of a concave function to show that y( t) y(0) f t 0 ( ) f ( ) This holds for all small t 0 Taking the limit it follows that 0 (0) ( ) ( ) y f f (b) Use the Chain Rule to show that 0 0 y (0) f ( )( ) (c) Hence show that the first definition imlies the second Given this result and the result in the revious eercise, the two definitions are equivalent
2 Eercise 3: Alternative definitions of differentiable concave functions Prove that the second definition of a concave function imlies the third Eercise 4: Maimizing a concave function (a) Show that the derivative of following function is continuous ( ), f ( ), ( ), (b) Deict the grah of the function (c) What values of are maimizers? Eercise 5: The sum of n strictly concave functions is strictly concave (a) Above we showed that if the two functions f ( ) and f ( ) are strictly concave then their sum g( ) f( ) f( ) is strictly concave (b) Elain why g( ) f3( ) is strictly concave if f ( ) 3 is strictly concave (c) Use this result to elain why the sum of n strictly concave functions is strictly concave Eercise 6: Profit-maimizing firm A monooly estimates that the market demand function is q( ) 6 q The cost of roduction is C( q) F 6q q 3 (a) What is the demand rice function? (b) Show that the revenue function is concave and the cost function is conve (c) Hence elain why the rofit function is concave
3 (d) Assuming that the firm is going to roduce, obtain an eression for total revenue Rq ( ) and hence the rofit function ( q) (e) Solve for the outut that maimizes ( q) You should elain why it is a maimum (e) What is the rofit-maimizing outut if (i) F 0 (ii) F 60 Eercise 7: Inut choice Using z units of an inut, the maimum outut of the firm is q f z z / ( ) 0 (a) If the rice of the inut is z and the rice of the outut is write down an eression for total rofit as a function of the number of units of the inut urchased (b) If (, ) (60, ) solve for the rofit maimizing inut and hence outut (c) Reeat this for any air of rices and show that maimized outut is q ( ) 5 Eercise 8: Price taking firm A firm sells roduces such a small fraction of the industry outut that its outut decision has a negligible effect on the outut rice The cost of roduction is C( q) q q q q (a) Write down an eression for total and marginal rofit if 8 (b) What condition must be satisfied for q to be a critical oint? (c) Show that this condition holds if q and q (d) Show that the condition can be written as follows: ( q)( q)(3 q) 0 (e) Which of the critical oints is a local maimum and which is a local minimum?
4 (f) Which is the rofit maimizing outut? (g) Deict the rofit function in a neat figure (h) In a searate figure deict the cost function and the revenue function (i) Use one of these figures to elain why there is an interval of oututs that will not be rofit maimizers at any outut rice Eercise 9: Discriminating monooly A monooly sells in the domestic market where the demand function is q 30 The cost of roduction is C( q) 0q (a) Solve for the demand rice function (the inverse maings from quantity to rice) (b) Solve for the rofit-maimizing outut and hence show that the rofit maimizing rice is 40 Euroe eliminates imort duties As a result the firm can sell in Euroe where the demand function is q 33 We ignore transortation costs so that the cost of roducing q units for sale in the USA and q units for sale in Euroe is C( q q) 0( q q) (c) Obtain an eression for the Euroean demand rice function (d) Write down an eression for the total rofit ( q, q) Elain why the rofit maimization roblem can be treated as two indeendent roblems, each with one variable (e) Solve for the rofit-maimizing oututs and hence show that the rofit-maimizing rices are 40 and 65 Eercise 0: Common rice This is a continuation of the revious eercise Manufacturers of cometing roducts comlain that the US is selling its eorts at a lower rice (This is called duming ) In resonse the 3
