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Lecture Notes
IUPUI— FALL SEMESTER 2013
Syllabus for E375/E504
Shahrokh Towfighi
e-mail: stowfig@iupui.edu
297-4442(H)
Please call between 6:00-9:00 pm.
278-7213 (O)
CA 513
MW—9:30-11:45
If you cannot meet me at these hours, please make an appointment.
Chiang, Alpha C., and Wainwright, Kevin, Fundamental Methods of Mathematical Economics
(McGraw-Hill/Irwin, 4th Edition)
Available on my website: iu.edu/~towfighi/E375-E504
A laptop computer with Microsoft Office 2010
We will use Microsoft Excel extensively in the course. Make sure to bring a laptop with you to
class.
Course Requirements
Homework Assignments
Two tests
Final
20%
50%
30%
The due dates for homework assignments and test dates will be announced in class. The test dates will depend on
the pace at which we cover the material.
The following is an outline of the course. I have used the chapter headings and the relevant sections that we will
follow as the guide for the topics we will cover in class.
Chapter 2—Economic Models
1. Ingredients of a Mathematical Model
1.1. Variables, Constants, and Parameters
2. The Real-Number System
3. Relations and Functions
4. Types of function
4.1. Constant Function
4.2. Polynomial Function
4.3. Rational Functions
4.4. Nonalgebraic Functions
5. Functions of Two or More Independent Variables
Chapter 3—Equilibrium Analysis in Economics
1. The meaning of Equilibrium
2. Partial Market Equilibrium—a Linear Model
2.1. Constructing the Model
2.2. Solution by Elimination of Variables
3. Partial Market Equilibrium—A Nonlinear Model
3.1. Quadratic Equation versus Quadratic Function
3.2. The Quadratic Formula
3.3. Another Graphical Solution
3.4. Higher Degree Polynomial Functions
4. General Market Equilibrium
4.1. Two-Commodity Market Model
4.2. Numerical Example
4.3. n-Commodity Case
4.4. Solution of a General-Equation System
Chapter 4—Linear Models and Matrix Algebra
1. Matrices and Vectors
1.1. Matrices as Arrays
2. Matrix Operations
2.1. Addition and Subtraction of Matrices
2.2. Scalar Multiplication
2.3. Multiplication of Matrices
3. Commutative, Associative, and Distributive Laws
4. Identity Matrices and Null Matrices
4.1. Identity Matrices
4.2. Null Matrices
4.3. Idiosyncrasies of Matrix Algebra
5. Transposes and Inverses
5.1. Properties of Transposes
5.2. Inverses and Their Properties
5.3. Inverse Matrix and Solution of Linear-Equation System
Chapter 5—Linear Models and Matrix Algebra (Continued)
1. Conditions for Nonsingularity of a Matrix
1.1. Necessary versus Sufficient Conditions
1.2. Conditions for Nonsingularity
1.3. Rank of a Matrix
2. Test of Nonsingularity by Use of Determinant
2.1. Determinants and Nonsingularity
2.2. Evaluating a Third-Order Determinant
2.3. Evaluating an nth-Order Determinant by Laplace Expansion
3. Basic Properties of Determinants
3.1. Determinantal Criterions for Nonsingularity
3.2. Rank of a Matrix Redefined
4. Finding the Inverse of Matrix
4.1. Matrix Inversion
5. Cramer’s Rule
5.1. Derivation of the Rule
6. Leontief Input-Output Models
Lecture Notes:
LN1-Matrix Algebra
Chapter 6—Comparative Statics and the Concept of Derivative
1. The Nature of Comparative Statics
2. Rate of Change and the Derivative
2.1. The Difference Quotient
3. The Derivative and the Slope of a Curve
Chapter 7—Rules of Differentiation and Their Use in Comparative Statics
1. Rules of Differentiation for a Function of One Variable
1.1. Constant-Function Rule
1.2. Power-Function Rule
1.3. Power-Function Rule Generalized
2. Rules of Differentiation Involving Two or More Functions of the Same Variable
