Bachelor
2020/2021

# Mathematical Methods for Economists

Type:
Elective course (HSE University and University of London Double Degree Programme in Economics)

Area of studies:
Economics

Delivered by:
International College of Economics and Finance

When:
2 year, 1-4 module

Mode of studies:
distance learning

Language:
English

ECTS credits:
10

Contact hours:
238

### Course Syllabus

#### Abstract

Mathematical Methods for Economists is a two-semester course for the second year students studying at ICEF which specialize in “Mathematics and Economics”. This course is an important part of the bachelor stage in education of the future economists. It has give students skills for implementation of the mathematical knowledge and expertise to the problems of economics. In the fall semester this course is split into two self-contained sub courses: namely “Multivariate Calculus and Optimization” (MCO) and “Linear Algebra” (LA). LA lasts for one fall module and completes with the final exam in the end of October. MCO continues beyond and from January onwards incorporates also the chapters of “Methods of Optimization” course. Also note that starting from January and through the end of April UoL “Algebra MT1 173” will be taught to the students of ‘Math and Econ”. The final assessment will be provided by the University of London (UoL) examinations in May. Students will sit two exams: on Calculus and Algebra.

#### Learning Objectives

- Students are expected to develop an understanding of basic algebraic concepts such as linear vector space, linear independence, bases, coordinate systems, dimension, matrix algebra, linear operators, dot product, orthogonality. On the practical side, among other skills, students are expected to be able to solve systems of linear equations, find fundamental system of solutions, invert matrices, find eigenvalues, and do orthogonal projections. Students are supposed: to acquire knowledge in the field of higher mathematics and become ready to analyze simulated as well as real economic situations; to develop ability to apply the knowledge of the differential and difference equations which will enable them to analyze dynamics of the processes.

#### Expected Learning Outcomes

- Solve equations by Gaussian elimination method
- Practice techniques of matrix operations
- Be able to invert a matrix either by finding cofactors or by Gaussian elimination method
- Be able to classify bilinear and quadratic forms
- Explain orthogonality of vectors, properties of a dot product, Gram-Schmidt procedure, eigenvalues, eigenvectors
- Classify the sets in n-dimensional space
- Apply the notion of level curve to microeconomics
- Be able to find a limit of a function at a point
- Be able to handle derivatives
- Explain and apply gradient and related directional derivative
- Find derivatives of implicit functions
- Apply IFT to microeconomic and macroeconomic problems
- Apply FOC to an objective function and checking definiteness of Hessian
- Apply Lagrange method for equality constrained type of problems
- Explain the meaning of a multiplier and be ready to demonstrate the applicability of envelope theorems
- Provide examples of differential equations especially in economics
- Apply the basic techniques of solving first-order equations
- Explain and apply Solow’s model
- Use integration mostly dealing with the linear equations with constant coefficients and quasipolynomials in the right side
- Outline complex numbers theory
- Apply method of undetermined coefficients for the search of a particular solution
- Apply difference equations to macroeconomics
- Use techniques of solving first-order equations
- Able to handle second-order difference equations
- Apply Euler’s equation to microeconomics
- Apply Kuhn-Tucker method for solving problems from primarily microeconomics
- Explain and apply linear programming
- Apply Nash equilibrium concept to economic problems
- Use maxmin/minmax techniques
- explain relationship between Cartesian, parametric equations for planes and lines, be able to find normal vector to a plane
- solve equations by Gaussian elimination method, be able to express solution as a sum of a particular solution and a general solution from the null space of the matrix
- practice techniques of matrix operations, be able to to invert a matrix either by finding cofactors or by Gaussian elimination method; use formula for determinants based on cofactor expansion, outline properties of determinants helping to reduce calculations
- define rank, range of a matrix, column, row spaces and null space of a matrix, be able to apply rank-nullity theorem
- define and give examples of spaces and subspaces, explain notions of basis and dimension, axioms of vector spaces
- explain and use linear independence, bases and dimension
- explain and use rotation, projection and basic properties of transformations, introduce a change of basis formula that enables to switch the basis and find the corresponding matrix of a transformation
- be able to classify forms and apply such a knowledge to conic sections
- explain and use properties of Markov chains along with the technique enabling to find steady-state solutions

