 # PROGRAM IN ENTREPRENEURIAL AND MANAGERIAL DEVELOPMENT

I. QUANTITATIVE METHODS I
7201
3 credits
No prerequisites

II. COURSE DESCRIPTION:

This course presents quantitative methods necessary for understanding mathematical models as used in quantitative research, entrepreneurial and managerial decision making. The course focuses on mathematical methods used in economic and managerial modeling like differential and integral calculus, optimizing techniques, linear algebra and linear programming. The models will be applied in various managerial areas like human resources, labor relations, marketing and organizational behavior.

III. COURSE OBJECTIVES
After completion of this course the student should be able to attain the following terminal and capacitating objectives:

1.Construct mathematical models of managerial situations applying functional mathematical analysis and set theory.
1.1Apply functional mathematical relations among variables.
1.2Apply in managerial situations polynomial, rational, linear, exponential and logarithmic functions.
1.3Explain the Fundamental Theorem of Algebra
1.4Apply set theory in the context of managerial situations involving groups and group behavior.
1.5Apply the concepts of union and intersection of sets to situations involving labor relations and collective bargaining.
1.6Solve simple economic models like breakeven and demand and supply.

2.Evaluate the use of linear algebra in managerial situations and to understand economic interrelations.
2.1Define the basic operations of matrix algebra including addition, multiplication and inverses of matrices.
2.2Use Excel to do basic matrix operations.
2.3Apply matrix algebra in the solution of linear multiple equations systems.
2.4Solve a system of linear equations using Excel .
2.5Use matrix algebra to apply it to the open Leontief input-output model of the economy.
2.6Explain the intersectoral transactions in the economy and the matrix of technical coefficients.
2.7Use Excel to derive the Leontief matrix and its inverse.
2.8Determine the final intersectoral output of the economy given a final demand vector using Excel.
2.9Explain Marcov chains and apply them in changing states in the context of collective bargaining, market shares, group behavior, population and preference dynamics.

3.Apply differential and integral calculus and optimization techniques to situations in managerial decision making.
3.1Define the concept of the limit of a function as it approaches a point in its domain.
3.2Apply the concept of the change quotient and apply it to estimate changes in the values of functions.
3.3Differentiate between continuity and differentiability.
3.4Define the concept of the derivative as the limit of the change quotient.
3.5Solve exercises utilizing the rules of differentiation of single and multiple variable functions: power, product, ratio, chain, implicit and inverse function rules.
3.6Solve exercises involving the differentiation of exponential and logarithmic functions.
3.7Apply the concepts of partial and cross differentiation.
3.8Define the differential of a function and the process of total differentiation.
3.9Construct channel maps of total differentiation detailing direct and indirect effects of variables and apply it in organizational behavior issues.
3.10Apply the differential calculus analysis to understand point price elasticity of demand and its relation to marginal revenue and optimum pricing.
3.11Apply the differential calculus analysis to understand total, marginal and average relations on functions.
3.12Solve optimization exercises of single and multiple variable functions given constraints utilizing the Lagrange multiplier technique.
3.13Solve profit maximizing and utility maximizing problems; solve cost minimization problems.
3.14Apply optimization techniques to choose optimum timing of projects.
3.15Apply the concepts of gradient vector, Hessian matrix and bordered Hessian matrix.
3.16Define comparative statics and the envelope theorems of comparative statics.
3.17Define the concept of the integral as an antiderivative and as a Riemann sum in the limit.
3.18Explain the Fundamental Theorem of Calculus.
3.19Derive the indefinite integral using the techniques of integration: power, substitution and by parts .
3.20Derive the definite integral using the techniques of integration and given values in the domain of a function.
3.21
3.22Apply integral calculus in investment, growth and capital accumulation issues of entrepreneurial and managerial importance.

4.Apply linear programming in entrepreneurial and managerial decisions.
4.1Construct a linear programming model.
4.2Differentiate between the objective function, decision variables, constraints and technical coefficients in a linear programming model.
4.3Solve linear programming models using Excel Solver.
4.4Define the concepts of shadow prices, viability and optimality ranges and be able to evaluate them in the context of managerial decisions.
4.5Apply linear programming to choose the optimum projects for the firm.
4.6Apply linear programming to choose the optimum asset allocation in investment situations with constraints.
4.7Apply linear programming in scheduling problems.
4.8Apply linear programming in plant location problems.

