DOCSLIB.ORG

Lecture Notes1 Mathematical Ecnomics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture Notes1 Mathematical Ecnomics Lecture Notes1 Mathematical Ecnomics Guoqiang TIAN Department of Economics Texas A&M University College Station, Texas 77843 ([email protected]) This version: August, 2020 1The most materials of this lecture notes are drawn from Chiang’s classic textbook Fundamental Methods of Mathematical Economics, which are used for my teaching and con- venience of my students in class. Please not distribute it to any others. Contents 1 The Nature of Mathematical Economics 1 1.1 Economics and Mathematical Economics . 1 1.2 Advantages of Mathematical Approach . 3 2 Economic Models 5 2.1 Ingredients of a Mathematical Model . 5 2.2 The Real-Number System . 5 2.3 The Concept of Sets . 6 2.4 Relations and Functions . 9 2.5 Types of Function . 11 2.6 Functions of Two or More Independent Variables . 12 2.7 Levels of Generality . 13 3 Equilibrium Analysis in Economics 15 3.1 The Meaning of Equilibrium . 15 3.2 Partial Market Equilibrium - A Linear Model . 16 3.3 Partial Market Equilibrium - A Nonlinear Model . 18 3.4 General Market Equilibrium . 19 3.5 Equilibrium in National-Income Analysis . 23 4 Linear Models and Matrix Algebra 25 4.1 Matrix and Vectors . 26 i ii CONTENTS 4.2 Matrix Operations . 29 4.3 Linear Dependance of Vectors . 32 4.4 Commutative, Associative, and Distributive Laws . 33 4.5 Identity Matrices and Null Matrices . 34 4.6 Transposes and Inverses . 36 5 Linear Models and Matrix Algebra (Continued) 41 5.1 Conditions for Nonsingularity of a Matrix . 41 5.2 Test of Nonsingularity by Use of Determinant . 43 5.3 Basic Properties of Determinants . 48 5.4 Finding the Inverse Matrix . 53 5.5 Cramer’s Rule . 58 5.6 Application to Market and National-Income Models . 63 5.7 Quadratic Forms . 66 5.8 Eigenvalues and Eigenvectors . 70 5.9 Vector Spaces . 73 6 Comparative Statics and the Concept of Derivative 81 6.1 The Nature of Comparative Statics . 81 6.2 Rate of Change and the Derivative . 82 6.3 The Derivative and the Slope of a Curve . 84 6.4 The Concept of Limit . 84 6.5 Inequality and Absolute Values . 88 6.6 Limit Theorems . 89 6.7 Continuity and Differentiability of a Function . 90 7 Rules of Differentiation and Their Use in Comparative Statics 95 7.1 Rules of Differentiation for a Function of One Variable . 95 7.2 Rules of Differentiation Involving Two or More Functions of the Same Variable . 98 CONTENTS iii 7.3 Rules of Differentiation Involving Functions of Different Vari- ables . 103 7.4 Integration (The Case of One Variable) . 107 7.5 Partial Differentiation . 110 7.6 Applications to Comparative-Static Analysis . 112 7.7 Note on Jacobian Determinants . 114 8 Comparative-Static Analysis of General-Functions 119 8.1 Differentials . 121 8.2 Total Differentials . 123 8.3 Rule of Differentials . 124 8.4 Total Derivatives . 126 8.5 Implicit Function Theorem . 130 8.6 Comparative Statics of General-Function Models . 136 8.7 Matrix Derivatives . 137 9 Optimization: Maxima and Minima of a Function of One Vari- able 139 9.1 Optimal Values and Extreme Values . 140 9.2 Existence of Extremum for Continuous Function . 141 9.3 First-Derivative Test for Relative Maximum and Minimum . 142 9.4 Second and Higher Derivatives . 145 9.5 Second-Derivative Test . 147 9.6 Taylor Series . 149 9.7 Nth-Derivative Test . 152 10 Exponential and Logarithmic Functions 153 10.1 The Nature of Exponential Functions . 153 10.2 Logarithmic Functions . 154 10.3 Derivatives of Exponential and Logarithmic Functions . 155 iv CONTENTS 11 Optimization: Maxima and Minima of a Function of Two or More Variables 159 11.1 The Differential Version of Optimization Condition . 159 11.2 Extreme Values of a Function of Two Variables . 160 11.3 Objective Functions with More than Two Variables . 167 11.4 Second-Order Conditions in Relation to Concavity and Con- vexity . 169 11.5 Economic Applications . 173 12 Optimization with Equality Constraints 177 12.1 Effects of a Constraint . 178 12.2 Finding the Stationary Values . 179 12.3 Second-Order Condition . 184 12.4 General Setup of the Problem . 186 12.5 Quasiconcavity and Quasiconvexity . 189 12.6 Utility Maximization and Consumer Demand . 195 13 Optimization with Inequality Constraints 199 13.1 Non-Linear Programming . 199 13.2 Kuhn-Tucker Conditions . 201 13.3 Economic Applications . 205 14 Differential Equations 211 14.1 Existence and Uniqueness Theorem of Solutions for Ordi- nary Differential Equations . 213 14.2 Some Common Ordinary Differential Equations with Ex- plicit Solutions . 