# Virtual University of Pakistan

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## ECO606 : Mathematical Economics I

### Course Overview

#### Course Synopsis

Mathematics is as old as man. Mathematical economics is an interesting approach to economics rather than its branch. It can include problems from microeconomics, macroeconomics, international economics, public finance etc. Mathematics for economics is comprehension of mathematical tools applied to economic situations. These tools include application of algebra on market equilibrium, application of matrices on national income analysis, application of differentiation on marginal utility/cost/product. Mathematical economics use symbols, numerical numbers, equations, identities, trigonometric ratios; calculus etc. that are more convenient to use in deductive reasoning. The use of symbols makes results concise and precise. Mathematical economics give liberty to use more and more variables in one situation. During current time, economics is highly mathematized. Lot of mathematics is now used in latest literature and research journals of economics.

#### Course Learning Outcomes

At the end of the course, students should be able to:

• Know about the introduction of mathematical economics, difference between mathematical & non-mathematical economics and difference between mathematical economics versus econometrics.
• Recognize variables and economic variables, constants and parameters, logic in mathematical economics and relevance of real-numbers in mathematical economics.
• Know set notation, set operators in economics, laws of set operations and Cartesian coordinates in economics.
• Understand functions, domains and ranges in a functions, constant functions, linear functions, quadratic functions, cubic functions, rational functions, rectangular hyperbolic functions, non-algebraic exponential functions, non-algebraic logarithmic functions and inverse functions.
• Differentiate between equations and identities in economics, definitional equations, behavioral equations and conditional equations in economics.
• Construct a partial equilibrium in linear market model and non-linear market model and general equilibrium.
• Understand matrices and their types, array and summation forms of matrices, matrices operations, matrix division, commutative, associative, and distributive laws, vector operations, transpose of a matrix, cofactors of a matrix, adjoint of a matrix, determinant of a matrix, inverse of a matrix, Cramer’s rule, input coefficient matrix etc.
• Know the concept of limit and continuity, rate of change, slope & derivatives.
• Know about rules of differentiation and to apply those rules on various economics concepts.
• Comprehend the concept of partial differential and their application on various economics concepts.
• Comprehend higher order derivatives and their application to economic situations.
• Learn about Young theorem and its presence in 2nd order conditions of differentiation.
• Learn the logic of envelope theorem, Roy's identity, Hotelling's lemma and Shephard's lemma in economics.
• Compare the differentials versus derivatives and their applications.
• Learn the concept of total differentials and their applications.
• Comprehend the concept of total derivatives and their applications.
• Comprehend the concept of implicit differentiation and their applications.
• Calculate instantaneous rate of growth, continuous vs. discrete growth,
• Understand discounting and negative growth continuous compounding, applications of instantaneous growth.
• Learn logarithms, their meaning and types, laws of logarithms etc.
• Transform CES production function using logarithms, rules of differentiation of exponential and logarithmic functions.
• Learn about concept of optimization, 1st test and 2nd test in the process of optimization, profit maximization analysis for various economic applications.
• Know about the rationale for constrained optimization, 2nd order condition: the bordered hessian, utility maximization and consumer demand.
• Apply optimization on production function, log-transformed CD production function, homogeneity of functions, homothetic functions etc.
• Recognize CES production function, homogeneity of CES production function, Euler's theorem and their applications etc.