5 Euroean Trade Commission rules that the firm can only sell in Euroe if the rice same as in the USA is the (a) Solve for the total demand in the USA and Euroe, q( ) q( ) q( ) and hence show that if the rice in both the USA and Euroe is, then the rice function for total demand is 4 q 3 (b) Solve for the quantity that equates marginal revenue and cost and show that the imlied rice is * 3 (c) What is the new rofit on sales in the USA and in Euroe? (d) Should the firm sell in Euroe? Eercise : Robinson Crusoe Robinson Crusoe is stranded on an island If he does nothing he has a daily suly of coconuts c0 0 If we works hours er day, he can increase the suly to c c0 Robinson does not like work If he works hours his utility is u( c, ) c(8 ) (a) Solve for the number of hours that maimizes his utility (i) if c0 4 (ii) c0 8 (b) For what values of c 0 will Robinson choose not to work? (c) Are there any values of c 0 for which Robinson will work all 8 hours Eercise 3: Necessary conditions for a maimum Suose that f( ) is twice differentiable and has a maimum at (a) Elain why the second derivative cannot be strictly ositive (b) Show that the function f ( ) ( ) 4 is concave (c) Show that f ( ) ( ) 4 has unique maimizer * that that the second derivative is not strictly negative at * 4
6 Eercise A4: Profit maimization A firm roducing outut has a rofit of ( ) 9 (a) Show that this function has critical oints maimum? (b) What is the rofit-maimizing outut and Which of these is a local 5
ε and ε > 0 we can find a δ > 0 such that
John Riley June 5, 3 ANSWERS TO EXERCISES IN APPENDIX A SECTION A: MAPPINGS OF A SINGLE VARIABLE Eercise A-: Rules of limits (a) Limit of the sum = the sum of the limits We wish to estalish that for any
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 informationEconomics 101. Lecture 7 - Monopoly and Oligopoly
Economics 0 Lecture 7 - Monooly and Oligooly Production Equilibrium After having exlored Walrasian equilibria with roduction in the Robinson Crusoe economy, we will now ste in to a more general setting.
More informationPROFIT MAXIMIZATION. π = p y Σ n i=1 w i x i (2)
PROFIT MAXIMIZATION DEFINITION OF A NEOCLASSICAL FIRM A neoclassical firm is an organization that controls the transformation of inuts (resources it owns or urchases into oututs or roducts (valued roducts
More informationEcon 401A: Economic Theory Mid-term. Answers
. Labor suly Econ 40: Economic Theory Mid-term nswers (a) Let be labor suly. Then x 4 The key ste is setting u the budget constraint. x w w(4 x ) Thus the budget constraint can be rewritten as follows:
More informationConvex Analysis and Economic Theory Winter 2018
Division of the Humanities and Social Sciences Ec 181 KC Border Conve Analysis and Economic Theory Winter 2018 Toic 16: Fenchel conjugates 16.1 Conjugate functions Recall from Proosition 14.1.1 that is
More informationCHAPTER 3: OPTIMIZATION
John Riley 8 February 7 CHAPTER 3: OPTIMIZATION 3. TWO VARIABLES 8 Second Order Conditions Implicit Function Theorem 3. UNCONSTRAINED OPTIMIZATION 4 Necessary and Sufficient Conditions 3.3 CONSTRAINED
More information5.1 THE ROBINSON CRUSOE ECONOMY
Essential Microeconomics -- 5 THE ROBINSON CRUSOE ECONOMY Ke ideas: Walrasian equilibrium allocation, otimal allocation, invisible hand at work A simle econom with roduction Two commodities, H consumers,
More informationProfit Maximization. Beattie, Taylor, and Watts Sections: 3.1b-c, 3.2c, , 5.2a-d
Proit Maimization Beattie Talor and Watts Sections:.b-c.c 4.-4. 5.a-d Agenda Generalized Proit Maimization Proit Maimization ith One Inut and One Outut Proit Maimization ith To Inuts and One Outut Proit
More informationECON 500 Fall Exam #2 Answer Key.
ECO 500 Fall 004. Eam # Answer Key. ) While standing in line at your favourite movie theatre, you hear someone behind you say: I like ocorn, but I m not buying any because it isn t worth the high rice.