214 14.3 Higher Order Linear Equations with Constant Coefficients . 218 14.4 System of Ordinary Differential Equations . 222 CONTENTS v 14.5 Simultaneous Differential Equations and Stability of Equi- librium . 227 14.6 The Stability of Dynamical System . 229 15 Difference Equations 231 15.1 First-order Difference Equations . 233 15.2 Second-order Difference Equation . 236 15.3 Difference Equations of Order n . 237 15.4 The Stability of nth-Order Difference Equations . 239 15.5 Difference Equations with Constant Coefficients . 240 Chapter 1 The Nature of Mathematical Economics The purpose of this course is to introduce the most fundamental aspects of the mathematical methods such as those matrix algebra, mathematical analysis, and optimization theory. 1.1 Economics and Mathematical Economics Economics is a social science that studies how to make decisions in face of scarce resources. Specifically, it studies individuals’ economic behavior and phenomena as well as how individuals, such as consumers, house- holds, firms, organizations and government agencies, make trade-off choic- es that allocate limited resources among competing uses. Mathematical economics is an approach to economic analysis, in which the economists make use of mathematical symbols in the statement of the problem and also draw upon known mathematical theorems to aid in rea- soning. Since mathematical economics is merely an approach to economic anal- 1 2 CHAPTER 1. THE NATURE OF MATHEMATICAL ECONOMICS ysis, it should not and does not differ from the nonmathematical approach to economic analysis in any fundamental way. The difference between these two approaches is that in the former, the assumptions and conclu- sions are stated in mathematical symbols rather than words and in the equations rather than sentences so that the interdependent relationship among economic variables and resulting conclusions are more rigorous and concise by using mathematical models and mathematical statistic- s/econometric methods. Also, to study, analyze, and solve practical social and economic prob- lems as well as arising your managemental skills and leadership well, I think the research methodology of "three dimensions and six natures” should be used in studying and solving realistic social economic problem- s. Three dimensions: theoretical logic, practical knowledge, and his- torical perspective; Six natures: being scientific, rigorous, realistic, pertinent, forward- looking and thought-provoking. Only by accomplishing the "three-dimensional" and “six natures", can we put forward logical, factual and historical insights, high perspicacity views and policy recommendations for the economic development of a nation and the world, as well as theoretical innovations and works that can be tested by data, practice and history. Only through the three dimen- sions, can your assessments and proposals be guaranteed to have these six natures and become a good economist. The study of economic social issues cannot simply involve real world in its experiment, so it not only requires theoretical analysis based on in- herent logical inference and, more often than not, vertical and horizontal comparisons from the larger perspective of history so as to draw experi- 1.2. ADVANTAGES OF MATHEMATICAL APPROACH 3 ence and lessons, but also needs tools of statistics and econometrics to do empirical quantitative analysis or test, the three of which are all indispens- able. When conducting economic analysis or giving policy suggestions in the realm of modern economics, the theoretical analysis often combines theory, history, and statistics, presenting not only theoretical analysis of inherent logic and comparative analysis from the historical perspective but also empirical and quantitative analysis with statistical tools for ex- amination and investigation. Indeed, in the final analysis, all knowledge is history, all science is logics, and all judgment is statistics. As such, it is not surprising that mathematics and mathematical statis- tics/econometrics are used as the basic and most important analytical tool- s in every field of economics. For those who study economics and conduct research, it is necessary to grasp enough knowledge of mathematics and mathematical statistics. Therefore, it is of great necessity to master suffi- cient mathematical knowledge if you want to learn economics well, con- duct economic research and become a good economist. 