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## ECO606 : Mathematical Economics I

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### Course Info

Economics

3

N/A

#### Instructor

Dr. Bilal Mehmood
PhD Economics
GC University, Lahore

### Course Contents

Demystifying Mathematics and Mathematical Economics. Mathematical versus Non-mathematical Economics. Mathematical Economics versus Econometrics. Ingredients of a Mathematical Model: Variables and Economic Variables. Ingredients of a Mathematical Model: Constants and Parameters. A Few Aspects of Logic: Propositions, Implications and Necessary and Sufficient Conditions. The Real-Number System. Use of Sets in economics: Set Notation. Use of Sets in economics: Operations of Sets. Use of Sets in economics: Laws of Operations of Sets. Cartesian Coordinates. What are Functions? Domain and Range in a Function. Difference between Function and Relations. Economic Example of Schultz Demand Function. Economic Example of Cost Function of Cleaning Impurities from a Lake. Economic Example of Function: Laffer Curve. Types of Functions: Constant Functions. Types of Functions: Polynomial Function: Linear Functions. Interpreting Linear Economic Functions.  Applications of Linear Functions: Population and Consumption Functions. Types of Functions: Polynomial Function: Quadratic Functions. Quadratic Cost function and Profit Function of a Monopoly. Quadratic Function and Production Possibilities Frontier. Types of Functions: Polynomial Function: Cubic Functions. Cubic Cost functions. Rational Functions. Other Types: Rectangular Hyperbolic Function. Other Types: Non-algebraic Exponential Functions. Population Growth using General Exponential Functions. Other Types: Non-algebraic Natural Exponential Functions. Population Growth using Natural Exponential Functions. Other Types: Non-algebraic Logarithmic Functions. Other Types: Non-algebraic Natural Logarithmic Functions. Rate of Growth of GNP using Logarithmic Functions. Inverse Functions. Functions with Two or More Independent Variables. Surfaces and Distance in Graphs of two or more independent variables. Equations and Identities. Types of equations in Economics: Definitional Equations. Fiscal Surplus and Fiscal Deficit Using Equations. Types of equations in Economics: Behavioral Equations. Types of equations in Economics: Conditional Equation. Structural and Reduced Form Equations. Constructing a Partial Linear Market Equilibrium. Solving using Elimination of Variable Method. Shifts in Demand in Market Equilibrium. Shifts in Supply in Market Equilibrium. Effect of Tax on Producer on Partial Market Equilibrium. Effect of Tax on Consumer on Partial Market Equilibrium. Partial Market Equilibrium-A Nonlinear Model. General Market Equilibrium: General form of Two Good Case. General Market Equilibrium: Numerical solution of Two Good Case. General Market Equilibrium: n-Good Case. National Income Equilibrium. National Income Equilibrium with Induced and Autonomous Tax. National Income Equilibrium with Proportion of Government Expenditure. Matrices and Vectors. Array and Summation forms of Matrices. Matrices Operations. Using Product of Matrices to Calculate Total Cost. Using Product of Matrices to Calculate Total Revenue and Profit. Question of 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. Sarrus''''s rule for 3x3 order Determinant of a Matrix. Inverse of a Matrix. Condition(s) for Non-singularity. Expression of National Income using Matrix Form. Minor and Cofactors. Market Model Analysis Using Matrix Inversion Method. National Income Analysis Using Matrix Inversion Method. Equilibrium Prices using Matrix Inversion Method. Solving Market Model using Cramer''''s Rule. Equilibrium Prices using Cramer''''s Rule. National Income Determination using Cramer''''s Rule. Input-Coefficient Matrix. Economic Meaning of Hawkins-Simon Condition. Input-Output Analysis in case of Open Economy. Input-Output Analysis in case of Closed Economy. The Need and Nature of Comparative Statics. Concept of Limit and continuity. Rate of Change, Slope & Derivative. Differentiation Rules for Single variable functions: Constant Function Rule and Power Function Rule. Sum-Difference Rule of Differentiation. Sum-Difference Rule: Numerical Analysis of Cost Function. Understanding graphs of Function and its derivatives. Product Rule of Differentiation. Relationship between Average Revenue and Marginal Revenue using Product Rule. Quotient Rule of Differentiation. Marginal Propensity to Consume via Differentiation with and without Tax. Relationship Between Marginal-Cost and Average-Cost Functions using Quotient Rule. Variable and Fixed Cost components in total cost function. Obtaining Marginal Cost function from Average Cost function. Marginal Cost Analysis. Marginal Revenue Analysis. Marginal Product Analysis. Rules of Differentiation: Functions with different variables: Chain Rule. Marginal Revenue Product of Labor (MRPL) Analysis. Marginal Analysis of Fishery Production Function. Inverse Function Rule. Partial Differentiation: The Concept. Marginal Physical Product of Labor and Capital using Partial Derivatives. Marginal Utility functions using Partial Derivatives. Output Elasticity of Labor and Capital using Partial Derivatives. Money Market Analysis using Partial Derivatives. Partial Market Model using Partial Differentiation. National-Income Model using Partial Differentiation. Second