#### Course Calendar

 Topic Lecture Resource Page Demystifying Mathematics and Mathematical Economics. 1 Lesson Video Topic 001 Mathematical versus Nonmathematical Economics 1 Lesson Video Topic 002 Mathematical Economics versus Econometrics. 1 Lesson Video Topic 003 Ingredients of a Mathematical Model: Variables and Economic Variables. 2 Lesson Video Topic 004 Ingredients of a Mathematical Model: Constants and Parameters 2 Lesson Video Topic 005 A Few Aspects of Logic: Propositions, Implications and Necessary and Sufficient Conditions. 2 Lesson Video Topic 006 The Real-Number System 2 Lesson Video Topic 007 Use of Sets in economics: Set Notation. 3 Lesson Video Topic 008 Use of Sets in economics: Operations of Sets. 3 Lesson Video Topic 009 Use of Sets in economics: Laws of Operations of Sets. 3 Lesson Video Topic 010 Cartesian Coordinates 3 Lesson Video Topic 011 What are Functions? 4 Lesson Video Topic 012 Domain and Range in a Function 4 Lesson Video Topic 013 Difference between Function and Relations. 4 Lesson Video Topic 014 Economic Example of Schultz Demand Function 4 Lesson Video Topic 015 Economic Example of Cost Function of Cleaning Impurities from a Lake 4 Lesson Video Topic 016 Economic Example of Function: Laffer Curve 4 Lesson Video Topic 017 Types of Functions: Constant Functions. 5 Lesson Video Topic 018 Types of Functions: Polynomial Function: Linear Functions. 5 Lesson Video Topic 019 Interpreting Linear Economic Functions 5 Lesson Video Topic 020 Applications of Linear Functions: Population and Consumption Functions. 5 Lesson Video Topic 021 Types of Functions: Polynomial Function: Quadratic Functions. 6 Lesson Video Topic 022 Quadratic Cost function and Profit Function of a Monopoly. 6 Lesson Video Topic 023 Quadratic Function and Production Possibilities Frontier 6 Lesson Video Topic 024 Types of Functions: Polynomial Function: Cubic Functions. 6 Lesson Video Topic 025 Cubic Cost functions. 6 Lesson Video Topic 026 Rational Functions 7 Lesson Video Topic 027 Other Types: Rectangular Hyperbolic Function 7 Lesson Video Topic 028 Other Types: Non-algebraic Exponential Functions 7 Lesson Video Topic 029 Population Growth using General Exponential Functions 7 Lesson Video Topic 030 Other Types: Non-algebraic Natural Exponential Functions 7 Lesson Video Topic 031 Population Growth using Natural Exponential Functions 7 Lesson Video Topic 032 Other Types: Non-algebraic Logarithmic Functions 8 Lesson Video Topic 033 Other Types: Non-algebraic Natural Logarithmic Functions 8 Lesson Video Topic 034 Rate of Growth of GNP using Logarithmic Functions 8 Lesson Video Topic 035 Inverse Functions 8 Lesson Video Topic 036 Functions with Two or More Independent Variables 8 Lesson Video Topic 037 Surfaces and Distance in Graphs of two or more independent variables. 8 Lesson Video Topic 038 QUIZ 01 Equations and Identities. 9 Lesson Video Topic 039 Types of equations in Economics: Definitional Equations. 9 Lesson Video Topic 040 Fiscal Surplus and Fiscal Deficit Using Equations 9 Lesson Video Topic 041 Types of equations in Economics: Behavioral Equations 9 Lesson Video Topic 042 Types of equations in Economics: Conditional Equation 9 Lesson Video Topic 043 Structural and Reduced Form Equations 9 Lesson Video Topic 044 Constructing a Partial Linear Market Equilibrium 10 Lesson Video Topic 045 Solving using Elimination of Variable Method 10 Lesson Video Topic 046 Shifts in Demand in Market Equilibrium 10 Lesson Video Topic 047 Shifts in Supply in Market Equilibrium 10 Lesson Video Topic 048 Effect of Tax on Producer on Partial Market Equilibrium. 10 Lesson Video Topic 049 Effect of Tax on Consumer on Partial Market Equilibrium. 10 Lesson Video Topic 050 Partial Market Equilibrium-A Nonlinear Model. 