More informationANSWERS TO ODD NUMBERED EXERCISES IN CHAPTER 5. The constraint is binding at the maximum therefore we can substitute for y
John Rile Aril 0 ANSWERS TO ODD NUMBERED EXERCISES IN CHAPTER 5 Section 5: The Robinson Crusoe Econom Eercise 5-: Equilibrium (a) = ( + ω) = ( + 47, ) Then = 47 Substituting or in the / roduction unction,
More informationTheory of Externalities Partial Equilibrium Analysis
Theory of Externalities Partial Equilibrium Analysis Definition: An externality is resent whenever the well being of a consumer or the roduction ossibilities of a firm are directly affected by the actions
More informationANSWERS TO ODD NUMBERED EXERCISES IN CHAPTER 3
John Riley 5 Setember 0 NSWERS T DD NUMERED EXERCISES IN CHPTER 3 SECTIN 3: Equilibrium and Efficiency Exercise 3-: Prices with Quasi-linear references (a) Since references are convex, an allocation is
More information4. CONTINUOUS VARIABLES AND ECONOMIC APPLICATIONS
STATIC GAMES 4. CONTINUOUS VARIABLES AND ECONOMIC APPLICATIONS Universidad Carlos III de Madrid CONTINUOUS VARIABLES In many games, ure strategies that layers can choose are not only, 3 or any other finite
More informationMicro I. Lesson 5 : Consumer Equilibrium
Microecono mics I. Antonio Zabalza. Universit of Valencia 1 Micro I. Lesson 5 : Consumer Equilibrium 5.1 Otimal Choice If references are well behaved (smooth, conve, continuous and negativel sloed), then
More informationCalculus One variable
Calculus One variable (f ± g) ( 0 ) = f ( 0 ) ± g ( 0 ) (λf) ( 0 ) = λ f ( 0 ) ( (fg) ) ( 0 ) = f ( 0 )g( 0 ) + f( 0 )g ( 0 ) f g (0 ) = f ( 0 )g( 0 ) f( 0 )g ( 0 ) f( 0 ) 2 (f g) ( 0 ) = f (g( 0 )) g
More information2x2x2 Heckscher-Ohlin-Samuelson (H-O-S) model with factor substitution
2x2x2 Heckscher-Ohlin-amuelson (H-O- model with factor substitution The HAT ALGEBRA of the Heckscher-Ohlin model with factor substitution o far we were dealing with the easiest ossible version of the H-O-
More informationEcon 101A Midterm 2 Th 8 April 2009.
Econ A Midterm Th 8 Aril 9. You have aroximately hour and minutes to answer the questions in the midterm. I will collect the exams at. shar. Show your work, and good luck! Problem. Production (38 oints).
More information(a) The isoquants for each of the three production functions are show below:
Problem Set 7: Solutions ECON 0: Intermediate Microeconomics Prof. Marek Weretka Problem (Production Functions) (a) The isoquants for each of the three roduction functions are show below: f(, ) = f (f
More information, αβ, > 0 is strictly quasi-concave on
John Riley 8 Setember 9 Econ Diagnostic Test Time allowed: 9 minutes. Attemt all three questions. Note that the last two arts of questions and 3 are marked with an asterisk (). These do not carry many
More informationHandout #3: Peak Load Pricing
andout #3: Peak Load Pricing Consider a firm that exeriences two kinds of costs a caacity cost and a marginal cost ow should caacity be riced? This issue is alicable to a wide variety of industries, including
More informationInternational Trade with a Public Intermediate Good and the Gains from Trade
International Trade with a Public Intermediate Good and the Gains from Trade Nobuhito Suga Graduate School of Economics, Nagoya University Makoto Tawada Graduate School of Economics, Nagoya University
More information14 March 2018 Module 1: Marginal analysis and single variable calculus John Riley. ( x, f ( x )) are the convex combinations of these two
4 March 28 Module : Marginal analysis single variable calculus John Riley 4. Concave conve functions A function f( ) is concave if, for any interval [, ], the graph of a function f( ) is above the line
More informationChapter 5 Notes. These notes correspond to chapter 5 of Mas-Colell, Whinston, and Green.
