Syllabus for MATHEMATICS FOR ECONOMISTS
Lecturers: Kirill Bukin, Boris Demeshev
Class teachers: Boris Demeshev, Daniil Esaulov, Artem Kalchenko, Elena
Kochegarova, Petr Lukianchenko, Pavel Zhoukov
Course description
Mathematics for Economists is a two-semester course for the second year students
studying at ICEF.
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. 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
linear equations.
The assessment of the students will be by the University of London (UL) examinations
at the end of the fourth semester. During the fourth semester students are specially
prepared to the University of London examination.
The course covers several variable calculus, optimization theory and the selected topics
drawn from the theory of differential and difference equations. That course is aimed at
teaching students to master comparative statics problems, optimization problems and
dynamic models using the acquired mathematical tools. The course is taught in English.
The students are also studying for Russian degree in Economics, and knowing Russian
terminology through reading in Russian is also a must.
The latter means the thinking over the theoretical material, working on the home
assignments given by the lecturer. During each semester there will be a midterm
examination. Since the major part of the students has the final UoL exam “MathematicsI” while the students specializing in Economics have the two UoL exams “Mathematics
I” and “Mathematics-II”, midterm exams resemble UoL examinations to a great extent.
Students specializing in Math and Econ will take Calculus UoL exam in lieu of Math
I,II.
Teaching objectives.
The objectives of the course are:
to acquire the students’ knowledge in the field of mathematics and to make them ready to
analyze simulated as well as real economic situations;
to develop the students' ability to apply the knowledge of the differential and difference
equations which will enable them to analyze dynamics of the processes.
Teaching Methods
The course program consists of:
- lectures,
- classes,
- regular self-study.
Assessment and grade determination
Control takes the following forms:
 written home assignments (25);
 midterm exam in the middle of fall semester (120 min.),
 mock exam (180 min) in the University of London examination format in the
middle of the spring semester;
 written exam (120 min) at the end of the fall semester,
 University of London exam by the end of the spring semester (180 min),
Mathematics - I and Mathematics – II or calculus (see above).
The fall semester will be determined from the following activities:
 average grade for the home assignments (20%);
 fall semester midterm exam (20%);
 fall exam (60%).
Course grade for those students who specialize in Economics is determined by
 University of London exam grade for “Mathematics – I” (20%),
 University of London exam grade for “Mathematics – II” (50%),
 fall term grade (20%),
 mock exam grade (10%).
Course grade for those students who specialize in Mathematics and Economics is
determined by
 University of London exam grade for “Calculus MT1 174” (40%),
 fall term grade (40%),
 mock exam grade (20%).
Course grade for the rest of the students is determined by
 University of London exam grade for “Mathematics – I” (40%),
 fall term grade (40%),
 mock exam grade (20%).
Main Reading
1. [SB] Carl P. Simon and Lawrence Blume. Mathematics for Economists, W.W.
Norton and Co, 1994.
2. [C] A.C. Chiang. Fundamental Methods of Mathematical Economics, McGraw-Hill,
1984, 2008.
Additional Reading
1. B. P. Demidovich. Collection of problems and exercises on calculus, Moscow,
“Nauka”, 1966.
2. A.F. Fillipov. Collection of problems on differential equations. Moscow, “Nauka”,
1973.
3. Anthony M., and Biggs N., Mathematics for Economics and Finance, Cambridge
University Press, UK, 1996.
2
4. Anthony M., Reader in Mathematics, LSE, University of London; Mathematics for
Economists, Study Guide, University of London.
5. [A] M. Anthony. Further mathematics for economists. University of London, 2005
6. [G] Robert Gibbons. A Primer in Game Theory. Harvester Wheatsheaf, 1992 Internet resources
University of London Exam papers and Examiners reports for the last three years
http://www.londonexternal.ac.uk/current_students/programme_resources/lse/index.shtm
l.
Current course materials are post at the ICEF information system http://mief.hse.ru
Course outline.
Part I. Multi-dimensional calculus
1. Main concepts of set theory. Operations on sets. Direct product of sets. Relations and
functions. Level sets and level curves.
(SB Sections 2.1-2.2; C Sections1.1-2.7)
2. Space R n . Metric in n-dimensional space. The triangle inequality. Euclidean spaces.
Neighborhoods and open sets in R n , Sequences and their limits. Close sets. The closure
and the boundary of a set.
(SB Sections 10.1-10.4; C Sections12.1-12.6)
3. Functions of several variables. Limits of functions. Continuity of functions.
(SB Sections 13.1-13.5; C Sections 6.4-6.7)
4. Partial differentiation. Economic interpretation, marginal products and elasticities.
Chain rule for partial differentiation.
