The Government of the Russian Federation

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The Government of the Russian Federation

The Federal State Autonomous Institution of Higher Education

"National Research University - Higher School of Economics"

Faculty of Business Informatics

Department of Innovation and Business in Information Technology

Course Title “Economic an Mathematic Modeling”

Master’s Program 38.04.05 “Big Data Systems”

Author:

Dr. Sci., Prof. Andrey Dmitriev, a.dmitriev@hse.ru

Moscow, 2015

This document may not be reproduced or redistributed by other Departments of the University without permission of the Authors.

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Field of Application and Regulations

The course "Economic and Mathematic Modeling" syllabus lays down minimum requirements for stu-dent’s knowledge and skills; it also provides description of both contents and forms of training and assess-ment in use. The course is offered to students of the Master’s Program "Big Data Systems" (area code 080500.68) in the Faculty of Business Informatics of the National Research University "Higher School of Economics". The course is a part of the curriculum pool of required courses (1st year, M.1.Б Core courses, M.1 Courses required by the standard 38.04.05 of the 2015-2016 academic year’s curriculum), and it is a one-module course (1st module). The duration of the course amounts to 32 class periods (both lecture and seminars) divided into 12 lecture hours and 20 practice hours. Besides, 82 academic hours are set aside to students for self-studying activity.

The syllabus is prepared for teachers responsible for the course (or closely related disciplines), teaching assistants, students enrolled on the course "Economic and Mathematic Modeling" as well as ex-perts and statutory bodies carrying out assigned or regular accreditations in accordance with

 educational standards of the National Research University – Higher School of Economics,

 curriculum ("Business Informatics", area code 38.04.05), Big Data Systems specialization, 1st year, 2015-2016 academic year.

1 Course Objectives

The main objective of the Course is to present, examine and discuss with students fundamentals and prin-ciples of economic and mathematic modeling. This course is focused on understanding the role of mathe-matic modeling for quantitative analysis of stochastic and dynamic economic systems.

Generally, the objective of the course can be thought as a combination of the following constituents:  familiarity with peculiarities of bifurcation theory, catastrophe theory, chaos theory, Levy random

walk and minority games theory as applied areas related to economic and mathematic modeling,  understanding of the main notions of dynamic and stochastic systems theory; the framework of

dy-namic and stochastic modeling as the most significant areas of economic systems studies, under-standing of the main notions of dynamic and stochastic systems theory; the framework of dynamic and stochastic modeling as the most significant areas of economic systems studies,

 understanding of the role of mathematic modeling in financial and economic modeling,  obtaining skills in utilizing nonlinear dynamic modeling in economic problem solving,  obtaining skills in utilizing stochastic modeling in financial problem solving,

 understanding of the role of equilibrium theory and instabilities in economic modeling,  understanding of the role of stable distributions in financial process modeling.

2 Students' Competencies to be Developed by the Course

While mastering the course material, the student will

 know main notions of the bifurcation theory, catastrophe theory, chaos theory, random walk theory and stochastic process theory,

 acquire skills of analyzing and solving economic and mathematic problems,

 gain experience in economic and mathematic modeling with use main notions of the bifurcation theory, catastrophe theory, chaos theory, random walk theory and stochastic process theory. In short, the course contributes to the development of the following professional competencies:

Ccompetencies

FSES/ HSE code

Descriptors – main mastering features (indicators of result

achievement)

Training forms and methods contributing

to the formation and development of

competence Ability to offer concepts,

models, invent and test

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Ccompetencies

FSES/ HSE code

Descriptors – main mastering features (indicators of result

achievement)

Training forms and methods contributing

to the formation and development of

competence thods and tools for

profes-sional work

Ability to apply the methods of system analysis and mod-eling to assess, design and strategy development of en-terprise architecture

PC-13 Owns and uses Lecture, practice, homeworks

Ability to develop and im-plement economic and ma-thematical models to justify the project solu-tions in the field of information and computer technology

PC-14 Owns and uses Lecture, practice, homeworks

Ability to organize self and collective research work in the enterprise and manage it

