| CODE | ECN3210 | |||||||||
| TITLE | Applied Mathematical Economics | |||||||||
| UM LEVEL | 03 - Years 2, 3, 4 in Modular Undergraduate Course | |||||||||
| MQF LEVEL | 6 | |||||||||
| ECTS CREDITS | 4 | |||||||||
| DEPARTMENT | Economics | |||||||||
| DESCRIPTION | 1. Equilibrium Analysis: Partial Market Equilibrium Linear/Non- Linear Economic Models Equilibrium in National Income Determination Models 2. Comparative Statics and the Derivative Concept: The Nature of Comparative Statics Rate of Change, the Slope of a Curvilinear Function & the Derivative Limits, Continuity & Differentiability of Functions Higher-Order Derivatives 3. Rules of Differentiation: Single Function of the Same Variable Several Functions of the Same Variable Inverse Functions and their rules 4. Multivariable Calculus I: Techniques of Partial Differentiation, Higher-Order Partial Derivatives Applications to Comparative Static Analysis: Single Commodity Market Model, Determination of Multipliers in National Income Model Income & Cross-Price Elasticities 5. Multi-Variable Calculus II: Total & Partial Differentials Analysis of Incremental Changes in Economics: National Income Equations Cost & Utility Functions Total Derivatives, Implicit Functions & their Derivatives Comparative Statics of General-Function Models: The Market Model Simultaneous Equation Approach IS-LM National Income Model 6. Exponential and Logarithmic Functions: Properties & Solutions of Equations Logarithmic Transformation of Non-Linear Functions Interest Compounding Effective versus Nominal Interest Rates Growth Rates & Discounting Derivatives (including Higher-Order and Partial Derivatives) of Exponential & Logarithmic Functions and their Applications in Economics Logarithmic Differentiation and Alternative Measures of Growth Evaluation of Point Elasticity 7. Optimisation of Economic Functions: Increasing & Decreasing Functions Concavity & Convexity Absolute & Relative Extrema, Inflection Points Successive Derivative Tests Total, Marginal & Average Concepts, and their Interrelationship Optimal Timing Problems for Exponential and Logarithmic Functions 8. Optimisation of Multi-Variable Functions: Use of the 1st and 2nd-Order Partial Derivatives The 1st and 2nd-Order Conditions for Total Differentials Polynomial (Quadratic) Forms and Total Differentials Economic Applications: Problem of a Multi-Product Firm Price Discrimination, A Firm's Input Decisions Comparative Static Aspects of Optimisation-Reduced Form Solutions and General Function Models 9. Optimisation with Equality Constraints: The Lagrangian Function-Constrained Optimisation with Lagrange Multipliers Economic Applications of Constrained Optimisation of Multi-Variable Functions: Utility Maximisation & Consumer Demand Homogenous Production Functions and Returns to Scale The Cobb-Douglas Production Function Least-Cost Combination of Inputs & The Expansion Path Homothetic Functions & The Constant Elasticity of Substitution (CES) Production Function The Cobb-Douglas Function as a Special Case of the CES Function 10. Advanced Topics in Optimisation: Non-Linear Programming & Kuhn-Tucker Conditions Effect of Nonnonnegativity Restrictions & Inequality Constraints Boundary Irregularities and the Constraint Qualification Economic Applications: War-time Rationing Peak-Load Pricing Sufficiency Theorems and Nonlinear Programming - Concave and Quasi-Concave Programming A Constraint Qualification Test Maximum Value Functions & The Envelope Theorem The Envelope Theorem for Constrained & Unconstrained Optimisation The Profit Function The Significance and Interpretation of The Lagrange Multiplier Duality and The Envelope Problem - The Primal and Dual Problems Roy's Identity & Shepard's Lemma 11. Fundamentals of Linear (Matrix) Algebra: Matrices & Vectors Matrix & Vector Operations Scalar Multiplication Commutative, Associative & Distributive Laws in Matrix Algebra Special Types of Matrices - Identity, Null, Transpose & Inverse Matrices The Inverse Matrix and Solution of a Linear Equation System 12. Linear Models and Matrix Algebra (contd): Determinants and Non-Singularity Evaluating nth Order Determinants by Laplace Expansion Use of Minors & Cofactors Basic Properties of Determinants Matrix Inversion & Adjoint Matrices Solving Linear Equation Systems using the Inverse Matrix Cramer's Rule for Matrix Solutions Solution Outcomes for Homogenous & Nonhomogenous Linear Equation Systems; Economic Applications: Two-commodity Market Model National Income Model, IS-LM Model in a closed economy Leontief Input-Output Model 13. Special Determinants and Matrices and their use in Economics: Comparative Statics, Partial Derivatives and Input-Output Analysis The Jacobian, the Hessian, and the Discriminant Higher-Order Hessians The Bordered Hessian for Constrained Optimisation Characteristic Roots & Vectors - Eigenvalues & Eigenvectors 14. Economic Dynamics and Integral Calculus: Indefinite Integrals - Basic Rules of Integration Rules of Operation - Integration by Substitution & by Parts Meaning & Properties of Definite Integrals The Definite Integral as the Area under a Curve Improper Integrals Economic Applications: From a Marginal