1.2 Advantages of Mathematical Approach Mathematical approach has the following advantages: (1) It makes the language more precise and the statement of as- sumptions more clear, which can deduce many unnecessary debates resulting from inaccurate verbal language. (2) It makes the analytical logic more rigorous and clearly s- tates the boundary, applicable scope and conditions for a conclusion to hold. Otherwise, the abuse of a theory may occur. (3) Mathematics can help obtain the results that cannot be eas- 4 CHAPTER 1. THE NATURE OF MATHEMATICAL ECONOMICS ily attained through intuition. (4) It helps improve and extend the existing economic theories. It is, however, noteworthy a good master of mathematics cannot guar- antee to be a good economist. It also requires fully understanding the an- alytical framework and research methodologies of economics, and having a good intuition and insight of real economic environments and econom- ic issues. The study of economics not only calls for the understanding of some terms, concepts and results from the perspective of mathematics (in- cluding geometry), but more importantly, even when those are given by mathematical language or geometric figure, we need to get to their eco- nomic meaning and the underlying profound economic thoughts and ide- als.
Load more

Recommended publications
  • 2.3: Modeling with Differential Equations Some General Comments: a Mathematical Model Is an Equation Or Set of Equations That Mi
    2.3: Modeling with Differential Equations Some General Comments: A mathematical model is an equation or set of equations that mimic the behavior of some phenomenon under certain assumptions/approximations. Phenomena that contain rates/change can often be modeled with differential equations. Disclaimer: In forming a mathematical model, we make various assumptions and simplifications. I am never going to claim that these models perfectly fit physical reality. But mathematical modeling is a key component of the following scientific method: 1. We make assumptions (a hypothesis) and form a model. 2. We mathematically analyze the model. (For differential equations, these are the techniques we are learning this quarter). 3. If the model fits the phenomena well, then we have evidence that the assumptions of the model might be valid. If the model fits the phenomena poorly, then we learn that some of our assumptions are invalid (so we still learn something). Concerning this course: I am not expecting you to be an expect in forming mathematical models. For this course and on exams, I expect that: 1. You understand how to do all the homework! If I give you a problem on the exam that is very similar to a homework problem, you must be prepared to show your understanding. 2. Be able to translate a basic statement involving rates and language like in the homework. See my previous posting on applications for practice (and see homework from 1.1, 2.3, 2.5 and throughout the other assignments). 3. If I give an application on an exam that is completely new and involves language unlike homework, then I will give you the differential equation (you won’t have to make up a new model).
    [Show full text]
  • An Introduction to Mathematical Modelling
    An Introduction to Mathematical Modelling Glenn Marion, Bioinformatics and Statistics Scotland Given 2008 by Daniel Lawson and Glenn Marion 2008 Contents 1 Introduction 1 1.1 Whatismathematicalmodelling?. .......... 1 1.2 Whatobjectivescanmodellingachieve? . ............ 1 1.3 Classificationsofmodels . ......... 1 1.4 Stagesofmodelling............................... ....... 2 2 Building models 4 2.1 Gettingstarted .................................. ...... 4 2.2 Systemsanalysis ................................. ...... 4 2.2.1 Makingassumptions ............................. .... 4 2.2.2 Flowdiagrams .................................. 6 2.3 Choosingmathematicalequations. ........... 7 2.3.1 Equationsfromtheliterature . ........ 7 2.3.2 Analogiesfromphysics. ...... 8 2.3.3 Dataexploration ............................... .... 8 2.4 Solvingequations................................ ....... 9 2.4.1 Analytically.................................. .... 9 2.4.2 Numerically................................... 10 3 Studying models 12 3.1 Dimensionlessform............................... ....... 12 3.2 Asymptoticbehaviour ............................. ....... 12 3.3 Sensitivityanalysis . ......... 14 3.4 Modellingmodeloutput . ....... 16 4 Testing models 18 4.1 Testingtheassumptions . ........ 18 4.2 Modelstructure.................................. ...... 18 i 4.3 Predictionofpreviouslyunuseddata . ............ 18 4.3.1 Reasonsforpredictionerrors . ........ 20 4.4 Estimatingmodelparameters . ......... 20 4.5 Comparingtwomodelsforthesamesystem . .........