2.1. Sum-Difference Rule
2.2. Product Rule
2.3. Finding Marginal-Revenue Function from Average Revenue Function
2.4. Quotient Rule
2.5. Relationship Between Marginal-Cost and Average-Cost Functions
3. Rules of Differentiation Involving Function of Different Variables
3.1. Chain Rule
3.2. Inverse-Function Rule
4. Partial Differentiation
4.1. Partial Derivatives
4.2. Techniques of Partial Differentiation
4.3. Geometric Interpretation of Partial Derivatives
5. Applications to Comparative-Static Analysis
5.1. Market Model
5.2. National Income Model
5.3. Input-Output Model
6. Note on Jacobian Determinants
Lecture Notes:
LN2-Derivatives
Chapter 8—Comparative-Static Analysis of General Function Models
1. Differentials
1.1. Differentials and Derivatives
1.2. Differentials and Point Elasticity
2. Total Differentials
3. Rules of Differentials
4. Total Derivatives
4.1. Finding the Total Derivative
4.2. A variation on the Theme
4.3. Another Variation on the Theme
5. Derivatives of Implicit Functions
5.1. Implicit Functions
5.2. Derivatives of Implicit Functions
5.3. Extension to the Simultaneous-Equation Case
6. Comparative Statics of General-Function Models
6.1. Market Model
6.2. Simultaneous-Equation Approach
6.3. Use of Total Derivatives
Lecture Notes:
LN3-Differentials
Chapter 9—Optimization: A Special Variety of Equilibrium Analysis
1. Optimum Values and Extreme Values
2. Relative Maximum and Minimum: First-Derivative Test
2.1. Relative versus Absolute Extremum
2.2. First-Derivative Test
3. Second and Higher Derivatives
3.1. Derivative of a Derivative
3.2. Interpretation of the Second Derivative
4. Second-Derivative Test
4.1. Conditions for Profit Maximization
4.2. Coefficients of a Cubic Total-Cost Function
Lecture Notes:
LN4-Optimization
Chapter 10-Exponential and Logarithmic Functions
1. The Nature of Exponential Functions
1.1. Simple Exponential Function
1.2. Graphical Form
2.
3.
4.
5.
6.
1.3. Generalized Exponential Function
1.4. A Preferred Base
Natural Exponential Functions and the Problem of Growth
2.1. The number e
2.2. An Economic Interpretation of e
2.3. Interest Compounding and the Function Aert
2.4. Instantaneous Rate of Growth
2.5. Continuous versus Discrete Growth
2.6. Discounting and Negative Growth
Logarithms
3.1. The Meaning of Logarithm
3.2. Common Log and Natural Log
3.3. Rules of Logarithms
3.4. An Application
Logarithmic Functions
4.1. Log Functions and Exponential Functions
4.2. The Graphical Form
4.3. Base Conversion
Derivatives of Exponential and Logarithmic Functions
5.1. Log-Function Rule
5.2. Exponential-Function Rule
5.3. The Rules Generalized
5.4. The Case of b
5.5. Higher Derivatives
5.6. An Application
Optimal Timing
6.1. A Problem of Wine Storage
6.2. Maximization Conditions
6.3. A Problem of Timber Cutting
Lecture Notes:
LN5-Exponential and Logarithmic Functions
An Excursion Into Econometrics
Lecture Notes: LN6-Applications in Econometrics
Chapter 11—The Case of More than One Choice Variable
1. The Differential Version of Optimization Conditions
1.1. First-Order Condition
1.2. Second-Order Condition
2. Extreme Values of a Function of Two Variables
2.1. First-Order Condition
2.2. Second-Order Partial Derivatives
2.3. Second-Order Total Differential
2.4. Second-Order Condition
Chapter 12—Optimization with Equality Constraints
1. Effect of a Constraint
2. Finding the Stationary Values
2.1. Lagrange-Multiplier Method
2.2. Total Differential Approach
2.3. An Interpretation of the Lagrange Multiplier
3. Second-Order Conditions
3.1. Second-Order Total Differential
3.2. Second-Order Conditions
3.3. The Bordered Hessian