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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.
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.
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.
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.
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.
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.
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.
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.
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.