    [Show full text]
  • Cryptocurrency: the Economics of Money and Selected Policy Issues
    Cryptocurrency: The Economics of Money and Selected Policy Issues Updated April 9, 2020 Congressional Research Service https://crsreports.congress.gov R45427 SUMMARY R45427 Cryptocurrency: The Economics of Money and April 9, 2020 Selected Policy Issues David W. Perkins Cryptocurrencies are digital money in electronic payment systems that generally do not require Specialist in government backing or the involvement of an intermediary, such as a bank. Instead, users of the Macroeconomic Policy system validate payments using certain protocols. Since the 2008 invention of the first cryptocurrency, Bitcoin, cryptocurrencies have proliferated. In recent years, they experienced a rapid increase and subsequent decrease in value. One estimate found that, as of March 2020, there were more than 5,100 different cryptocurrencies worth about $231 billion. Given this rapid growth and volatility, cryptocurrencies have drawn the attention of the public and policymakers. A particularly notable feature of cryptocurrencies is their potential to act as an alternative form of money. Historically, money has either had intrinsic value or derived value from government decree. Using money electronically generally has involved using the private ledgers and systems of at least one trusted intermediary. Cryptocurrencies, by contrast, generally employ user agreement, a network of users, and cryptographic protocols to achieve valid transfers of value. Cryptocurrency users typically use a pseudonymous address to identify each other and a passcode or private key to make changes to a public ledger in order to transfer value between accounts. Other computers in the network validate these transfers. Through this use of blockchain technology, cryptocurrency systems protect their public ledgers of accounts against manipulation, so that users can only send cryptocurrency to which they have access, thus allowing users to make valid transfers without a centralized, trusted intermediary.
    [Show full text]
  • Limitations on the Use of Mathematical Models in Transportation Policy Analysis
    Limitations on the Use of Mathematical Models in Transportation Policy Analysis ABSTRACT: Government agencies are using many kinds of mathematical models to forecast the effects of proposed government policies. Some models are useful; others are not; all have limitations. While modeling can contribute to effective policyma king, it can con tribute to poor decision-ma king if policymakers cannot assess the quality of a given application. This paper describes models designed for use in policy analyses relating to the automotive transportation system, discusses limitations of these models, and poses questions policymakers should ask about a model to be sure its use is appropriate. Introduction Mathematical modeling of real-world use in analyzing the medium and long-range ef- systems has increased significantly in the past fects of federal policy decisions. A few of them two decades. Computerized simulations of have been applied in federal deliberations con- physical and socioeconomic systems have cerning policies relating to energy conserva- proliferated as federal agencies have funded tion, environmental pollution, automotive the development and use of such models. A safety, and other complex issues. The use of National Science Foundation study established mathematical models in policy analyses re- that between 1966 and 1973 federal agencies quires that policymakers obtain sufficient infor- other than the Department of Defense sup- mation on the models (e-g., their structure, ported or used more than 650 models limitations, relative reliability of output) to make developed at a cost estimated at $100 million informed judgments concerning the value of (Fromm, Hamilton, and Hamilton 1974). Many the forecasts the models produce.