(SB Sections 14.1-14.3; C Section 7.4)
5. Total differential. Geometric interpretation of partial derivatives and the differential.
Linear approximation. Differentiability. Smooth functions. Directional derivatives and
gradient.
(SB Sections 14.4-14.6; C Sections 8.1-8.7)
6. Higher-order derivatives. Young’s theorem. Hessian matrix. Economic applications.
(SB Sections 14.8-14.9; C Sections 7.6, 9.3)
7. Implicit functions. Implicit function theorem.
(SB Sections 15.1-15.2; C Section 8.5)
8. Vector-valued functions. Jacobian.
(SB Section 14.7; C Section 8.5)
9. Implicit function theorem for the vector-valued functions.
(SB Sections 15.3, 15.5; C Section 8.5)
10. Economic applications of the IFT for the comparative statics problems.
(SB Section 15.4; C Section 8.6)
Part II. Optimization
11. Unconstrained optimization of the multi-dimensional functions. Stationary points.
First-order conditions.
3
(SB Sections17.1-17.2; C Sections11.1-11.2)
12. Second differential. Quadratic forms and the associated matrices. Definiteness and
semi-definiteness of the quadratic forms. Sylvester criterion. Second-order conditions
for extrema.
(SB Sections 16.1-16.2, 17.3-17.4; C Sections11.3-11.7)
13. Constrained optimization. Lagrangian function and multiplier. First-order conditions
for constrained optimization.
(SB Sections 18.1-18.2; C Sections 12.1-12.2)
14. 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.
(SB Sections 16.3-16.4, 19.3; C Section 12.3)
15. Economic meaning of a multiplier. Applications of the Lagrange approach in
economics. Smooth dependence on the parameters. Envelope theorem.
(SB Sections 18.7-19.2, 19.4; C Section 12.5)
Part III. Differential and difference equations
16. Dynamics 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.
(SB Sections 24.1-24.2; C Sections13.6, 14.1-14.3)
17. Qualitative theory of differential equations. Solow’s growth model. Phase diagram.
(Section 24.5; C Sections14.6-14.7)
18. Second-order linear differential equations with constant coefficients.
(SB Section 24.3; C Section15.1)
19. Complex numbers and operations on them. Representation of a number. De Moivre
and Euler formulae.
(SB Appendix A3; C Section 15.2)
20. 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).
(SB Section 24.3; C Sections15.3-15.7)
21. Discrete time economic systems. Difference equations. Method of solving firstorder equations. Convergence and oscillations of a solution. Cobweb model. Partial
equilibrium model with the inventory.
(SB Section 23.2; C Sections 16.2-16.6)
22. Second-order difference equations.
(C Sections 17.1-17.3)
23. Higher-order difference equations. Characteristic equation. Undetermined
coefficients method. Conditions for the stability of solutions.
(C Section 17.4)
Part IV. Optimization: further topics
24. Homogeneous functions. Cobb-Douglas production function. Properties of
homogeneous functions. Euler’s theorem. (SB, 20.1, pp. 483–493; C, 12.6, pp. 410–418)
25. Inequality constrained maximization problem of the two-variable function.
4
Modification of the first-order necessary conditions for the Lagrangian. The
complementary slackness condition. (SB, 18.3, pp. 424–430; C, Ch. 21: 21.1, pp. 716–722)
26. First-order necessary conditions for the Lagrangian in the multidimensional case
(inequality constraints). Constraints qualification. (S 18.3, pp. 430–434; C, 21.3, 21.4, pp. 731–738; A, pp. 144– 146)
27. Inequality constrained minimization problem in the multidimensional case. Mixed
constraints. Kuhn-Tucker formulation of the first-order necessary conditions for the
Lagrangian function under non-negativity of instrumental variables. (SB, 18.4–18.6, pp. 434–442; C, 21.2, 21.4, pp. 722–731, 738– 744; A, pp. 146–150) 28. Economic applications of non-linear programs. Utility maximization subject to the
budget constraint. Maximization of sales taken into account the advertising costs. (SB, 18.4–18.7, pp. 442–447; C, 21.6, pp. 747–754)
29. The economic meaning of Lagrange multipliers. The envelope theorem. Smooth
dependence of the optimal value on parameters. (SB, 19.1–19.2, 19.4, pp. 448–457; A, pp. 150–156)
30. Linear programming. The diet problem. Optimal production under resources
constraints. Graphical solution of a linear program for two instrumental variables.
