METHODS OF OPTIMIZATION
Lecturer: Kirill A. Bukin
Class teachers: Boris B. Demeshev, Daniil Esaulov, Artem Kalchenko
Course description
The course Methods of Optimization is one-semester course for Year 2 ICEF students.
It is a part of the Bachelor programme in economics. The course has to give skills of
implementation of mathematical knowledge and practice to both theoretical and applied
economic problems. Its prerequisites are knowledge and skills of one-variable calculus, linear
algebra including general theory of systems of linear equations and matrix algebra.
The course covers inequality-constraints optimization, linear programming and selected
topics of game theory. The contents of the course have to teach students to solve and analyze
different optimization problems using developed mathematical tools.
Teaching objectives
By the end of semester a student should have skills of implementation of mathematical concepts
to solution of microeconomic and macroeconomic problems.
A student has to be able to investigate optimization problems in different economic applications.
Teaching methods
The course program consists of lectures, classes and regular students’ work without assistance.
The latter means thinking over lectures’ material, their extension and doing of assignments given
by the teacher. During the course there will be a mid-term exam.
Assessment
•Final exam (120 min.)
•Home assignments (11)
• Mid-term exam (120 min.)
Grade determination
The final grade is determined by:
•Final exam 60%
•Home assignments 20%
• Mid-term exam 20%
Main reading
Carl P. Simon and Lawrence Blume. Mathematics for Economists, W.W. Norton and Co, 1994.
A.C. Chiang. Fundamental Methods of Mathematical Economics, McGrow-Hill, 1984, 2008.
Additional reading
1. Б.П. Демидович. Сборник задач и упражнений по математическому анализу, М.,
«Наука», 1966.
2. А.Ф. Филиппов. Сборник задач по дифференциальным уравнениям. М., «Наука»,
1973.
3. Anthony M., and Biggs N., Mathematics for Economics and Finance, Cambridge
University Press, UK, 1996.
4. Anthony M., Reader in Mathematics, LSE, University of London; Mathematics for
Economists, Study Guide, University of London.
5. Robert Gibbons. A Primer in Game Theory. Harvester Wheatsheaf, 1992.
6. M. Anthony. Further Mathematics for Economists. University of London, 2005.
Course outline
1. Homogeneous functions. Cobb-Douglas production function. Property of homogeneous
functions. Euler’s theorem.
(SL Section 20.1; C Section 12.6)
2. Inequality constrained maximization problem of two-variable function. Modification
of the first-order necessary conditions for the Lagrangian function. The
complementary slackness condition.
(SL Section 18.3; C Section 21.1)
3. Several variables and several inequality constraints generalization of the first-order
necessary conditions for the Lagrangian function. Constraints qualification.
(SL Section 18.3; C Sections 6, 21.3-21.4)
4. Inequality constrained minimization problem of several variables function. Mixed
constraints. Kuhn-Tucker formulation of the first-order necessary conditions for the
Lagrangian function under non-negativity of instrumental variables.
(SL Sections 18.4-18.6; C Sections 6, 21.2-21.4)
5. Economic applications of non-linear programs. Utility maximization subject to the
budget constraint. Maximization of sales taking into account advertising costs.
(SL Sections 18.4-1876; C Section 21.6)
6. The economic meaning of Lagrange multipliers. The envelope theorem. Smooth
dependence of the optimal value on parameters.
(SL Sections 19.1-19.2, 19.4; C Section 6)
7. Linear programming. The diet problem. Optimal production under re- sources
constraints. Graphical solution of a linear program for two instrumental variables
case.
(C Section 19.1)
8. Standard formulation of a general linear program. The first order conditions for a
linear program, a solution’s property. Concept of a simplex method. A dual program
for a linear program.
( C Sections 6, 19.2-19.6)
9. Theorems of linear programming.
Complementary slackness theorem.
(C Sections 6, 20.2)
Existence
theorem.
Duality
theorem.
10. Economic interpretation of the dual program. Dual variables and shadow prices.
Profits maximization and costs minimization.
( C Sections19.2-19.6)
11. Prisoners dilemma. Normal form representation of a static game. Eliminating of
strictly dominated strategies. Solution of a game.
(R. Gibbons, Section 1.1)
12. Nash equilibrium. Bertrand model. Cournot model. Nash theorem. Pure and mixed
strategies. Searching for Nash equilibria in 2*2 games.
(R. Gibbons, Sections 1.1-1.3)
13. Zero-sum games. Von Neumann equilibrium. Optimal strategies in zero- sum games
and duality in linear programming.
(M. Anthony. Further Mathematics for Economists. pp. 167-171)
Distribution of hours
№
Topic
Total
In-class hours
Self-study
Lectures
Seminars
9
2
1
6
1.
Homogeneous functions
2.
Inequality-constrained maximization of a
function of two variables. Several
constraints generalisation of the firstorder necessary conditions for the
Lagrangian function
16
6
3
7
3.
Kuhn-Tucker formulation.
Economic applications of nonlinear programs
16
6
3
7
4.
Economic meaning of Lagrange
multipliers. Envelope theorem
9
2
1
6
5.
Linear programming. Theorems of linear
programming. Economic interpretation of
the dual program.
31
8
4
19
6.
Normal form representation of
a static game.
9
2
1
6
7.
Nash equilibrium
12
4
2
6
8.
Zero-sum games.
6
2
2
2
Total:
108
32
17
59