Chater 5 Notes These notes corresond to chater 5 of Mas-Colell, Whinston, and Green. 1 Production We now turn from consumer behavior to roducer behavior. For the most art we will examine roducer behavior
More informationExercises - SOLUTIONS UEC Advanced Microeconomics, Fall 2018 Instructor: Dusan Drabik, de Leeuwenborch 2105
Eercises - SOLUTIONS UEC-5806 Advanced Microeconomics, Fall 08 Instructor: Dusan Drabik, de Leeuwenborch 05. A consumer has a preference relation on R which can be represented by the utility function u()
More informationCHAPTER 1-2: SHADOW PRICES
Essential Microeconomics -- CHAPTER -: SHADOW PRICES An intuitive approach: profit maimizing firm with a fied supply of an input Shadow prices 5 Concave maimization problem 7 Constraint qualifications
More informationMODELING A RANDOM YIELD IN- HOUSE PRODUCTION SET UP IN A NEWSVENDOR PROBLEM
www.araress.com/volumes/vol16issue3/ijrras_16_3_10.df MODEING A RANDOM YIED IN- HOSE PRODCTION SET P IN A NEWSVENDOR PROBEM Krishna Solanki 1 & Ravi Gor 1.D.R.P. Institute of Technology and Research, Gandhinagar,
More informationMonopolist s mark-up and the elasticity of substitution
Croatian Oerational Research Review 377 CRORR 8(7), 377 39 Monoolist s mark-u and the elasticity of substitution Ilko Vrankić, Mira Kran, and Tomislav Herceg Deartment of Economic Theory, Faculty of Economics
More informationCONSUMPTION. (Lectures 4, 5, and 6) Remark: (*) signals those exercises that I consider to be the most important
CONSUMPTION (Lectures 4, 5, and 6) Remark: (*) signals those eercises that I consider to be the most imortant Eercise 0 (MWG, E. 1.B.1, 1.B.) Show that if is rational, then: 1. if y z, then z;. is both
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 informationQUADRATIC FUNCTIONS. ( x 7)(5x 6) = 2. Exercises: 1 3x 5 Sum: 8. We ll expand it by using the distributive property; 9. Let s use the FOIL method;
QUADRATIC FUNCTIONS A. Eercises: 1.. 3. + = + = + + = +. ( 1)(3 5) (3 5) 1(3 5) 6 10 3 5 6 13 5 = = + = +. ( 7)(5 6) (5 6) 7(5 6) 5 6 35 4 5 41 4 3 5 6 10 1 3 5 Sum: 6 + 10+ 3 5 ( + 1)(3 5) = 6 + 13 5
More informationSchool of Economics and Management
School of Economics and Management TECHNICAL UNIVERSITY OF LISBON Deartment of Economics Carlos Pestana Barros & Nicolas Peyoch José Pedro Pontes A Comarative Analysis of Productivity Change in Italian
More informationSection A (Basic algebra and calculus multiple choice)
BEE1 Basic Mathematical Economics Dieter Balkenborg January 4 eam Solutions Department of Economics 2.2.4 University of Eeter Section A (Basic algebra and calculus multiple choice) Question A1 : The function
More informationChapter 3 - The Concept of Differentiation
alculus hapter - The oncept o Dierentiation Applications o Dierentiation opyright 00-004 preptests4u.com. All Rights Reserved. This Academic Review is brought to you ree o charge by preptests4u.com. Any
More informationPart 6A. 4. Tax and Monopoly Taxes in Monopoly vs Taxes
Part 6A. Monooly 4. Tax and Monooly 租稅與獨佔 Taxes in Monooly vs. Cometitive Markets Lum-Sum Tax Secific Taxes Ad Valorem Taxes Proortional Profit Taxes 2015.5.21 1 Taxes in Monooly vs. Cometitive Markets
More informationThe Supply Side of the Economy. 1 The Production Function: What Determines the Total Production
Lecture Notes 3 The Suly Side of the Economy (Mankiw, Cht. 3) 1 The Production Function: What Determines the Total Production of Goods and Services? An economy's outut of goods and services - its GDP -
More informationPretest (Optional) Use as an additional pacing tool to guide instruction. August 21
Trimester 1 Pretest (Otional) Use as an additional acing tool to guide instruction. August 21 Beyond the Basic Facts In Trimester 1, Grade 7 focus on multilication. Daily Unit 1: The Number System Part
More information1 Entropy 1. 3 Extensivity 4. 5 Convexity 5
Contents CONEX FUNCIONS AND HERMODYNAMIC POENIALS 1 Entroy 1 2 Energy Reresentation 2 3 Etensivity 4 4 Fundamental Equations 4 5 Conveity 5 6 Legendre transforms 6 7 Reservoirs and Legendre transforms
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 informationMicroeconomics Fall 2017 Problem set 1: Possible answers