PC-16 Demonstrates Lecture, practice, homeworks

3 The Course within the Program’s Framework

The course "Economic and Mathematic Modeling" syllabus lays down minimum requirements for stu-dent’s knowledge and skills; it also provides description of both contents and forms of training and assess-ment in use. The course is offered to students of the Master’s Program "Big Data Systems" (area code 080500.68) in the Faculty of Business Informatics of the National Research University "Higher School of Economics". The course is a part of the curriculum pool of required courses (1st year, M.1.Б Core courses, M.1 Courses required by the standard 080500.68 of the 2015-2016 academic year’s curriculum), and it is a one-module course (1st module). The duration of the course amounts to 32 class periods (both lecture and seminars) divided into 12 lecture hours and 20 practice hours. Besides, 82 academic hours are set aside to students for self-studying activity.

Academic control forms include

 1 class assignment, which implies problems solving in the end of 1st module; material to be covered by class assignment is fully determined by both course schedule and topics discussed by the corres-ponding date,

 8 homeworks are done by students individually, herewith each student has to prepare electronic (PDF format solely) report; all reports have to be submitted in LMS; all reports are checked and graded by the instructor on ten-point scale by the end of the 1st module,

 pass-final examination, which implies written test and computer-based problem solving. The Course is to be based on the acquisition of the following courses:

 Calculus  Linear Algebra

 Probability Theory and Mathematical Statistics  Macroeconomics

 Microeconomics

The Course requires the following students' competencies and knowledge:

 main definitions, theorems and properties from Calculus, Linear Algebra, Probability Theory and Mathematical Statistics, Macroeconomics and Microeconomics courses,

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 ability to search for, process and analyze information from a variety of sources. Main provisions of the course should be used to further the study of the following courses:

 Advanced Data Analysis and Big Data for Business  Predictive Modeling

 Applied Machine Learning

4 Thematic Course Contents

№ Title of the topic / lecture

Hours (total number) Class hours Indepen-dent work Lec-tures Semi-nars Practice

1 Economical dynamics, growth and equili-brium

2 2 8

2 Systems of first-order differential equations and economic dynamics

2 2 10

3 Bifurcation theory and economic dynamics 2 4 12

4 Catastrophe theory and economic dynamics 2 2 12

5 Chaos theory and economic dynamics 1 4 10

6 Probability distribution function of stock market instruments. Correlation in the stock market

1 2 10

7 Random walk in financial market models 1 2 10

8 Minority games 1 2 10

TOTAL 12 20 82

5 Forms and Types of Testing

Type of

control

Form of con-trol

1 year Department Parameters

1 2 3 4 Current (week) Class as-signment week 8 Innovation and Busi-ness in In-formation Technology

problems solving, written report (paper) – 120 minute

Homework week 1 to week 8

LMS electronic report Resultant Pass-fail

ex-am

week 9 written test (paper) and

computer-based problem solving

Evaluation Criteria

Current and resultant grades are made up of the following components:

 1 class assignment

implies problems solving in the end of 1st module; material to be covered by class assignment is fully de-termined by both course schedule and topics discussed by the corresponding date. The class assignment (CA) is assessed on the ten-point scale.

 8 homeworks

are done by students individually, herewith each student has to prepare electronic (PDF format solely) re-port. All reports have to be submitted in LMS. All reports are checked and graded by the instructor on ten-point scale by the end of the 1st module. All homeworks (HW) is assessed on the ten-point scale summary.

 pass-final examination

implies written test (WT) and computer-based problem solving (CS). Finally, the total course grade on ten-point scale is obtained as

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O(Total) = 0,2 * O(HW) + 0,4 * O(CA) + 0,1 * O(WT) + 0,3 * O(CS).

A grade of 4 or higher means successful completion of the course ("pass"), while grade of 3 or low-er means unsuccessful result ("fail"). Convlow-ersion of the concluding rounded grade O(Total) to five-point scale grade.