    [Show full text]
  • Mathematical Economics - B.S
    College of Mathematical Economics - B.S. Arts and Sciences The mathematical economics major offers students a degree program that combines College Requirements mathematics, statistics, and economics. In today’s increasingly complicated international I. Foreign Language (placement exam recommended) ........................................... 0-14 business world, a strong preparation in the fundamentals of both economics and II. Disciplinary Requirements mathematics is crucial to success. This degree program is designed to prepare a student a. Natural Science .............................................................................................3 to go directly into the business world with skills that are in high demand, or to go on b. Social Science (completed by Major Requirements) to graduate study in economics or finance. A degree in mathematical economics would, c. Humanities ....................................................................................................3 for example, prepare a student for the beginning of a career in operations research or III. Laboratory or Field Work........................................................................................1 actuarial science. IV. Electives ..................................................................................................................6 120 hours (minimum) College Requirement hours: ..........................................................13-27 Any student earning a Bachelor of Science (BS) degree must complete a minimum of 60 hours in natural,
    [Show full text]
  • Course Syllabus ECN101G
    Course Syllabus ECN101G - Introduction to Economics Number of ECTS credits: 6 Time and Place: Tuesday, 10:00-11:30 at VeCo 3 Thursday, 10:00-11:30 at VeCo 3 Contact Details for Professor Name of Professor: Abdel. Bitat, Assistant Professor E-mail: [email protected] Office hours: Monday, 15:00-16:00 and Friday, 15:00-16:00 (Students who are unable to come during these hours are encouraged to make an appointment.) Office Location: -1.63C CONTENT OVERVIEW Syllabus Section Page Course Prerequisites and Course Description 2 Course Learning Objectives 2 Overview Table: Link between MLO, CLO, Teaching Methods, 3-4 Assignments and Feedback Main Course Material 5 Workload Calculation for this Course 6 Course Assessment: Assignments Overview and Grading Scale 7 Description of Assignments, Activities and Deadlines 8 Rubrics: Transparent Criteria for Assessment 11 Policies for Attendance, Later Work, Academic Honesty, Turnitin 12 Course Schedule – Overview Table 13 Detailed Session-by-Session Description of Course 14 Course Prerequisites (if any) There are no pre-requisites for the course. However, since economics is mathematically intensive, it is worth reviewing secondary school mathematics for a good mastering of the course. A great source which starts with the basics and is available at the VUB library is Simon, C., & Blume, L. (1994). Mathematics for economists. New York: Norton. Course Description The course illustrates the way in which economists view the world. You will learn about basic tools of micro- and macroeconomic analysis and, by applying them, you will understand the behavior of households, firms and government. Problems include: trade and specialization; the operation of markets; industrial structure and economic welfare; the determination of aggregate output and price level; fiscal and monetary policy and foreign exchange rates.
    [Show full text]
  • 2020-2021 Bachelor of Arts in Economics Option in Mathematical
    CSULB College of Liberal Arts Advising Center 2020 - 2021 Bachelor of Arts in Economics Option in Mathematical Economics and Economic Theory 48 Units Use this checklist in combination with your official Academic Requirements Report (ARR). This checklist is not intended to replace advising. Consult the advisor for appropriate course sequencing. Curriculum changes in progress. Requirements subject to change. To be considered for admission to the major, complete the following Major Specific Requirements (MSR) by 60 units: • ECON 100, ECON 101, MATH 122, MATH 123 with a minimum 2.3 suite GPA and an overall GPA of 2.25 or higher • Grades of “C” or better in GE Foundations Courses Prerequisites Complete ALL of the following courses with grades of “C” or better (18 units total): ECON 100: Principles of Macroeconomics (3) MATH 103 or Higher ECON 101: Principles of Microeconomics (3) MATH 103 or Higher MATH 111; MATH 112B or 113; All with Grades of “C” MATH 122: Calculus I (4) or Better; or Appropriate CSULB Algebra and Calculus Placement MATH 123: Calculus II (4) MATH 122 with a Grade of “C” or Better MATH 224: Calculus III (4) MATH 123 with a Grade of “C” or Better Complete the following course (3 units total): MATH 247: Introduction to Linear Algebra (3) MATH 123 Complete ALL of the following courses with grades of “C” or better (6 units total): ECON 100 and 101; MATH 115 or 119A or 122; ECON 310: Microeconomic Theory (3) All with Grades of “C” or Better ECON 100 and 101; MATH 115 or 119A or 122; ECON 311: Macroeconomic Theory (3) All with
    [Show full text]
  • International Baccalaureate Diploma Programme Subject Brief Individuals and Societies: Economics—Higher Level First Assessments 2022—Last Assessments 2029
    International Baccalaureate Diploma Programme Subject Brief Individuals and societies: Economics—higher level First assessments 2022—last assessments 2029 The Diploma Programme (DP) is a rigorous pre-university course of study designed for students in the 16 to 19 age range. It is a broad-based two-year course that aims to encourage students to be knowledgeable and inquiring, but also caring and compassionate. There is a strong emphasis on encouraging students MA PROG PLO RAM to develop intercultural understanding, open-mindedness, and the attitudes necessary for DI M IB S IN LANG E UDIE UAG ST LITERAT E them to respect and evaluate a range of points of view. AND URE A IN The course is presented as six academic areas enclosing a central core. Students study E E N D N DG G E D IV A O L E I W X S ID U IT O T O two modern languages (or a modern language and a classical language), a humanities G E U IS N N C N K ES TO T I A U CH EA D E A F A C L O H E T or social science subject, an experimental science, mathematics and one of the creative L Q R S O I I C P N D P G E A Y S A E R arts. Instead of an arts subject, students can choose two subjects from another area. S O S E A Y H It is this comprehensive range of subjects that makes the Diploma Programme a T demanding course of study designed to prepare students effectively for university A P G entrance.
    [Show full text]
  • The Rise and Fall of Walrasian General Equilibrium Theory: the Keynes Effect
    The Rise and Fall of Walrasian General Equilibrium Theory: The Keynes Effect D. Wade Hands Department of Economics University of Puget Sound Tacoma, WA USA [email protected] Version 1.3 September 2009 15,246 Words This paper was initially presented at The First International Symposium on the History of Economic Thought “The Integration of Micro and Macroeconomics from a Historical Perspective,” University of São Paulo, São Paulo, Brazil, August 3-5, 2009. Abstract: Two popular claims about mid-to-late twentieth century economics are that Walrasian ideas had a significant impact on the Keynesian macroeconomics that became dominant during the 1950s and 1060s, and that Arrow-Debreu Walrasian general equilibrium theory passed its zenith in microeconomics at some point during the 1980s. This paper does not challenge either of these standard interpretations of the history of modern economics. What the paper argues is that there are at least two very important relationships between Keynesian economics and Walrasian general equilibrium theory that have not generally been recognized within the literature. The first is that influence ran not only from Walrasian theory to Keynesian, but also from Keynesian theory to Walrasian. It was during the neoclassical synthesis that Walrasian economics emerged as the dominant form of microeconomics and I argue that its compatibility with Keynesian theory influenced certain aspects of its theoretical content and also contributed to its success. The second claim is that not only did Keynesian economics contribute to the rise Walrasian general equilibrium theory, it has also contributed to its decline during the last few decades. The features of Walrasian theory that are often suggested as its main failures – stability analysis and the Sonnenschein-Mantel-Debreu theorems on aggregate excess demand functions – can be traced directly to the features of the Walrasian model that connected it so neatly to Keynesian macroeconomics during the 1950s.