Microeconomics Fall 07 Problem set Possible answers Each answer resents only one way of solving the roblem. Other right answers are ossible and welcome. Exercise For each of the following roerties, draw
More informationMATH 2710: NOTES FOR ANALYSIS
MATH 270: NOTES FOR ANALYSIS The main ideas we will learn from analysis center around the idea of a limit. Limits occurs in several settings. We will start with finite limits of sequences, then cover infinite
More informationEntrepreneurship and new ventures finance. Designing a new business (3): Revenues and costs. Prof. Antonio Renzi
Entrereneurshi and new ventures finance Designing a new business (3): Revenues and costs Prof. Antonio Renzi Agenda 1. Revenues analysis 2. Costs analysis 3. Break even analysis Revenue Model Primary Demand
More informationVoting and Lobbying - 3 Models
Voting and obbying - 3 Models Series of 3 aers eloring the effects of olitical actions on market outcomes. Current theories of regulation unsatisfying (to me!: Toulouse School: Agency Model regulators
More informationMANAGEMENT SCIENCE doi /mnsc ec
MANAGEMENT SCIENCE doi 0287/mnsc0800993ec e-comanion ONLY AVAILABLE IN ELECTRONIC FORM informs 2009 INFORMS Electronic Comanion Otimal Entry Timing in Markets with Social Influence by Yogesh V Joshi, David
More information3 Additional Applications of the Derivative
3 Additional Applications of the Derivative 3.1 Increasing and Decreasing Functions; Relative Etrema 3.2 Concavit and Points of Inflection 3.4 Optimization Homework Problem Sets 3.1 (1, 3, 5-9, 11, 15,
More informationSolution Week 75 (2/16/04) Hanging chain
Catenary Catenary is idealized shae of chain or cable hanging under its weight with the fixed end oints. The chain (cable) curve is catenary that minimizes the otential energy PHY 322, Sring 208 Solution
More informationf(x) p(x) =p(b)... d. A function can have two different horizontal asymptotes...
Math Final Eam, Fall. ( ts.) Mark each statement as either true [T] or false [F]. f() a. If lim f() =and lim g() =, then lim does not eist......................!5!5!5 g() b. If is a olynomial, then lim!b
More informationIf C(x) is the total cost (in dollars) of producing x items of a product, then
Supplemental Review Problems for Unit Test : 1 Marginal Analysis (Sec 7) Be prepared to calculate total revenue given the price - demand function; to calculate total profit given total revenue and total
More informationCompetitive Equilibrium
Competitive Equilibrium Econ 2100 Fall 2017 Lecture 16, October 26 Outline 1 Pareto Effi ciency 2 The Core 3 Planner s Problem(s) 4 Competitive (Walrasian) Equilibrium Decentralized vs. Centralized Economic
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 informationSolved Problems. (a) (b) (c) Figure P4.1 Simple Classification Problems First we draw a line between each set of dark and light data points.
Solved Problems Solved Problems P Solve the three simle classification roblems shown in Figure P by drawing a decision boundary Find weight and bias values that result in single-neuron ercetrons with the
More informationIMPORTANT NOTES HERE IS AN EXAMPLE OF A SCANTRON FORM FOR YOUR EXAM.
IMPORTANT NOTES HERE IS AN EXAMPLE OF A SCANTRON FORM FOR YOUR EXAM. YOU NEED TO MAKE SURE YOU PROPERLY FILL OUT THE SCANTRON FORM.. Write and bubble in your first and last name.. VERY important, write
More informationElectronic Companion to Tax-Effective Supply Chain Decisions under China s Export-Oriented Tax Policies
Electronic Companion to Tax-Effective Supply Chain Decisions under China s Export-Oriented Tax Policies Optimality Equations of EI Strategy In this part, we derive the optimality equations for the Export-Import
More informationCOMMUNICATION BETWEEN SHAREHOLDERS 1
COMMUNICATION BTWN SHARHOLDRS 1 A B. O A : A D Lemma B.1. U to µ Z r 2 σ2 Z + σ2 X 2r ω 2 an additive constant that does not deend on a or θ, the agents ayoffs can be written as: 2r rθa ω2 + θ µ Y rcov
More informationYou are responsible for upholding the University of Maryland Honor Code while taking this exam.