6 Detailed Course Contents

Lecture 1. Economical dynamics, growth and equilibrium Lecture’s content:

 Solow-Swan model. Assumptions of the model. Mathematics of the model. Balanced-growth equi-librium. Golden rule. Production function. Dynamical system. Stability of the dynamical system: Lyapunov stability, asymptotic stability, orbital stability. Dynamic equilibrium. Koopmans theorem (existence of an equilibrium in the Solow model). Arrow-Gurwicz theorem (existence of stable equilibrium). Asymptotic stability of the model.

 Attractors and repellers.

 Walrasian and Marshallian equilibriums. Open and closed systems. Dynamic and static equilibrium. Practice 1. Ramsey-Cass-Koopmans model. Problem of consumer choice.

Pon-tryagin’s maximum principle. General economic equilibrium. Modified golden rule. At-tractor of Ramsey-Cass-Koopmans model.

Materials required

1. Zang W.B. synergetic Economics. Time and Change in Nonlinear Economics. Springer, 1991. Recommended readings

1. Shone R. Economic Dynamic. Phase Diagrams and Their Economic Applications. 2nd Edition. Cambridge Press, 2002

2. Zang W.B. Differential Equations, Bifurcations, and Chaos in Economics. World Scientific Pub-lishing Company, 2005.

3. Puu T. Attractors, Bifurcations, and Chaos: Nonlinear Phenomena in Economics, 2nd Edition, Springer, 2003.

4. Rosser J.B. From Catastrophe to Chaos: A General Theory of Economic Discontinuities. 2nd Edi-tion, Springer, 2000.

5. Stachurski D. Economic Dynamics: Theory and Computation. MIT Press, 2009.

Lecture 2. Systems of first-order differential equations and economic dynamics Lecture’s content:

 Goodwin model. Lotka-Volterra equation. Equilibrium points. Linearization. Jacobian matrix. Ei-genvalues of the matrix. Equilibrium points of linear autonomous system: saddle point, stable node, unstable node, stable focus, unstable focus, centre. Phase diagram. The principle of linearized sta-bility. Equilibrium points of Lotka-Volterra equation. Structural stasta-bility. Structural instability of Lotka-Volterra model. Conservative system. Conservative Lotka-Volterra system.

 Slow and fast variables. Tikhonov theorem on dynamical systems.

Practice 2. The simplest model of competition between two firms. Competition model with a limited production growth. Bazykin model. Mankiw-Romer-Weil model.

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5. Stachurski D. Economic Dynamics: Theory and Computation. MIT Press, 2009. Lecture 3. Bifurcation theory and economic dynamics Lecture’s content:

 Modeling regional dynamics. Local bifurcations: saddle-node (fold) bifurcation, transcritical bifur-cation, Pitchfork bifurbifur-cation, period-doubling (flip) bifurbifur-cation, Hopf bifurbifur-cation, Neimark bifurca-tion.

 Global bifurcations: homoclinic bifurcation, heteroclinic bifurcation, infinite-period bifurcation, blue sky catastrophe. Codimension of a bifurcation. Bifurcation diagram. Flows. Hopf theorem. The Poincare-Bendixson theorem.

 Limit cycle (attractor). Stable, unstable and semi-stable limit cycle.

Practice 3. Dynamic transportation modal choice. Oscillations in van der Ploeg’s hybrid growth model. Periodic optimal employment policy. Optimal economic growth

associated with endogenous fluctuations. Materials required

5. Stachurski D. Economic Dynamics: Theory and Computation. MIT Press, 2009. Lecture 4. Catastrophe theory and economic dynamics

Lecture’s content:

 Business cycles in the Kaldor model. Structural stability. Morse lemma. Thom theorem. Morse crit-ical points. Degenerate critcrit-ical points. Thom elementary catastrophes.

 Potential functions of one active variable: fold catastrophe, cusp catastrophe, Swallowtail phe, butterfly catastrophe. Potential functions of two active variables: hyperbolic umbilic catastro-phe, elliptice umbilic catastrocatastro-phe, parabolic umbilic catastrophe.