to a Total Cost Function e.g. Estimation of the Savings Function from a Marginal Propensity to Save (MPS) Function Estimation of the Total Cost Function from a Marginal Cost (MC) Function Investment & Capital Formation Present Value of a Cash-Flow Framework and Solution of Domar Growth Model Consumers' & Producers' Surplus 15. Continuous Time - First-Order Differential Equations: General Formula for First-Order Linear Differential Equations Using a Constant Coefficient and Constant Term The Homogenous and Non-Homogenous Case Verification of Solution Dynamics of Market Price: The Framework, Time-Path and the Dynamic Stability of Equilibrium Using a Variable Coefficient and Variable Term The Homogenous and Non-Homogenous Case Exact Differential Equations and Partial Integration Integrating Factor & Solution of First-Order Differential Equation Non-Linear Differential Equations of the First-Order and the First Degree: Exact Differential Equations - Separable of Variables - Equations reducible to the Linear Form Economic Applications: Qualitative-Graphic Analysis The Phase Diagrams Types of Time-Path The Framework of the Solow Growth Model - Qualitative & Quantitative Analysis 16. Higher-Order Differential Equations: Second-Order Linear Differential Equations with Constant Coefficients & Constant Term The Particular Integral and the Complementary Function The Dynamic Stability of Equilibrium Characteristic Roots Cojugate Complex Numbers/Complex Roots Alternative Representations of Complex Numbers - Transformation of Imaginary and Complex Numbers Analysis of the Complex Root Case: The Complementary Function and an example of a Solution The Time Path and the Dynamic Stability of Equilibrium Economic Applications: A Market Model with Price Expectations - Price Trend and Price Expectations, The Time Path of Price The Interaction of Inflation and Unemployment - The Phillips Relation, The Expectations-Augmented Phillips Relation, The Feedback from Inflation to Unemployment, The Time Path of the Expected Rate of Inflation Differential Equations with a Variable Term Method of Undetermined Coefficients and related modification Higher-Order Linear Differential Equations Finding the Solution Convergence and the Routh Theorem 17. Discrete Time - First-Order Difference Equations: Solving a First-Order Difference Equation Iterative Method General Method/Formula for a First-Order Difference Equation Dynamic Stability Conditions: Significance & the Role of the various Parameters Convergence towards Equilibrium Economic Applications: Lagged Income Determination Model The Harrod Growth Model A Market Model with Inventory Non-linear Difference Equations - the Qualitative-Graphic Approach and the use of Phase Diagrams Types of Time Path A Market with a Price Ceiling 18. Higher-Order Difference Equations: Second Order Linear Difference Equations with Constant Coefficients & Constant Term Particular Solution Complementary Function The Convergence of the Time Path Samuelson Multiplier-Acceleration Interaction Model: The Framework The Solution Convergence versus Divergence A Graphical Summary Inflation and Unemployment in Discrete Time: The Model The Difference Equation in the endogenous variable-the actual inflation rate The Time Path of the actual inflation rate The Analysis of the Unemployment Rate The Long-Run Phillips Relation Generalisations to Variable-Term and Higher-Order Equations: Different forms of the Variable Term Higher-Order Linear Difference Equations Convergence and the Schur Theorem 19. Simultaneous Differential and Difference Equations: The Genesis of Dynamic Systems: Interacting Patterns of Change The Transformation of a High-Order Dynamic Equation Solving Simultaneous Dynamic Equations: Matrix Solution of Simultaneous Differential Equations Matrix Solution of Simultaneous Difference Equations Further Comments on the Characteristic Equation Dynamic Input-Output Models: Time Lag in Production Excess Demand and Output Adjustment Capital Formation The Inflation-Unemployment Model Revisited: Simultaneous Differential Equations and their Solution Paths Simultaneous Difference Equations and their Solution Paths Stability and Phase Diagrams for Simultaneous Differential Equations: Two-Variable Phase Diagrams - the Phase Space The Demarcation Curves Streamlines Types of Equilibrium Inflation and Monetary Rule a' la Obst Linearisation of a Nonlinear Differential Equation System: Taylor Expansion and Linearisation The Reduced Linearisation Local Stability Analysis 20. The Calculus of Variations: Dynamic Optimisation Distance Between Two Points on a Plane Euler's Equation and the Necessary Condition for Dynamic Optimisation Finding Candidates for Extremals The Sufficiency Conditions for the Calculus of Variations Dynamic Optimisation Subject to Functional Constraints Variational Notation Applications to Economics 21. Optimal Control Theory: The Nature of Optimal Control and Terminology Illustration using a Simple Macroeconomic Model Pontryagin's Maximum Principle The Hamiltonian and the Necessary Conditions for Maximisation Sufficiency Conditions for Maximisation in Optimal Control Alternative Terminal Conditions: Fixed Terminal Point and Free End Point Inequality Constraints in the Endpoints Autonomous Problems Economic Applications: Lifetime Utility Maximisation Exhaustible Resource Infinite Time Horizon: Neoclassical Optimal Growth Model The Current-Value Hamiltonian Constructing a Phase Diagram Analysing a Phase Diagram Study-unit Aims An even progression in the level of material is to be ensured throughout the whole duration of this study-unit so as to induce students' motivation in tackling more advanced mathematical issues during the later stages of the unit. A portfolio of mathematical techniques is built upon a carefully graduated schedule designed with the idea that elementary tools serve as stepping stones to more complex analytical tools discussed later on in this study-unit. Wherever possible, graphic illustrations emphasise visual reinforcement to the algebraic results. The exercises and problems which will be assigned continuously to the students are designed as drills to help solidify grasp of content and bolster confidence, rather than exact challenges that may unwittingly frustrate and intimidate the novice. The pedagogy of the course is also constructed to incorporate fundamental economic concepts at each stage of the course, underscoring the importance and of economic applications and contributing to a holistic understanding of the subject. Demonstrations of economic relevance will therefore ensure that student motivation is not impaired. By underlining the importance of the appropriate mathematical methods to the relevant economic models, the student of economics becomes equipped with a comprehensive mathematical tool kit; the latter, besides contributing to a more deeper and synergistic understanding of topics covered in other finance and economics study-units, will also sufficiently equip the student to tackle more difficult material at Honours Level (and perhaps even for eventual postgraduate study) Learning Outcomes 1. Knowledge & Understanding: By the end of the study-unit the student will be able to: - have a firm grasp of a wide array of mathematical topics economists need to master today, such as linear algebra, differential and integral calculus, nonlinear programming,differential and difference equations, the calculus of variations and optimal control theory - apply the knowledge they have gained to gain a better understanding of other economics and finance topics in their programme of studies, both at the general degree level as well as at Honours and postgraduate level - appreciate that mathematics is a fundamental language of economics and finance, and that through mastering this language, students will be more capable of both communicating effectively economic concepts; also, expertise in understanding mathematical expressions will increase the exposure of students to the more recent and rather complex research literature which would otherwise be rendered incomprehensible to those not adequately trained in mathematical and analytical techniques. 2. Skills: By the end of the study-unit the student will be able to: - express economic issues and applications in terms of the relevant mathematical notation and equations, both within this subject as well as in other topics of economics and finance which easily lend themselves to mathematical expression. - manipulate effectively mathematical expressions in accordance with the dynamic requirements of the economic/financial system under investigation. - interpret in a coherent and consistent manner mathematical expressions whenever they crop up in economic literature - apply their grasp of numeracy and intuitive knowledge of analytical techniques to their work during the rest of their academic studies, whether this consists of assignments, dissertations, etc. Main Text/s and any supplementary readings Main Text Alpha C. Chiang & Kevin Wainwright, Fundamental Methods of Mathematical Economics, McGraw-Hill International Edition 2005 Supplementary Text Michael Hoy, John Livernois, Chris McKenna, Ray Rees, Thanasis Stengos, Mathematics for Economics Massachusetts Institute of Technology (MIT) Press 2001 |
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| ADDITIONAL NOTES | Pre-Requisite qualifications: Successful completion of Years 1 & 2 of B.Com | |||||||||
| STUDY-UNIT TYPE | Lecture, Independent Study & Tutorial | |||||||||
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| LECTURER/S | Ian P. Cassar |
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The University makes every effort to ensure that the published Courses Plans, Programmes of Study and Study-Unit information are complete and up-to-date at the time of publication. The University reserves the right to make changes in case errors are detected after publication.
The availability of optional units may be subject to timetabling constraints. Units not attracting a sufficient number of registrations may be withdrawn without notice. It should be noted that all the information in the description above applies to study-units available during the academic year 2025/6. It may be subject to change in subsequent years. |
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