11 Lesson Video Topic 051 General Market Equilibrium: General form of Two Good Case. 11 Lesson Video Topic 052 General Market Equilibrium: Numerical solution of Two Good Case. 11 Lesson Video Topic 053 General Market Equilibrium: n-Good Case. 11 Lesson Video Topic 054 National Income Equilibrium 11 Lesson Video Topic 055 National Income Equilibrium with Induced and Autonomous Tax 11 Lesson Video Topic 056 National Income Equilibrium with Proportion of Government Expenditure 11 Lesson Video Topic 057 Matrices and Vectors 12 Lesson Video Topic 058 Matrices Operations 12 Lesson Video Topic 059 Using Product of Matrices to Calculate Total Cost 12 Lesson Video Topic 060 Using Product of Matrices to Calculate Total Revenue and Profit 12 Lesson Video Topic 061 Question of Matrix Division. 12 Lesson Video Topic 062 Commutative, Associative, and Distributive Laws. 13 Lesson Video Topic 063 Vector Operations 13 Lesson Video Topic 064 Transpose of a Matrix 13 Lesson Video Topic 065 Cofactors of a Matrix 13 Lesson Video Topic 066 Adjoint of a Matrix 13 Lesson Video Topic 067 Determinant of a Matrix 14 Lesson Video Topic 068 Sarrus''s rule for 3x3 order Determinant of a Matrix 14 Lesson Video Topic 069 Inverse of a Matrix 14 Lesson Video Topic 070 Condition(s) for Non-singularity 14 Lesson Video Topic 071 Expression of National Income using Matrix Form 14 Lesson Video Topic 072 Minor and Cofactors. 14 Lesson Video Topic 073 Market Model Analysis Using Matrix Inversion Method. 15 Lesson Video Topic 074 National Income Analysis Using Matrix Inversion Method. 15 Lesson Video Topic 075 Equilibrium Prices using Matrix Inversion Method 15 Lesson Video Topic 076 ASSIGNMENT Solving Market Model using Cramer''s Rule. 16 Lesson Video Topic 077 Equilibrium Prices using Cramer''s Rule 16 Lesson Video Topic 078 National Income Determination using Cramer''s Rule. 16 Lesson Video Topic 079 Input Coefficient Matrix 17 Lesson Video Topic 080 Economic Meaning of Hawkins-Simon Condition 17 Lesson Video Topic 081 Input-Output Analysis in case of Open Economy. 17 Lesson Video Topic 082 Input-Output Analysis in case of Closed Economy. 17 Lesson Video Topic 083 The Need and Nature of Comparative Statics. 17 Lesson Video Topic 084 MID TERM EXAM Concept of Limit and continuity. 18 Lesson Video Topic 085 Rate of Change, Slope & Derivative. 18 Lesson Video Topic 086 Differentiation Rules for Single variable functions: Constant Function Rule and Power Function Rule. 19 Lesson Video Topic 087 Sum-Difference Rule of Differentiation. 19 Lesson Video Topic 088 Sum-Difference Rule: Numerical Analysis of Cost Function 19 Lesson Video Topic 089 Understanding graphs of Function and its derivatives. 19 Lesson Video Topic 090 Product Rule of Differentiation. 20 Lesson Video Topic 091 Relationship between Average Revenue and Marginal Revenue using Product Rule. 20 Lesson Video Topic 092 Quotient Rule of Differentiation. 20 Lesson Video Topic 093 Marginal Propensity to Consume via Differentiation with and without Tax 20 Lesson Video Topic 094 Relationship Between Marginal-Cost and Average-Cost Functions using Quotient Rule. 20 Lesson Video Topic 095 Variable and Fixed Cost components in total cost function. 21 Lesson Video Topic 096 Obtaining Marginal Cost function from Average Cost function 21 Lesson Video Topic 097 Marginal Cost Analysis 21 Lesson Video Topic 098 Marginal Revenue Analysis 21 Lesson Video Topic 099 Marginal Product Analysis 21 Lesson Video Topic 100 Rules of Differentiation: Functions with different variables: Chain Rule. 22 Lesson Video Topic 101 Marginal Revenue Product of Labor (MRPL) Analysis. 22 Lesson Video Topic 102 Marginal Analysis of Fishery Production Function 22 Lesson Video Topic 103 Inverse Function Rule 22 Lesson Video Topic 104 ASSIGNMENT Partial Differentiation: The Concept. 23 Lesson Video Topic 105 Marginal Physical Product of Labor and Capital using Partial Derivatives. 23 Lesson Video Topic 106 Marginal Utility functions using Partial Derivatives. 23 Lesson Video Topic 107 Output Elasticity of Labor and Capital using Partial Derivatives. 23 Lesson Video Topic 108 Money Market Analysis using Partial Derivatives. 23 Lesson Video Topic 109 Partial Market Model using Partial Differentiation. 24 Lesson Video Topic 110 National-Income Model using Partial Differentiation. 