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Practice 4. Resource management. Multiple equilibria in Wilson’s retail model. Stock market forecasting.

5. Stachurski D. Economic Dynamics: Theory and Computation. MIT Press, 2009. Lecture 5. Chaos theory and economic dynamics

Lecture’s content:

 Chaotic dynamic price formation. Lorenz system. Properties of the Lorenz system: homogeneity, symmetry, dissipativity, bounded trajectories. Equilibrium points of the Lorenz system. Lyapunov stability of equilibrium points. Lorenz attractor. Lorenz map.

 Measures of chaos: Lyapunov exponent, correlation function, Hausdorff-Besicovitch fractal dimen-sion, Renyi fractal dimension. Fractal. Sensitivity to initial conditions. Strange attractors.

 Chaos and economic forecasting. Deterministic systems and time series.

Practice 5. Chaotic dynamic of cities. Chaos in an international economic mod-el.

5. Stachurski D. Economic Dynamics: Theory and Computation. MIT Press, 2009. Lecture 6. Probability distribution function of stock market instruments.

Correlation in the stock market Lecture’s content:

 Empirical distributions of stock returns and number of shares. Stable distribution. Characteristic function. Properties of stable distribution: infinitely divisible, leptokurtotic, closure under convolu-tion. A generalized central limit theorem.

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 Levy distribution. Probability density function and characteristic function of the Levy distribution. Autocorrelation function and spectral density. Higher-order correlation: the volatility. Stationarity of price changes.

Practice 6. Empirical distributions of share volumes. Empirical distributions of the number of transactions. Empirical distributions of time interval between

transac-tions. Materials required

1. Mantegna R.N., Stanley H.E. An Introduction to Econophysics, Correlation and Complexity in Finance. Cambridge University Press, 2000.

Recommended readings

1. Malliaris A.G., Brock W.A. Stochastic Methods in Economics and Finance. North Holland, 1988.

2. Rachev S.T., Kim Y.S. Financial Models with Levy processes and Volatility Clustering. Wiley, 2011.

3. Schoutens W. Levy Processes in Finance: Pricing Financial Derivatives. Wiley, 2003. 4. Bottasso A. Stable Levy Processes in Finance: Economics. Lambert, 2011.

5. Challet D., Marsili M. Minority Games: Interacting Agents in Financial Markets. Oxford Press, 2013

Lecture 7. Random walk in financial market models Lecture’s content:

 Options pricing. Lattice random walk. One-dimensional random walk. Gaussian random walk. The speed of convergence. Berry-Essen theorem 1. Berry-Essen theorem 2. Basin of attraction.

 Levy flight. Mandelbrot survival function. Scale invariant. Truncated Levy flights and fluctuations of stock market instruments. Functional Levy flight.

 Holtsmark distribution.

Practice 7. Black-Scholes equation. Materials required

1. Mantegna R.N., Stanley H.E. An Introduction to Econophysics, Correlation and Complexity in Finance. Cambridge University Press, 2000.

Lecture 8. Minority games Lecture’s content:

Price dynamics. Formulation of the minority game. Thermal minority game. Minority game without infor-mation. Grand-canonical minority game. Analytic approach.

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Practice 8. Speculative trading. Materials required

7 Educational Technology

During classes various types of active methods are used: analysis of practical problems, group work, computer simulations in computational software program Mathematica, distance learning with use LMS.

8 Methods and Materials for Current Control and Attestation

8.1 Example of Problems for Class Assignment

Problem 1. Solve the nonhomogeneous differential equation = ( ) − (1 − )

for (0) = .

Show that for > 0 and 0 < < 1 the equilibrium ∗_{is asymptotically stable.} Problem 2. Given the following parameters for the Solow growth model

= 4, = 0.25, = 0.1, = 0.4, = 0.03 (i) to plot the graph of ( )

(ii) plot the function ̇ = − ( + )

(iii) Linearise ̇ about the equilibrium in (ii) and establish whether it is stable or unstable. Problem 3. For the following Holling–Tanner predatory–prey model

̇ = 1 −

6 −

6 8 + 8 ̇ = 0.2 1 −0.4

(i) Find the fixed points.