and Higher Order Derivatives. Economic Applications of Second Derivative: Profit Maximization Condition. Numerical Example of Profit Maximization Condition using Second Derivative. Young''''s Theorem. Demand for Money Function Analysis using Partial Derivatives. Income Elasticity of Demand using Partial Derivatives. Cross Price Elasticity of Demand using Partial Derivatives. Partial Derivatives: Application on Herring Production Function. Partial Derivatives: Application on Three Input Production Function. Envelope Theorem. Roy''''s Identity. Hotelling''''s Lemma. Shephard Lemma. Differentials versus Derivatives. Point Elasticity using Differentials. Elasticity of Rectangular Hyperbolic Demand Curve. Income and Price Elasticity of Demand using Differentials. Income and Price Elasticity of Import Function using Differentials. Income Elasticity of Consumption using Differentials. Output Elasticity of Cost. Concept of Total Differentials. Savings Function and Total Differentials. General Utility Function and Total Differentials. Specific Utility Function and Total Differentials. Price and Rain Elasticity of Supply using Total Differentials. Local Price Elasticity of Foreign Demand of Exports using Total Differentials. Concept of Total Derivatives. Complementarity between Coffee and Sugar Using total Derivative. General Production Function with time-dependent Labor and Capital. Specific Production Function with time-dependent Labor and Capital. Concept of Implicit Differentiation. Production Function Analysis using Implicit Differentiation. Marginal Rate of Technical Substitution using Implicit Differentiation. Marginal Utilities and Marginal Rate of Substitution using Implicit Differentiation. Nerlove-Ringstad Production Function using Implicit Differentiation. Marginal Products of Three Input Logarithmic Production Function. Exponential Functions and Growth. Instantaneous Rate of Growth. Numerical Examples of Instantaneous Rate of Growth. Continuous vs. Discrete Growth. Discounting and Negative Growth. Applications of Continuous Compounding. Applications of Instantaneous Growth. Logarithms: Meaning and Types. Laws of Logarithms. Laws of Logarithms for Transformation of CES Production Function. Rules of Differentiation of Exponential and Logarithmic Functions. Optimal Timing: A Problem of Wine Storage. Optimal Timing: A Problem of Timber Cutting. Optimal Timing: Land Purchase for Speculation. Optimal Timing: Art Collection. Optimal Timing: Diamond Purchase. Finding the Rate of Growth using Exponential and Logarithmic Functions. Growth of Exports of a Country. Finding the Point Elasticity. Rates of Growth of Population, Consumption, and Per Capita Consumption. Rate of Growth of Per Capita Employment. Rate of Growth of Export Earnings of a Country. Rate of Growth of Sales. Rate of Growth of Profit. Concept of Optimization. Calculus Approach to Optimization: 1st Order Test. Average Cost Analysis. Calculus Approach to Optimization: 2nd Order Test. Matrix Approach to Optimization: 2nd Order Test – Hessian. Profit Maximization Analysis. Numerical Example of Profit Maximization. Profit Maximization of Technically Related Goods. Profit Maximization of Monopolistic Firm producing Related Goods. Profit Maximization of Firm producing Substitute Goods. Price Discrimination and Elasticity of Demand. Cubic Total Cost Analysis. Marginal and Average Revenue Analysis. Short Run Production Function Analysis. Total Cost, Total Revenue and Profit Maximization. Quadratic Profit Function Analysis. Optimization of Exponential Revenue Function. Optimization of More than one Choice variable. Economic Application on Multi-Product Firm. Economic Application on Multi-Plant Firm. Price Discrimination by Monopoly. Price Discrimination by Monopsony. Input Decision of a Firm. Profit Maximization of two-product firm. Comparative-Static Aspects of Optimization. Rationale for Constrained Optimization. Finding Stationary Values using Substitution/Elimination Method. Finding Stationary Values using Method of Lagrange Multiplier. Interpretation of the Lagrange Multiplier. Second Order Condition: The Bordered Hessian. Utility Maximization and Consumer Demand. Two Period Model of Utility. Convexity and Concavity using Second Order Derivative. Utility Maximization and Consumer Demand: First Order Condition. Utility Maximization and Consumer Demand: Second Order Condition. Numerical Example of Utility Maximization. Law of Equi-Marginal Utility using Lagrangian Multiplier. Economic Application of Production Function Maximization. Economic Application on Logarithmically Transformed Production Function. Homogeneous Functions. Homogeneous Production Function, Average Products and Capital-Labor ratio. Homogeneous Production Function, Marginal Products and Capital-Labor ratio. Homogeneous Production Function and Euler''''s Theorem. Cobb-Douglas Production Function and Returns to Scale. Homogeneity and Returns to Scale of Three input Production Function. Least-Cost Combination in Cobb-Douglas Production Function. Expansion Path using First Order Condition. Homothetic Functions. Homotheticity of Cobb-Douglas Production Function. Introducing CES Production Function. Homogeneity of CES Production Function. Marginal Products of CES Production Function. Share of Labor and Capital in CES Production Function. CES Production Function and Euler''''s Theorem. Numerical CES Production Function Calculation.