24 Lesson Video Topic 111 Second and Higher Order Derivatives 24 Lesson Video Topic 112 Economic Applications of Second Derivative: Profit Maximization Condition. 24 Lesson Video Topic 113 Numerical Example of Profit Maximization Condition using Second Derivative 24 Lesson Video Topic 114 Young''s Theorem 25 Lesson Video Topic 115 Demand for Money Function Analysis using Partial Derivatives. 25 Lesson Video Topic 116 Income Elasticity of Demand using Partial Derivatives 25 Lesson Video Topic 117 Cross Price Elasticity of Demand using Partial Derivatives 25 Lesson Video Topic 118 Partial Derivatives Application on Herring Production Function 25 Lesson Video Topic 119 Partial Derivatives Application on Three Input Production Function 25 Lesson Video Topic 120 Envelope Theorem 26 Lesson Video Topic 121 Roy''s Identity 26 Lesson Video Topic 122 Hotelling''s Lemma 26 Lesson Video Topic 123 Shephard Lemma 26 Lesson Video Topic 124 Differentials versus Derivatives 27 Lesson Video Topic 125 Point Elasticity using Differentials. 27 Lesson Video Topic 126 Elasticity of Rectangular Hyperbolic Demand Curve. 27 Lesson Video Topic 127 Income and Price Elasticity of Demand using Differentials. 27 Lesson Video Topic 128 Income Elasticity of Consumption using Differentials. 27 Lesson Video Topic 129 Income and Price Elasticity of Import Function using Differentials. 27 Lesson Video Topic 130 Output Elasticity of Cost 27 Lesson Video Topic 131 Concept of Total Differentials. 28 Lesson Video Topic 132 Savings Function and Total Differentials. 28 Lesson Video Topic 133 General Utility Function and Total Differentials. 28 Lesson Video Topic 134 Specific Utility Function and Total Differentials. 28 Lesson Video Topic 135 Price and Rain Elasticity of Supply using Total Differentials. 28 Lesson Video Topic 136 Local Price Elasticity of Foreign Demand of Exports using Total Differentials. 28 Lesson Video Topic 137 Concept of Total Derivatives 29 Lesson Video Topic 138 Complementarity between Coffee and Sugar Using total Derivative 29 Lesson Video Topic 139 General Production Function with time-dependent Labor and Capital. 29 Lesson Video Topic 140 Specific Production Function with time-dependent Labor and Capital. 29 Lesson Video Topic 141 Concept of Implicit Differentiation. 30 Lesson Video Topic 142 Production Function Analysis using Implicit Differentiation. 30 Lesson Video Topic 143 Marginal Rate of Technical Substitution using Implicit Differentiation 30 Lesson Video Topic 144 Marginal Utilities and Marginal Rate of Substitution using Implicit Differentiation. 30 Lesson Video Topic 145 Nerlove-Ringstad Production Function using Implicit Differentiation. 30 Lesson Video Topic 146 Marginal Products of Three Input Logarithmic Production Function 30 Lesson Video Topic 147 ASSIGNMENT Exponential Functions and Growth 31 Lesson Video Topic 148 Instantaneous Rate of Growth 31 Lesson Video Topic 149 Numerical Examples of Instantaneous Rate of Growth 31 Lesson Video Topic 150 Continuous vs Discrete Growth 31 Lesson Video Topic 151 Discounting and Negative Growth 31 Lesson Video Topic 152 Applications of Continuous Compounding 31 Lesson Video Topic 153 Logarithms: Meaning and Types 32 Lesson Video Topic 154 Laws of Logarithms 32 Lesson Video Topic 155 Laws of Logarithms for Transformation of CES Production Function 32 Lesson Video Topic 156 Rules of Differentiation of Exponential and Logarithmic Functions. 33 Lesson Video Topic 157 Optimal Timing: A Problem of Wine Storage 33 Lesson Video Topic 158 Optimal Timing: A Problem of Timber Cutting 33 Lesson Video Topic 159 Optimal Timing: Land Purchase for Speculation 33 Lesson Video Topic 160 Optimal Timing: Art Collection 33 Lesson Video Topic 161 Optimal Timing: Diamond Purchase 33 Lesson Video Topic 162 Finding the Rate of Growth using Exponential and Logarithmic Functions. 