(ii) Do any of the fixed points exhibit a stable limit cycle? Problem 4. The value of a share at time is t

( ) = + ( ),

where > 0 and [ ( ), ≥ 0] is a Brownian motion with positive drift parameter μ and variance parame-ter σ2. At time point t=0 a speculator acquires an American call option on this share with finite expiry date τ. Assume that

+ > 3 √ , 0 ≤ ≤ .

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(ii) When should the speculator exercise to make maximal mean undiscounted profit? 8.2 Questions for Pass-Final Examination

Economic Models

1. Solow-Swan model. Assumptions of the model. Mathematics of the model. Balanced-growth equi-librium. Golden rule. Production function.

2. Ramsey-Cass-Koopmans model. Problem of consumer choice. Pontryagin’s maximum principle. General economic equilibrium. Modified golden rule.

3. Goodwin model.

4. The simplest model of competition between two firms. 5. Competition model with a limited production growth. 6. Bazykin model.

7. Mankiw-Romer-Weil model. Accounting for external. 8. Modeling regional dynamics.

9. Dynamic transportation modal choice.

10.Oscillations in van der Ploeg’s hybrid growth model. 11.Periodic optimal employment policy.

12.Optimal economic growth associated with endogenous fluctuations. 13.Business cycles in the Kaldor model.

14.Resource management.

15.Multiple equilibria in Wilson’s retail model. 16.Stock market forecasting.

17.Chaotic dynamic price formation. 18.Chaotic dynamic of cities.

19.Chaos in an international economic model.

20.Empirical distributions of stock returns and number of shares. 21.Empirical distributions of share volumes.

22.Empirical distributions of the number of transactions.

23.Empirical distributions of time interval between transactions. 24.Options pricing.

25.Black-Scholes equation. 26.Price dynamics.

27.Speculative trading.

Mathematical Foundations

1. Dynamical system. Stability of the dynamical system: Lyapunov stability, asymptotic stability, or-bital stability.

2. Dynamic equilibrium. Koopmans theorem (existence of an equilibrium in the Solow model). 3. Arrow-Gurwicz theorem (existence of stable equilibrium). Asymptotic stability of the model. 4. Attractor of Ramsey-Cass-Koopmans model.

5. Attractors and repellers.

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7. Lotka-Volterra equation. Equilibrium points. Linearization. Jacobian matrix. Eigenvalues of the matrix. Equilibrium points of linear autonomous system: saddle point, stable node, unstable node, stable focus, unstable focus, centre. Phase diagram.

8. The principle of linearized stability. Equilibrium points of Lotka-Volterra equation. Structural sta-bility. Structural instability of Volterra model. Conservative system. Conservative Lotka-Volterra system. Attrractors. Slow and fast variables.

9. Tikhonov theorem on dynamical systems.

10.Local bifurcations: saddle-node (fold) bifurcation, transcritical bifurcation, Pitchfork bifurcation, period-doubling (flip) bifurcation, Hopf bifurcation, Neimark bifurcation.

11.Global bifurcations: homoclinic bifurcation, heteroclinic bifurcation, infinite-period bifurcation, blue sky catastrophe. Codimension of a bifurcation. Bifurcation diagram. Flows. Hopf theorem. The Poincare-Bendixson theorem.

12.Limit cycle (attractor). Stable, unstable and semi-stable limit cycle.

13.Structural stability. Morse lemma. Thom theorem. Morse critical points. Degenerate critical points. Thom elementary catastrophes.

14.Potential functions of one active variable: fold catastrophe, cusp catastrophe, Swallowtail catastro-phe, butterfly catastrophe.

15.Potential functions of two active variables: hyperbolic umbilic catastrophe, elliptice umbilic catas-trophe, parabolic umbilic catastrophe.