(C, 19.1, pp. 651–661)
31. 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, 19.2–19.6, pp. 661–687; A, pp. 146–150)
32. Theorems of linear programming. Existence theorem. Duality theorem.
Complementary slackness theorem. (C, 20.2, pp. 696–700; A, pp. 146–150)
33. Economic interpretation of the dual program. Dual variables and shadow prices.
Profits maximization and costs minimization. (C, 19.2–19.6, pp. 661–687) 34. An example of game theory analysis of a real battle state during the World War II.
Prisoners dilemma. Games, players and strategies. Normal form representation of a
static game. Eliminating of strictly dominated strategies. Solution of a game. (G, 1.1.A–1.1.B, pp. 1–8)
35. Nash equilibrium. Bertrand model. Cournot model. Nash theorem. Pure and mixed
strategies. Searching for Nash equilibria. (G, 1.1.C–1.3.B, pp. 8–48)
36. Zero-sum games. Von Neumann equilibrium. Optimal strategies in zero-sum games
and duality in linear programming.
(A, p. 167–171)
Distribution of hours
№ Topic
Part I. Multi-dimensional calculus
1. Main concepts of set theory. Operations on sets.
Direct product of sets. Relations and functions.
Level sets and level curves.
Total
Lectures
14
4
Classes Self study
4
6
5
2.
3.
4
5
6
7
8
9
10
11
12
13
14
15
Space R n . Metric in n-dimensional space. The
triangle
inequality.
Euclidean
spaces.
n
Neighborhoods and open sets in R , 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
14
4
4
6
14
4
4
6
12
4
2
6
12
4
2
6
12
4
2
6
12
4
2
6
Vector-valued functions. Jacobian
Implicit function theorem for the vector-valued
functions
Economic applications of the IFT for the
comparative statics problems.
Part II. Optimization
Unconstrained optimization of the multidimensional functions. Stationary points. Firstorder conditions
Second differential. Quadratic forms and the
associated matrices. Definiteness and semidefiniteness 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
Part III. Differential and difference equations
12
12
4
4
2
2
6
6
12
4
2
6
12
4
2
6
10
2
2
6
10
2
2
6
10
2
2
6
10
2
2
6
6
16 Dynamics 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.
17 Qualitative theory of differential equations.
Solow’s growth model. Phase diagram
18 Second-order linear differential equations with
constant coefficients
19 Complex numbers and operations on them.
Representation of a number. De Moivre and Euler
formulae
20 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).
21 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
22 Second-order difference equations
23 Higher-order difference equations. Characteristic
equation. Undetermined coefficients method.
Conditions for the stability of solutions
Part IV. Optimization: further topics
functions.
Cobb-Douglas
24 Homogeneous
production function. Property of homogeneous
functions. Euler’s theorem.
25 Inequality constrained maximization problem in
two-variables. Modification of the first-order
necessary conditions for the Lagrangian function.
Complementary slackness condition.
26 Several variables and several inequality
constraints generalization of the first-order
necessary conditions for the Lagrangian function.
Constraints qualification.
27 Inequality constrained minimization problem.
Mixed constraints. Kuhn-Tucker formulation.
28 Economic applications of non-linear programs.
Utility maximization subject to the budget
constraint. Maximization of sales.
29 The economic meaning of Lagrange multipliers.
The envelope theorem. Smooth dependence of the
optimal value on parameters.
10
2
2
6
10
2
2
6
10
2
2
6
10
2
2
6
12
2
2
8
12
2
2
8
14
14
2
2
2
2
10
10
8
2
2
4
8
2
2
4
8
2
2
4
16
4
4
8
8
2
2
4
8
2
2
4
7
30 Linear programming. The diet problem. Optimal
production under resources constraints. Graphical
solution of a linear program for two instrumental
variables case.
31 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.
32 Theorems of linear programming. Existence
theorem. Duality theorem. Complementary
slackness theorem.
33 Economic interpretation of the dual program. Dual
variables and shadow prices. Profits maximization
and costs minimization.
34 Prisoners dilemma. Games, players and strategies.
Normal form representation of a static game.
Eliminating of strictly dominated strategies.
Solution of a game.
35 Nash equilibrium. Bertrand model. Cournot
model. Nash theorem. Pure and mixed strategies.
Searching for Nash equilibria.
36 Zero-sum games. Von Neumann equilibrium.
Optimal strategies in zero-sum games and duality
in linear programming.
Total:
8
8
2
2
4
8
2
2
4
8
2
2
4
8
2
2
4
8
2
2
4
16
4
4
8
8
2
2
4
390
98
82
210