Econ300 Spring 2014 Second Midterm Eam version T Answers This eam consists of 25 multiple choice questions. The maimum duration of the eam is 50 minutes. 1. In the spaces provided on the scantron, write
More informationYou are responsible for upholding the University of Maryland Honor Code while taking this exam.
Econ300 Spring 2014 Second Midterm Eam version W Answers This eam consists of 25 multiple choice questions. The maimum duration of the eam is 50 minutes. 1. In the spaces provided on the scantron, write
More informationGSOE9210 Engineering Decisions
GSOE9 Engineering Decisions Problem Set 5. Consider the river roblem described in lectures: f f V B A B + (a) For =, what is the sloe of the Bayes indifference line through A? (b) Draw the Bayes indifference
More informationPart I Analysis in Economics
Part I Analysis in Economics D 1 1 (Function) A function f from a set A into a set B, denoted by f : A B, is a correspondence that assigns to each element A eactly one element y B We call y the image of
More informationAdding Production to the Theory
Adding Production to the Theory We begin by considering the simplest situation that includes production: two goods, both of which have consumption value, but one of which can be transformed into the other.
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 informationCalculus in Business. By Frederic A. Palmliden December 7, 1999
Calculus in Business By Frederic A. Palmliden December 7, 999 Optimization Linear Programming Game Theory Optimization The quest for the best Definition of goal equilibrium: The equilibrium state is defined
More informationLimiting Price Discrimination when Selling Products with Positive Network Externalities
Limiting Price Discrimination when Selling Products with Positive Network Externalities Luděk Cigler, Wolfgang Dvořák, Monika Henzinger, Martin Starnberger University of Vienna, Faculty of Comuter Science,
More informationEssential Microeconomics EXISTENCE OF EQUILIBRIUM Core ideas: continuity of excess demand functions, Fixed point theorems
Essetial Microecoomics -- 5.3 EXISTENCE OF EQUILIBRIUM Core ideas: cotiuity of excess demad fuctios, Fixed oit teorems Two commodity excage ecoomy 2 Excage ecoomy wit may commodities 5 Discotiuous demad
More informationQuestion 1. (8 points) The following diagram shows the graphs of eight equations.
MAC 2233/-6 Business Calculus, Spring 2 Final Eam Name: Date: 5/3/2 Time: :am-2:nn Section: Show ALL steps. One hundred points equal % Question. (8 points) The following diagram shows the graphs of eight
More informationPractice Problems **Note this list of problems is by no means complete and to focus solely on these problems would be unwise.**
Topics for the Final Eam MATC 100 You will be allowed to use our MATC 100 calculator. The final eam is cumulative (Sections.-., Sections 3.1-3.5, Sections.1-.5) - see the details below. Sections.-. & 3.1-3.3:
More informationMath 140 Review for Quiz 13 page 1
Math Review for Quiz age. Solve each of the following sstems of equations over the real numbers. a) + x + x + = x = x + = x d) x + = x =. Solve each of the following equations. a) x x + = x + x = x + x
More informationExcerpt from "Intermediate Algebra" 2014 AoPS Inc.
Ecert from "Intermediate Algebra" 04 AoPS Inc. www.artofroblemsolving.com for which our grah is below the -ais with the oints where the grah intersects the -ais (because the ineuality is nonstrict), we
More informationName: MA 160 Dr. Katiraie (100 points) Test #3 Spring 2013
Name: MA 160 Dr. Katiraie (100 points) Test #3 Spring 2013 Show all of your work on the test paper. All of the problems must be solved symbolically using Calculus. You may use your calculator to confirm
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 informationAdvance Selling in the Presence of Experienced Consumers
Advance Selling in the Presence of Eerienced Consumers Oksana Loginova X. Henry Wang Chenhang Zeng June 30, 011 Abstract The advance selling strategy is imlemented when a firm offers consumers the oortunity
More information1. Sets A set is any collection of elements. Examples: - the set of even numbers between zero and the set of colors on the national flag.