34 Lesson Video Topic 163 Growth of Exports of a Country 34 Lesson Video Topic 164 Finding the Point Elasticity 34 Lesson Video Topic 165 Rates of Growth of Population, Consumption, and Per Capita Consumption 34 Lesson Video Topic 166 Rate of Growth of Per Capita Employment 34 Lesson Video Topic 167 Rate of Growth of Export Earnings of a Country 34 Lesson Video Topic 168 Rate of Growth of Sales 34 Lesson Video Topic 169 Rate of Growth of Profit 34 Lesson Video Topic 170 Concept of Optimization 35 Lesson Video Topic 171 Calculus Approach to Optimization: 1st Order Test. 35 Lesson Video Topic 172 Average Cost Analysis 35 Lesson Video Topic 173 Calculus Approach to Optimization: 2nd Order Test 35 Lesson Video Topic 174 Matrix Approach to Optimization: 2nd Order Test - Hessian 35 Lesson Video Topic 175 Profit Maximization Analysis 36 Lesson Video Topic 176 Numerical Example of Profit Maximization 36 Lesson Video Topic 177 Profit Maximization of Technically Related Goods 36 Lesson Video Topic 178 Profit Maximization of Monopolistic Firm producing Related Goods 36 Lesson Video Topic 179 Profit Maximization of Firm producing Substitute Goods 36 Lesson Video Topic 180 Marginal and Average Revenue Analysis 37 Lesson Video Topic 181 Short Run Production Function Analysis 37 Lesson Video Topic 182 Total Cost, Total Revenue and Profit Maximization 37 Lesson Video Topic 183 Quadratic Profit Function Analysis 38 Lesson Video Topic 184 Optimization of Exponential Revenue Function 38 Lesson Video Topic 185 Optimization of More than one Choice variable. 38 Lesson Video Topic 186 Economic Application on Multi-product Firm 38 Lesson Video Topic 187 Economic Application on Multi-plant Firm 38 Lesson Video Topic 188 Price Discrimination by Monopoly 39 Lesson Video Topic 189 Price Discrimination by Monopsony 39 Lesson Video Topic 190 Input Decision of a Firm 39 Lesson Video Topic 191 Profit Maximization of two-product firm. 39 Lesson Video Topic 192 FINAL TERM EXAM Comparative-Static Aspects of Optimization 40 Lesson Video Topic 193 Rationale for Constrained Optimization. 40 Lesson Video Topic 194 Finding Stationary Values using Substitution/Elimination Method. 40 Lesson Video Topic 195 Finding Stationary Values using Method of Lagrange Multiplier. 40 Lesson Video Topic 196 Interpretation of the Lagrange Multiplier. 40 Lesson Video Topic 197 Second Order Condition: The Bordered Hessian 40 Lesson Video Topic 198 Two Period Model of Utility. 41 Lesson Video Topic 199 Convexity and Concavity using Second Order Derivative. 41 Lesson Video Topic 200 Utility Maximization and Consumer Demand: First Order Condition 41 Lesson Video Topic 201 Utility Maximization and Consumer Demand: Second Order Condition 41 Lesson Video Topic 202 Numerical Example of Utility Maximization 41 Lesson Video Topic 203 Law of Equi-marginal Utility using Lagrangian Multiplier 41 Lesson Video Topic 204 Economic Application of Production Function Maximization 42 Lesson Video Topic 205 Economic Application on Logarithmically Transformed Production Function 42 Lesson Video Topic 206 Homogeneous Functions 42 Lesson Video Topic 207 Homogeneous Production Function, Average Products and Capital-Labor ratio. 42 Lesson Video Topic 208 Homogeneous Production Function, Marginal Products and Capital-Labor ratio. 42 Lesson Video Topic 209 Homogeneous Production Function and Euler''s Theorem 42 Lesson Video Topic 210 Cobb-Douglas Production Function and Returns to Scale. 43 Lesson Video Topic 211 Homogeneity and Returns to Scale of Three input Production Function. 43 Lesson Video Topic 212 Least-Cost Combination in Cobb-Douglas Production Function. 43 Lesson Video Topic 213 Expansion Path using First Order Condition. 43 Lesson Video Topic 214 Homothetic Functions 43 Lesson Video Topic 215 Homotheticity of Cobb-Douglas Production Function. 43 Lesson Video Topic 216 Introducing CES Production Function 44 Lesson Video Topic 217 Homogeneity of CES Production Function 44 Lesson Video Topic 218 Marginal Products of CES Production Function 44 Lesson Video Topic 219 Share of Labor and Capital in CES Production Function 45 Lesson Video Topic 220 CES Production Function and Euler''s Theorem 45 Lesson Video Topic 221 Numerical CES Production Function Calculation 45 Lesson Video Topic 222