Informações sobre o curso: This course is an important part of the undergraduate stage in education for future economists. It's also useful for graduate students who would like to gain knowledge and skills in an important part of math. It gives students skills for implementation of the mathematical knowledge and expertise to the problems of economics. Its prerequisites are both the knowledge of the single variable calculus and the foundations of linear algebra including operations on matrices and the general theory of systems of simultaneous equations. Some knowledge of vector spaces would be beneficial for a student. The course covers several variable calculus, both constrained and unconstrained optimization. The course is aimed at teaching students to master comparative statics problems, optimization problems using the acquired mathematical tools. Home assignments will be provided on a weekly basis. The objective of the course is to acquire the students’ knowledge in the field of mathematics and to make them ready to analyze simulated as well as real economic situations. Students learn how to use and apply mathematics by working with concrete examples and exercises. Moreover this course is aimed at showing what constitutes a solid proof. The ability to present proofs can be trained and improved and in that respect the course is helpful. It will be shown that math is not reduced just to “cookbook recipes”. On the contrary the deep knowledge of math concepts helps to understand real life situations.
Para quem é direcionado este curso: Undergraduate students (all years of study) and masters degree students (first year of study)
Desenvolvido por: National Research University Higher School of Economics
Ministrado por: Kirill Bukin, Associate Professor, Candidate of sciences (phys.-math.)
Faculty of Economic Sciences, Department of Theoretical Economics
| Nível | Intermediário |
| Compromisso | 8 weeks of study |
| Idioma | Inglês |
| Como ser aprovado | Seja aprovado em todas as tarefas para concluir o curso. |
Programa
SEMANA 1
The basics of the set theory. Functions in Rn
Week 1 of the Course is devoted to the main concepts of the set theory, operation on sets and functions in Rn. Of special attention will be level curves. Also in this week introduced definitions of sequences, bounded and compact sets, domain and limit of the function. Also from this week students will grasp the concept of continuous function.
11 videos
- Vídeo: 1.1. Definitions and examples of sets
- Vídeo: 1.2. Operations on sets
- Vídeo: 1.3. Open balls in Rn
- Vídeo: 1.4. Sequences in Rn. Closed sets.
- Vídeo: 1.5. Bounded and compact sets
- Vídeo: 1.6. Functions and level curves in Rn
- Vídeo: 1.7. Domain and limit of a function. Continuous functions.
- Vídeo: 1.8. Continuity of a function. Weierstrass theorem.
- Vídeo: 1.9. Composite function.
- Vídeo: 1.10. Continuity of a composite function
Nota atribuída: The basics of the set theory. Functions in Rn
SEMANA 2
Differentiation. Gradient. Hessian.
Week 2 of the Course is devoted to the main concepts of differentiation, gradient and Hessian.
Of special attention is the chain rule. Also students will understand economic applications of the
gradient.
13 videos
- Vídeo: 2.2. Example of differentiation. Cobb-Douglas function.
- Vídeo: 2.3. Tangent plane.
- Vídeo: 2.4. Total differential.
- Vídeo: 2.5. Chain rule for multivariate functions.
- Vídeo: 2.6. Gradient of the function.
- Vídeo: 2.7. Economic applications of the gradient.
- Vídeo: 2.8. Equation of a circumference. Smooth curves.
- Vídeo: 2.9. Chain rule for differentiation.
- Vídeo: 2.10. Linear approximation. Example of tangent plane for particular function.
- Vídeo: 2.11. Second-order derivatives.
- Vídeo: 2.12. Young's Theorem.
- Vídeo: 2.13. Hessian matrix.
Nota atribuída: Open and closed sets. Limits. Level curves.
Nota atribuída: Limits. Derivatives. Continuity
SEMANA 3
Implicit Function Theorems and their applications.
Week 3 of the Course is devoted to implicit function theorems. In this week three different
implicit function theorems are explained. This week students will grasp how to apply IFT
concept to solve different problems.
12 videos
- Vídeo: 3.2. Implicit Function Theorem.
- Vídeo: 3.3. Applications of the Implicit Function Theorem (part 1).
- Vídeo: 3.4. Applications of the Implicit Function Theorem (part 2).
- Vídeo: 3.5. Gradient is perpendicular to a level curve of a function.
- Vídeo: 3.6. Implicit function theorem for the function of many variables.
- Vídeo: 3.7. Example of application of the IFT for the function of many variables.
- Vídeo: 3.8. Implicit Function Theorem for the system of implicit functions. Jacobian matrix.
- Vídeo: 3.9. Example of application IFT for the system of implicit functions (part 1).
- Vídeo: 3.10. Example of application IFT for the system of implicit functions (part 2).
- Vídeo: 3.11. Example of application in microeconomics.
- Vídeo: 3.12. Cramer's rule.
Nota atribuída: Gradient.
SEMANA 4
Unconstrained and constrained optimization.
Week 4 of the Course is devoted to the problems of constrained and unconstrained optimization.
Of special attention are quadratic forms, critical points and their classification.
15 videos
- Vídeo: 4.2. Global max. Local max. Saddle point.
- Vídeo: 4.3. Unconstrained optimization.
- Vídeo: 4.4. Critical point. Taylor's formula.
- Vídeo: 4.5. Quadratic forms. Positive definiteness. Negative definiteness.
- Vídeo: 4.6. Sylvester's criterion (part 1).
- Vídeo: 4.7. Sylvester's criterion (part 2).
- Vídeo: 4.8. Examples of Hessians (part 1).
- Vídeo: 4.9. Example of Hessians (part 2).