IV.COURSE CONTENT

Part I. Foundations of mathematical analysis

A. The number system

B. Economic models
1. Variables, parameters, constants and coefficients
2. Equations
3. Endogenous and exogenous variables

C. Set theory
1. Basic set operations
a. Union
b. Intersection
c. Complement
d. Joint vs. disjoint sets

D. Mathematical Framework of Analysis
1. Functions
a. Introduction to functions
b. Polynomial and rational functions
b. Exponential and Logarithmic Functions
2. Mathematical Models and Data Analysis
3. Equilibrium Analysis

E. Applications
1. Breakeven model
2. Demand and supply model

Part II. Linear Models and Matrix Algebra

A. Introduction to Matrix algebra
1. Matrices and vectors
2. Matrix operations and concepts
b. Determinants, minors and cofactors
b. Matrix inverse
d. Singularity
5. Excel functions for matrices operations

B. Solving System of Equations
1. Solving 2 x 2 systems
2. Generalization to higher dimensions
3. The Leontief System
a. Intersectoral transactions matrix
b. Coefficient matrix
c. Leontief matrix
d. Inverse of Leontief matrix
e. Final demand and final output
4. Marcov chains
1. State matrix
1. Transition matrix
3. Applications

Part III. Differential Calculus and Integral Calculus

A. Comparative Static Analysis and Derivatives
1. The Nature of Comparative Statics
2. Limit of a function
3. Continuity and differentiability

B. Rules of differentiation; univariate functions.
1. Power
2. Product
3. Quotient
4. Chain
5. Inverse function
6. Logarithmic
7. Exponential

C. Rules of differentiation; multivariate functions
1. Derivatives of higher order
2. Cross derivatives

D. Total Differentials and Total Derivatives
1. Channel maps
2. Direct and indirect effects

E. Applications
1. Consumers Utility Functions
2. Cost Functions
3. Revenue Functions
4. The Market Model
5. Market penetration

Part IV. Optimization Analysis

A. Derivatives and Extreme Values
1.Stationary points
2.First and second order conditions for maximum or minimum
3.The nth derivative test for extreme points

B. Constrained Optimization
1. Using the substitution technique
2. Using the Lagrange multiplier technique
4. Hessian matrix
5. Bordered Hessian
6. Comparative statics and envelope theorem

E. Applications
1. Cost minimization for a given output
2. Profit Maximization
3. Optimum timing
4. Utility maximization and consumer demand
5. Least cost combination of inputs with Cobb Douglas production function

Part V. Integral Calculus

A.Definition of Integrals
1.Antiderivative
2.Riemann sum
3.Fundamental Theorem of Calculus
4.Properties of integrals

B.Rules for Integration: Indefinite Integrals
1.Power
2.Substitution
3.By parts

C.Definite Integrals

D. Improper integrals

E.Applications
1.Investment and capital accumulation
2. Compound interest and present value
3.Economic growth: Domar Model
4.Population accumulation

Part V. Linear Programming

A.Mathematical Optimization
1.Decision variables
2.Objective function
3.Restrictions
4.Boundaries of decision variables

B. Graphical Solution
1. Inequalities
2.Isocurves
3.Viability Set
4.Corner solutions

C. General Solution sing Excel Solver
1.Optimization of objective function
3.Sensitivity Analysis
a.Optimality Range
b.Viability Range

D.Applications
1. Production decisions
2.Resource allocation
3.Optimal factor combination
4.Diet problem
5.Transportation
6.Investment planning
7.Capital Budgeting

V.COURSE ACTIVITIES

A. Power Point presentations by professor and student feedback
B.Class discussion of exercises
C.Communication among students and professor via e-mail

VI. EVALUATION CRITERIA
A.Partial Examinations: There will be two partial examinations.
B.Final Examination
C. Model construction project

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