    [Show full text]
  • Nine Lives of Neoliberalism
    A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Plehwe, Dieter (Ed.); Slobodian, Quinn (Ed.); Mirowski, Philip (Ed.) Book — Published Version Nine Lives of Neoliberalism Provided in Cooperation with: WZB Berlin Social Science Center Suggested Citation: Plehwe, Dieter (Ed.); Slobodian, Quinn (Ed.); Mirowski, Philip (Ed.) (2020) : Nine Lives of Neoliberalism, ISBN 978-1-78873-255-0, Verso, London, New York, NY, https://www.versobooks.com/books/3075-nine-lives-of-neoliberalism This Version is available at: http://hdl.handle.net/10419/215796 Standard-Nutzungsbedingungen: Terms of use: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Documents in EconStor may be saved and copied for your Zwecken und zum Privatgebrauch gespeichert und kopiert werden. personal and scholarly purposes. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle You are not to copy documents for public or commercial Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich purposes, to exhibit the documents publicly, to make them machen, vertreiben oder anderweitig nutzen. publicly available on the internet, or to distribute or otherwise use the documents in public. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, If the documents have been made available under an Open gelten abweichend von diesen Nutzungsbedingungen die in der dort Content Licence (especially Creative
    [Show full text]
  • THE NEOLIBERAL THEORY of SOCIETY Simon Clarke
    THE NEOLIBERAL THEORY OF SOCIETY Simon Clarke The ideological foundations of neo-liberalism Neoliberalism presents itself as a doctrine based on the inexorable truths of modern economics. However, despite its scientific trappings, modern economics is not a scientific discipline but the rigorous elaboration of a very specific social theory, which has become so deeply embedded in western thought as to have established itself as no more than common sense, despite the fact that its fundamental assumptions are patently absurd. The foundations of modern economics, and of the ideology of neoliberalism, go back to Adam Smith and his great work, The Wealth of Nations. Over the past two centuries Smith’s arguments have been formalised and developed with greater analytical rigour, but the fundamental assumptions underpinning neoliberalism remain those proposed by Adam Smith. Adam Smith wrote The Wealth of Nations as a critique of the corrupt and self-aggrandising mercantilist state, which drew its revenues from taxing trade and licensing monopolies, which it sought to protect by maintaining an expensive military apparatus and waging costly wars. The theories which supported the state conceived of exchange as a ‘zero-sum game’, in which one party’s gain was the other party’s loss, so the maximum benefit from exchange was to be extracted by force and fraud. The fundamental idea of Smith’s critique was that the ‘wealth of the nation’ derived not from the accumulation of wealth by the state, at the expense of its citizens and foreign powers, but from the development of the division of labour. The division of labour developed as a result of the initiative and enterprise of private individuals and would develop the more rapidly the more such individuals were free to apply their enterprise and initiative and to reap the corresponding rewards.
    [Show full text]
  • Introduction to Mathematical Modeling Difference Equations, Differential Equations, & Linear Algebra (The First Course of a Two-Semester Sequence)
    Introduction to Mathematical Modeling Difference Equations, Differential Equations, & Linear Algebra (The First Course of a Two-Semester Sequence) Dr. Eric R. Sullivan [email protected] Department of Mathematics Carroll College, Helena, MT Content Last Updated: January 8, 2018 1 ©Eric Sullivan. Some Rights Reserved. This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. You may copy, distribute, display, remix, rework, and per- form this copyrighted work, but only if you give credit to Eric Sullivan, and all deriva- tive works based upon it must be published under the Creative Commons Attribution- NonCommercial-Share Alike 4.0 United States License. Please attribute this work to Eric Sullivan, Mathematics Faculty at Carroll College, [email protected]. To view a copy of this license, visit https://creativecommons.org/licenses/by-nc-sa/4.0/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, Cali- fornia, 94105, USA. Contents 0 To the Student and the Instructor5 0.1 An Inquiry Based Approach...........................5 0.2 Online Texts and Other Resources........................7 0.3 To the Instructor..................................7 1 Fundamental Notions from Calculus9 1.1 Sections from Active Calculus..........................9 1.2 Modeling Explorations with Calculus...................... 10 2 An Introduction to Linear Algebra 16 2.1 Why Linear Algebra?............................... 16 2.2 Matrix Operations and Gaussian Elimination................. 20 2.3 Gaussian Elimination: A First Look At Solving Systems........... 23 2.4 Systems of Linear Equations........................... 28 2.5 Linear Combinations............................... 35 2.6 Inverses and Determinants............................ 38 2.6.1 Inverses................................... 40 2.6.2 Determinants..............................
    [Show full text]