- Vídeo: 4.10. Sufficient condition for a critical point to be a local maximum, a local minimum and neither of both.
- Vídeo: 4.11. Examples of finding and classification of critical points (part 1).
- Vídeo: 4.12. Examples of finding and classification of critical points (part 2).
- Vídeo: 4.13. Constrained optimization.
- Vídeo: 4.14. Lagrangian.
- Vídeo: 4.15. Example of constrained optimization problem.
Nota atribuída: Implicit function theorems
Nota atribuída: Partial derivatives and unconstrained optimization.
SEMANA 5
Constrained optimization for n-dim space. Bordered Hessian.
Week 5 of the Course is devoted to the extension of the constrained optimization problem to the
n-dimensional space. This week students will grasp how to apply bordered Hessian concept to
classification of critical points arising in different constrained optimization problems.
13 videos
- Vídeo: 5.2. Weierstrass theorem. Compact sets.
- Vídeo: 5.3. Bordered Hessian.
- Vídeo: 5.4. Constrained optimization in general case (part 1).
- Vídeo: 5.5. Constrained optimization in general case (part 2).
- Vídeo: 5.6. Application of the bordered Hessian in the constrained optimization.
- Vídeo: 5.7. Generalization of the constrained optimization problem for the n variables case.
- Vídeo: 5.8. Example of constrained optimization for the case of more than two variables (part 1).
- Vídeo: 5.9. Example of constrained optimization for the case of more than two variables (part 2).
- Vídeo: 5.10. Example of constrained optimization problem on non-compact set.
- Vídeo: 5.11. Theorem for determining definiteness (positive or negative) or indefiniteness of the bordered matrix.
- Vídeo: 5.12. Example of application bordered Hessian technique for the constrained optimization problem.
- Vídeo: 5.13. Example of violating NDCQ.
Nota atribuída: Unconstrained optimization
SEMANA 6
Envelope theorems. Concavity and convexity.
Week 6 of the Course is devoted to envelope theorems, concavity and convexity of functions.
This week students will understand how to interpret Lagrange multiplier and get to learn the
criteria of convexity and concavity of functions in n-dimensional space.
11 videos
- Vídeo: 6.2. Envelope Theorem for constrained optimization.
- Vídeo: 6.3. Examples of the Envelope Theorem application (part 1).
- Vídeo: 6.4. Examples of the Envelope Theorem application (part 2).
- Vídeo: 6.5. Interpretation of the Lagrangian multiplier.
- Vídeo: 6.6. Relaxing assumptions using second order conditions.
- Vídeo: 6.7. Concave and convex functions.
- Vídeo: 6.8. Concave and convex functions in n-dimensional case.
- Vídeo: 6.9. Inequality for concave function in n-dimensional space.
- Vídeo: 6.10. Criteria of concavity and convexity of the function in n-dimensional space.
- Vídeo: 6.11. Properties of concave functions.
Nota atribuída: Constrained optimization
SEMANA 7
Global extrema. Constrained optimization with inequality constraints.
Week 7 of the Course is devoted to identification of global extrema and constrained optimization
with inequality constraints. This week students will grasp the concept of binding constraints and
complementary slackness conditions.
11 videos
- Vídeo: 7.2. How to identify global extrema? (part 2)
- Vídeo: 7.3. How to identify global extrema? (part 3)
- Vídeo: 7.4. How to identify global extrema? (part 4)
- Vídeo: 7.5. Constrained optimization with inequality constraints.
- Vídeo: 7.6. Binding constraints. Complementary slackness conditions.
- Vídeo: 7.7. Constrained optimization problem with inequalities for n-dimensional space.
- Vídeo: 7.8. Constrained optimization problem with inequalities. Theorem.
- Vídeo: 7.9. Example of solving constrained optimization problem with inequalities (part 1).
- Vídeo: 7.10. Example of solving constrained optimization problem with inequalities (part 2).
- Vídeo: 7.11. Example of solving constrained optimization problem with inequalities (part 3).
Nota atribuída: Inequality constraints. Envelope theorem.
SEMANA 8
Kunh-Tucker conditions. Homogeneous functions.
Week 8 of the Course is devoted to Kuhn-Tucker conditions and homogenous functions. This
week students will find out how to use Kuhn-Tucker conditions for solving various economic
problems.
9 videos
- Vídeo: 8.2. Solving consumer choice problem using Kuhn-Tucker conditions (part 1).
- Vídeo: 8.3. Solving consumer choice problem using Kuhn-Tucker conditions (part 2).
- Vídeo: 8.4. Solving minimization costs problem using Kuhn-Tucker conditions (part 1).
- Vídeo: 8.5. Solving minimization costs problem using Kuhn-Tucker conditions (part 2).
- Vídeo: 8.6. Homogeneous functions.
- Vídeo: 8.7. Homogeneous functions. Two propositions.
- Vídeo: 8.8. Income-consumption curve.
- Vídeo: 8.9. Euler's Theorem.
Nota atribuída: Kuhn-Tucker conditions
Nota atribuída: Kuhn-Tucker conditions. Concavity, convexity.
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Desenvolvedores
National Research University Higher School of Economics
National Research University - Higher School of Economics (HSE) is one of the top research universities in Russia. Established in 1992 to promote new research and teaching in economics and related disciplines, it now offers programs at all levels of university education across an extraordinary range of fields of study including business, sociology, cultural studies, philosophy, political science, international relations, law, Asian studies, media and communications, IT, mathematics, engineering, and more.
Learn more on www.hse.ru
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