#### Course Contents

- Linear AlgebraSystems of linear equations in matrix form Linear space. Linear independence Linear subspace Matrix as a set of columns and as a set of rows Determinant of a set of vectors Inverse matrix Linear operator as a geometric object Eigenvalues, eigenvectors and their properties Bilinear and quadratic forms Dot product in linear spaces
- OptimizationUnconstrained optimization of the multi-dimensional functions. Stationary points. First-order conditions Second differential. Quadratic forms and the associated matrices. Definiteness and semi-definiteness of the quadratic forms. Sylvester criterion. Second-order conditions for extrema Constrained optimization. Lagrangian function and multiplier. First-order conditions for constrained optimization Second differential for the function with the dependent variables. Definiteness of quadratic form under a linear constraint. Bordered Hessian. Second-order conditions for the constrained optimization Economic meaning of a multiplier. Applications of the Lagrange approach in economics. Smooth dependence on the parameters. Envelope theorem
- Algebra1. Lines, planes in R2 and R3. Lines and hyperplanes in Rn 2. Homogeneous systems and null space. Consistent and inconsistent systems. Linear systems with free variables. Solution sets 3. Matrix inversion and determinants 4. Rank, range and linear equations 5. Vector spaces 6. Linear independence, bases and dimension 7. Linear transformations, change of basis 8. Diagonalization 9. Markov chains
- Systems of linear equations in matrix form. Linear space. Linear independence. Linear subspace. Matrix as a set of columns and as a set of rows. Determinant of a set of vectors Inverse matrix. Linear operator as a geometric object. Eigenvalues, eigenvectors and their properties. Bilinear and quadratic form.s Dot product in linear spaces.
- 1. Lines, planes in R2 and R3. Lines and hyperplanes in Rn 2. Homogeneous systems and null space. Consistent and inconsistent systems. Linear systems with free variables. Solution sets 3. Matrix inversion and determinants 4. Rank, range and linear equations 5. Vector spaces 6. Linear independence, bases and dimension 7. Linear transformations, change of basis 8. Diagonalization 9. Markov chains
- Methods of optimizationHomogeneous functions Optimization in 2 variables with the inequality constraints. First order conditions, generalization on the n-dimensional case Kuhn-Tucker formulation, applications from economics Meaning of Lagrange multipliers, envelope theorems (refreshment) Linear programming Introduction to the game theory. Bimatrix games. The notion of Nash equilibrium. Dominant and dominated strategies. Equilibrium in mixed strategies. Methods of finding equilibria in the zero sum games
- Differential and difference equationsDynamics in economics. Simple first-order equations. Separable equations. Concept of stability of the solution of ODE. Exact equations. General solution as a sum of a general solution of homogeneous equation and a particular solution of a nonhomogeneous equation. Bernoulli equation. Qualitative theory of differential equations. Solow’s growth model. Phase diagram Second-order linear differential equations with constant coefficients Complex numbers and operations on them. Representation of a number. De Moivre and Euler formulae Higher-order linear differential equation with constant coefficients. Characteristic equation. Method of undetermined coefficients for the search of a particular solution. Stability of solutions. Routh theorem (without proof). Discrete time economic systems. Difference equations. Method of solving first-order equations. Convergence and oscillations of a solution. Cobweb model. Partial equilibrium model with the inventory Second-order difference equations Higher-order difference equations. Characteristic equation. Undetermined coefficients method. Conditions for the stability of solutions
- Multi-dimensional calculusMain concepts of set theory. Operations on sets. Direct product of sets. Relations and functions. Level sets and level curves. Space . Metric in n-dimensional space. The triangle inequality. Euclidean spaces. Neighborhoods and open sets in , Sequences and their limits. Close sets. The closure and the boundary of a set. Functions of several variables. Limits of functions. Continuity of functions Partial differentiation. Economic interpretation, marginal products and elasticities. Chain rule for partial differentiation Total differential. Geometric interpretation of partial derivatives and the differential. Linear approximation. Differentiability. Smooth functions. Directional derivatives and gradient Higher-order derivatives. Young’s theorem. Hessian matrix. Economic applications Implicit functions. Implicit function theorem Vector-valued functions. Jacobian Implicit function theorem for the vector-valued functions Economic applications of the IFT for the comparative statics problems.

#### Assessment Elements

- home assignments
- exam on linear algebra
- fall mock
- December exam
- spring mock on math methods
- spring mock on algebra
- UoL Calculus exam
- UoL Algebra exam
- home assignments (Linear algebra)
- midterm (linear algebra)

#### Interim Assessment

- Interim assessment (2 module)0.42 * December exam + 0.15 * exam on linear algebra + 0.14 * fall mock + 0.14 * home assignments + 0.03 * home assignments (Linear algebra) + 0.12 * midterm (linear algebra)
- Interim assessment (4 module)0.1 * home assignments + 0.1 * Interim assessment (2 module) + 0.15 * spring mock on algebra + 0.15 * spring mock on math methods + 0.25 * UoL Algebra exam + 0.25 * UoL Calculus exam