San Francisco State University Math Review Notes Michael Bar Sets A set is any collection of elements Eamples: a A {,,4,6,8,} - the set of even numbers between zero and b B { red, white, bule} - the set
More informationSlides Prepared by JOHN S. LOUCKS St. Edward s s University Thomson/South-Western. Slide
s Preared by JOHN S. LOUCKS St. Edward s s University 1 Chater 11 Comarisons Involving Proortions and a Test of Indeendence Inferences About the Difference Between Two Poulation Proortions Hyothesis Test
More informationOperations Management
Universidade Nova de Lisboa Faculdade de Economia Oerations Management Winter Semester 009/010 First Round Exam January, 8, 009, 8.30am Duration: h30 RULES 1. Do not searate any sheet. Write your name
More informationLecture Notes October 18, Reading assignment for this lecture: Syllabus, section I.
Lecture Notes October 18, 2012 Reading assignment for this lecture: Syllabus, section I. Economic General Equilibrium Partial and General Economic Equilibrium PARTIAL EQUILIBRIUM S k (p o ) = D k k (po
More informationM112 Short Course In Calculus V. J. Motto Spring 2013 Applications of Derivatives Worksheet
M11 Short Course In Calculus V. J. Motto Spring 01 Applications of Derivatives Worksheet 1. A tomato is thrown from the top of a tomato cart its distance from the ground in feet is modeled by the equation
More informationA note on the preferred hedge instrument
ingnan University Digital Commons @ ingnan University ong ong Institute o Business tudies Working aer eries ong ong Institute o Business tudies 香港商學研究所 6-5 A note on the reerred hedge instrument Arthur
More informationEconomics 205 Exercises
Economics 05 Eercises Prof. Watson, Fall 006 (Includes eaminations through Fall 003) Part 1: Basic Analysis 1. Using ε and δ, write in formal terms the meaning of lim a f() = c, where f : R R.. Write the
More informationTHE INSTITUTE OF FINANCE MANAGEMENT (IFM) Department of Mathematics. Mathematics 01 MTU Elements of Calculus in Economics
THE INSTITUTE OF FINANCE MANAGEMENT (IFM) Department of Mathematics Mathematics 0 MTU 070 Elements of Calculus in Economics Calculus Calculus deals with rate of change of quantity with respect to another
More informationAn Optimization Model for Multi-period Multi- Product Multi-objective Production Planning
International Journal of Engineering & Technology IJET-IJENS Vol:16 No:01 43 An Otimization Model for Multi-eriod Multi- Product Multi-objective Production Planning M. S. Al-Ashhab Design & Production
More informationHomework Set #3 Rates definitions, Channel Coding, Source-Channel coding
Homework Set # Rates definitions, Channel Coding, Source-Channel coding. Rates (a) Channels coding Rate: Assuming you are sending 4 different messages using usages of a channel. What is the rate (in bits
More informationSystems of Linear Equations in Two Variables. Break Even. Example. 240x x This is when total cost equals total revenue.
Systems of Linear Equations in Two Variables 1 Break Even This is when total cost equals total revenue C(x) = R(x) A company breaks even when the profit is zero P(x) = R(x) C(x) = 0 2 R x 565x C x 6000
More informationGraphing and Optimization
BARNMC_33886.QXD //7 :7 Page 74 Graphing and Optimization CHAPTER - First Derivative and Graphs - Second Derivative and Graphs -3 L Hôpital s Rule -4 Curve-Sketching Techniques - Absolute Maima and Minima
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Linear equations 1 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) Find the slope of the line passing through the points (, -3) and (2, -1). 1)
More informationThe Euler Phi Function
The Euler Phi Function 7-3-2006 An arithmetic function takes ositive integers as inuts and roduces real or comlex numbers as oututs. If f is an arithmetic function, the divisor sum Dfn) is the sum of the
More informationUniversidad Carlos III de Madrid
Universidad Carlos III de Madrid Eercise 1 2 3 4 5 6 Total Points Department of Economics Mathematics I Final Eam January 22nd 2018 LAST NAME: Eam time: 2 hours. FIRST NAME: ID: DEGREE: GROUP: 1 (1) Consider
More informationChapter Four. Chapter Four
Chapter Four Chapter Four CHAPTER FOUR 99 ConcepTests for Section 4.1 1. Concerning the graph of the function in Figure 4.1, which of the following statements is true? (a) The derivative is zero at two
More informationLecture 7: Linear Classification Methods
Homeork Homeork Lecture 7: Linear lassification Methods Final rojects? Grous oics Proosal eek 5 Lecture is oster session, Jacobs Hall Lobby, snacks Final reort 5 June. What is linear classification? lassification
More informationSample Final Exam 4 MATH 1110 CALCULUS I FOR ENGINEERS
Dept. of Math. Sciences, UAEU Sample Final Eam Fall 006 Sample Final Eam MATH 0 CALCULUS I FOR ENGINEERS Section I: Multiple Choice Problems [0% of Total Final Mark, distributed equally] No partial credit
More informationMath 116: Business Calculus Chapter 4 - Calculating Derivatives
Math 116: Business Calculus Chapter 4 - Calculating Derivatives Instructor: Colin Clark Spring 2017 Exam 2 - Thursday March 9. 4.1 Techniques for Finding Derivatives. 4.2 Derivatives of Products and Quotients.