16.Lorenz system. Properties of the Lorenz system: homogeneity, symmetry, dissipativity, bounded trajectories.

17.Equilibrium points of the Lorenz system. Lyapunov stability of equilibrium points. Lorenz attractor. Lorenz map.

18.Measures of chaos: Lyapunov exponent, correlation function, Hausdorff-Besicovitch fractal dimen-sion, Renyi fractal dimension.

19.Fractal. Sensitivity to initial conditions. Strange attractors. 20.Chaos and economic forecasting.

21.Deterministic systems and time series.

22.Stable distribution. Characteristic function. Properties of stable distribution: infinitely divisible, lep-tokurtotic, closure under convolution.

23.A generalized central limit theorem.

24.Levy distribution. Probability density function and characteristic function of the Levy distribution. 25.Autocorrelation function and spectral density. Higher-order correlation: the volatility.

26.Lattice random walk. One-dimensional random walk. Gaussian random walk. The speed of conver-gence.

27.Berry-Essen theorem 1. Berry-Essen theorem 2. 28.Basin of attraction.

29.Levy flight.

30.Mandelbrot survival function. Scale invariant.

31.Truncated Levy flights and fluctuations of stock market instruments. 32.Functional Levy flight.

33.Holtsmark distribution.

34.Formulation of the minority game. 35.Thermal minority game.

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36.Minority game without information.

37. Grand-canonical minority game.

Example of Problem Consider the following Walrasian price and quantity adjustment model

( ) = 0.5 + 0.25

( ) = −0.025 + 0.75 − 6 + 40 ̇ = 0.75 , ( ) −

̇ = 2[ − ′( )]

(i) What is the economically meaningful fixed point of this system? (ii) Does this system have a stable limit cycle?

9 Teaching Methods and Information Provision

9.1 Core Textbook

Mantegna R.N., Stanley H.E. An Introduction to Econophysics, Correlation and Complexity in Finance. Cambridge University Press, 2000.

Zang W.B. Synergetic Economics. Time and Change in Nonlinear Economics. Springer, 1991 Challet D., Marsili M. Minority Games: Interacting Agents in Financial Markets. Oxford Press, 2013 9.2 Required Reading

Shone R. Economic Dynamic. Phase Diagrams and Their Economic Applications. 2nd Edition. Cambridge Press, 2002

Zhang W.B. Differential Equations, Bifurcations, and Chaos in Economics. World Scientific Publishing Company, 2005.

Malliaris A.G., Brock W.A. Stochastic Methods in Economics and Finance. North Holland, 1988. Rachev S.T., Kim Y.S. Financial Models with Levy processes and Volatility Clustering. Wiley, 2011. 9.3 Supplementary Reading

Puu T. Attractors, Bifurcations, and Chaos: Nonlinear Phenomena in Economics, 2nd Edition, Springer, 2003.

Rosser J.B. From Catastrophe to Chaos: A General Theory of Economic Discontinuities. 2nd Edition, Springer, 2000.

Stachurski D. Economic Dynamics: Theory and Computation. MIT Press, 2009. Schoutens W. Levy Processes in Finance: Pricing Financial Derivatives. Wiley, 2003. Bottasso A. Stable Levy Processes in Finance: Economics. Lambert, 2011.

Challet D., Marsili M. Minority Games: Interacting Agents in Financial Markets. Oxford Press, 2013 Ian Jacques. Mathematics for Economics and Business. Pearson, 7 edition, 2012.

Kenneth Shaw. Mathematical Modeling in Business and Economics: A Data-Driven Approach. Business Expert Press, September, 2014

9.4 Handbooks

Gardiner C. Stochastic Methods: A Handbook for the Natural and Social Sciences. Springer, 2009 Schemedders K., Judd K. Handbook of Computational Economics. V.3. Elsevier, 2014

9.5 Software Mathematica v. 9.0

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9.6 Distance Learning

HSE Electronic Library access: Books24*7, Scopus, EBSCOHost, Science Direct, Web of Knowledge HSE Learning Management System

The Government of the Russian Federation