Module Code and Title:       QME101         Mathematics for Economics

Programme:                          BA in Development Economics

Credit Value:                         12

Module Tutor:                       Tshering Lhamo Dukpa

General objective: This is the first of a compulsory sequence of quantitative modules related to economics. The objective of this module is to provide the knowledge of basic mathematics that enables the study of economic theory at the undergraduate level, specifically the modules on microeconomics, macroeconomics, statistics, and econometrics. In this module, particular economic models are not the ends, but the means for illustrating the method of applying mathematical techniques to economic theory in general.

Learning outcomes – On completion of this module, learners should be able to:

1. Apply the notions of a set and basic operations on sets.
2. Solve matrix operations and systems of linear equations.
3. Apply the Leontief input-output model to solve real-world problems.
4. Apply the concepts of functions, continuity and limit.
5. Calculate the derivatives of a function.
6. Calculate derivatives to find extreme points.
7. Define the concepts of definite and indefinite integral.
8. Apply the notions of a partial derivative of a function of several variables.
9. Interpret gradient and level curves.

Learning and Teaching Approach: This module will use a mix of procedural and conceptual approaches to teaching mathematics. The module tutor shall combine direct instruction of rules and procedures for solving problems in the classroom lectures, and use a conceptual approach to convey why particular formulae and processes of solutions work. The lectures will be complemented with tutorials focusing on self-discovery, use of manipulative problem solving, and group work.

Assessment Approach:

A. Individual Assignment: Portion of Final Marks: 15%

To be given before the midterm examination, this assignment will test problem-solving skills, ability to identify a problem, and decide why and how a particular mathematical device can be applied to find solution. The assignment shall have maximum limit of 200 words.

• 3%       Ability to understand a problem
• 3%       Identify appropriate mathematical device to solve the problem
• 6%       Finding solution
• 3%       Interpretation of the findings

B. Class Tests: Portion of Final Marks: 20%

Four written tests will be conducted (at least one test every month, each test worth 5%), that will comprise 45 min duration and cover 2-3 weeks of material. The tests will contain 4 questions (2 on the conceptual understanding and 2 on problem solving).

C. Group Work: Portion of Final Mark: 10%

Group size: 4 students. The task is to test conceptual understanding of given mathematical devices, and identify situations in which they can be applied. The group work should have one component each for every student to be individually responsible for, while effective groups would also cross-check each team member’s work. The assignment should have a maximum limit of 200 words.

• 1%       Group work plan
• 3%       Individual work
• 2%       Review of individual work by team members
• 4%       Synthesis of individual work in a joint report

D. Midterm Examination: Portion of Final Mark: 15%

Students will take a written exam of 1.5 hr duration covering topics up to the mid-point of the semester.

Pre-requisites:

Subject matter:

1. Preliminaries
   • Logic and proof techniques: propositions, implications, necessary and sufficient conditions, mathematical proof, deductive and inductive reasoning
   • Sets and set operations: Set notation, relationship between sets, operation of sets, laws of set operation, Venn diagrams
   • Real number systems: natural numbers, positive integers, rational numbers, irrational numbers, decimal system, real numbers
2. Functions of one real variable
   • Graphs
   • Elementary types of functions: linear, quadratic, polynomial, power, exponential, logarithmic
   • Sequences and series: convergence, algebraic properties and applications
   • Continuous functions: characterizations, properties with respect to various operations and applications
   • Differentiable functions: characterizations, properties with respect to various operations and applications; second and higher order derivatives: properties and applications
3. Single variable differentiation
   • Slope
   • Simple rules for differentiation
   • Second and higher order differentiation, Chain rule, polynomial approximations, elasticities
   • Limit, continuity, continuity and differentiation
   • Value theorems, Taylor’s formula, Indeterminate forms, inverse functions
4. Optimization
   • Single variable Optimisation: geometric properties of functions, first derivative test, convex functions, their characterizations and applications
   • Multivariable optimisation: local and global optima: value theorem, geometric characterizations (concave/convex functions, conditions for concavity /convexity, quasi concave/convex functions), characterizations using calculus and application (constrained optimisation through Lagrangean multiplier method)
   • Linear programming: Duality theory, complementary slackness
5. Integration of functions
   • Areas under curves
   • Indefinite integrals
   • The definite integral, integration by parts, Integration by substitution
6. Difference equations
   • First order difference equations
   • Compound interest and present discounted values
   • Linear equations with variable coefficient
   • Second order equations, with constant coefficients
7. Differential equations
   • First-order differential equations
   • Integral curve, direction diagram and slope field
   • Qualitative theory and stability
   • Second order differential equations, with constant coefficients
8. Linear algebra
   • Vector spaces; algebraic and geometric properties, scalar products, norms, orthogonality
   • Linear transformations: properties, matrix representations and elementary operations
   • Systems of linear equations: properties of their solution sets; determinants: characterization, properties and applications

Reading List:

1. Essential Reading
   • Chiang, A.C. (1984). Fundamental Methods of Mathematical Economics. New York: McGraw Hill.
   • Sydsaeter, K., & Hammond, P. (2000). Mathematics for Economic Analysis. Delhi: Pearson Educational Asia.
2. Additional Reading
   • Allen, R.G.D. (1974). Mathematical methods for Economics. McGraw Hill.
   • Simon, K.P. & Blume, L. (1994). Mathematics for Economists. W.W. Norton, New York. London. Retrieved from http://academia.edu/4797403/Mathmatics_for_Economists_e_book_Simeone_and_Blume_
   • Warner, S. & Costenoble, S.R. (2010). Finite Mathematics and Applied Calculus. Thomson, Brooks/Cole

Date: January 15, 2016