More informationMeasuring the Market Power of the Portuguese Milk Industry
International Journal of the Economics of Business, Vol. 6, No. 2, 1999,. 209± 222 Measuring the Market Power of the Portuguese Milk Industry MARGARIDA DE MELLO and ANT ÂONIO BRAND ÄAO ABSTRACT This aer
More informationApproximate Market Equilibrium for Near Gross Substitutes
Aroximate Market Equilibrium for Near Gross Substitutes Chinmay Karande College of Comuting, Georgia Tech ckarande@cc.gatech.edu Nikhil Devanur College of Comuting, Georgia Tech nikhil@cc.gatech.edu Abstract
More informationute measures of uncertainty called standard errors for these b j estimates and the resulting forecasts if certain conditions are satis- ed. Note the e
Regression with Time Series Errors David A. Dickey, North Carolina State University Abstract: The basic assumtions of regression are reviewed. Grahical and statistical methods for checking the assumtions
More informationProof: We follow thearoach develoed in [4]. We adot a useful but non-intuitive notion of time; a bin with z balls at time t receives its next ball at
A Scaling Result for Exlosive Processes M. Mitzenmacher Λ J. Sencer We consider the following balls and bins model, as described in [, 4]. Balls are sequentially thrown into bins so that the robability
More informationTrading OTC and Incentives to Clear Centrally
Trading OTC and Incentives to Clear Centrally Gaetano Antinolfi Francesca Caraella Francesco Carli March 1, 2013 Abstract Central counterparties CCPs have been art of the modern financial system since
More information1.4 FOUNDATIONS OF CONSTRAINED OPTIMIZATION
Essential Microeconomics -- 4 FOUNDATIONS OF CONSTRAINED OPTIMIZATION Fundamental Theorem of linear Programming 3 Non-linear optimization problems 6 Kuhn-Tucker necessary conditions Sufficient conditions
More informationSection 0.10: Complex Numbers from Precalculus Prerequisites a.k.a. Chapter 0 by Carl Stitz, PhD, and Jeff Zeager, PhD, is available under a Creative
Section 0.0: Comlex Numbers from Precalculus Prerequisites a.k.a. Chater 0 by Carl Stitz, PhD, and Jeff Zeager, PhD, is available under a Creative Commons Attribution-NonCommercial-ShareAlike.0 license.
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 informationBasic mathematics of economic models. 3. Maximization
John Riley 1 January 16 Basic mathematics o economic models 3 Maimization 31 Single variable maimization 1 3 Multi variable maimization 6 33 Concave unctions 9 34 Maimization with non-negativity constraints
More informationStatics and dynamics: some elementary concepts
1 Statics and dynamics: some elementary concets Dynamics is the study of the movement through time of variables such as heartbeat, temerature, secies oulation, voltage, roduction, emloyment, rices and
More informationMath 99 Review for Exam 3
age 1 1. Simlify each of the following eressions. (a) ab a b + 1 b 1 a 1 b + 1 Solution: We will factor both numerator and denominator and then cancel. The numerator can be factored by grouing ab {z a
More information