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1 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 Author: Dr. Sci., Prof. Andrey Dmitriev, [email protected] Moscow, 2015 This document may not be reproduced or redistributed by other Departments of the University without permission of the Authors.

2 Field of Application and Regulations The course "Economic and Mathematic Modeling" syllabus lays down minimum requirements for student s knowledge and skills; it also provides description of both contents and forms of training and assessment in use. The course is offered to students of the Master s Program "Big Data Systems" (area code ) 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 (1 st year, M.1.Б Core courses, M.1 Courses required by the standard of the academic year s curriculum), and it is a one-module course (1 st 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 experts 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 ), Big Data Systems specialization, 1 st year, academic year. 1 Course Objectives The main objective of the Course is to present, examine and discuss with students fundamentals and principles of economic and mathematic modeling. This course is focused on understanding the role of mathematic 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 dynamic and stochastic modeling as the most significant areas of economic systems studies, understanding 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: Training forms and Ccompetencies FSES/ HSE code Descriptors main mastering features (indicators of result achievement) methods contributing to the formation and development of competence Ability to offer concepts, models, invent and test me- SC-2 Demonstrates Lecture, practice, homeworks

3 Ccompetencies thods and tools for professional work Ability to apply the methods of system analysis and modeling to assess, design and strategy development of enterprise architecture Ability to develop and implement economic and mathematical models to justify the project solu-tions in the field of information and computer technology Ability to organize self and collective research work in the enterprise and manage it FSES/ HSE code Descriptors main mastering features (indicators of result achievement) Training forms and methods contributing to the formation and development of competence PC-13 Owns and uses Lecture, practice, homeworks PC-14 Owns and uses Lecture, practice, homeworks 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 student s knowledge and skills; it also provides description of both contents and forms of training and assessment in use. The course is offered to students of the Master s Program "Big Data Systems" (area code ) 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 (1 st year, M.1.Б Core courses, M.1 Courses required by the standard of the academic year s curriculum), and it is a one-module course (1 st 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 1 st module; material to be covered by class assignment is fully determined by both course schedule and topics discussed by the corresponding 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 1 st 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, ability to communicate both orally and in written form in English language,

4 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 Independent work Lecturenars Semi- Practice Economical dynamics, growth and equilibrium 2 Systems of first-order differential equations and economic dynamics 3 Bifurcation theory and economic dynamics Catastrophe theory and economic dynamics Chaos theory and economic dynamics Probability distribution function of stock market instruments. Correlation in the stock market Random walk in financial market models Minority games TOTAL Forms and Types of Testing Type of control Current (week) Form of control Class assignment Homework Resultant Pass-fail exam 1 year Department Parameters week 8 Innovation problems solving, written and Business report (paper) 120 in In- minute week 1 to formation LMS electronic report week 8 Technology 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 1 st module; material to be covered by class assignment is fully determined 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) report. All reports have to be submitted in LMS. All reports are checked and graded by the instructor on tenpoint scale by the end of the 1 st 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

5 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 lower means unsuccessful result ("fail"). Conversion of the concluding rounded grade O(Total) to five-point scale grade. 6 Detailed Course Contents Lecture 1. Economical dynamics, growth and equilibrium Solow-Swan model. Assumptions of the model. Mathematics of the model. Balanced-growth equilibrium. 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. Pontryagin s maximum principle. General economic equilibrium. Modified golden rule. Attractor of Ramsey-Cass-Koopmans model. 1. Zang W.B. synergetic Economics. Time and Change in Nonlinear Economics. Springer, Shone R. Economic Dynamic. Phase Diagrams and Their Economic Applications. 2 nd Edition. Cambridge Press, Zang W.B. Differential Equations, Bifurcations, and Chaos in Economics. World Scientific Publishing Company, Puu T. Attractors, Bifurcations, and Chaos: Nonlinear Phenomena in Economics, 2 nd Edition, Springer, Rosser J.B. From Catastrophe to Chaos: A General Theory of Economic Discontinuities. 2 nd Edition, Springer, Stachurski D. Economic Dynamics: Theory and Computation. MIT Press, Lecture 2. Systems of first-order differential equations and economic dynamics Goodwin model. 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. The principle of linearized stability. Equilibrium points of Lotka-Volterra equation. Structural stability. 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. Accounting for external.

6 1. Zang W.B. synergetic Economics. Time and Change in Nonlinear Economics. Springer, Shone R. Economic Dynamic. Phase Diagrams and Their Economic Applications. 2 nd Edition. Cambridge Press, Zang W.B. Differential Equations, Bifurcations, and Chaos in Economics. World Scientific Publishing Company, Puu T. Attractors, Bifurcations, and Chaos: Nonlinear Phenomena in Economics, 2 nd Edition, Springer, Rosser J.B. From Catastrophe to Chaos: A General Theory of Economic Discontinuities. 2 nd Edition, Springer, Stachurski D. Economic Dynamics: Theory and Computation. MIT Press, Lecture 3. Bifurcation theory and economic dynamics Modeling regional dynamics. Local bifurcations: saddle-node (fold) bifurcation, transcritical bifurcation, Pitchfork bifurcation, period-doubling (flip) bifurcation, Hopf bifurcation, Neimark bifurcation. 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. 1. Zang W.B. synergetic Economics. Time and Change in Nonlinear Economics. Springer, Shone R. Economic Dynamic. Phase Diagrams and Their Economic Applications. 2 nd Edition. Cambridge Press, Zang W.B. Differential Equations, Bifurcations, and Chaos in Economics. World Scientific Publishing Company, Puu T. Attractors, Bifurcations, and Chaos: Nonlinear Phenomena in Economics, 2 nd Edition, Springer, Rosser J.B. From Catastrophe to Chaos: A General Theory of Economic Discontinuities. 2 nd Edition, Springer, Stachurski D. Economic Dynamics: Theory and Computation. MIT Press, Lecture 4. Catastrophe theory and economic dynamics Business cycles in the Kaldor model. Structural stability. Morse lemma. Thom theorem. Morse critical points. Degenerate critical points. Thom elementary catastrophes. Potential functions of one active variable: fold catastrophe, cusp catastrophe, Swallowtail catastrophe, butterfly catastrophe. Potential functions of two active variables: hyperbolic umbilic catastrophe, elliptice umbilic catastrophe, parabolic umbilic catastrophe.

7 Practice 4. Resource management. Multiple equilibria in Wilson s retail model. Stock market forecasting. 1. Zang W.B. synergetic Economics. Time and Change in Nonlinear Economics. Springer, Shone R. Economic Dynamic. Phase Diagrams and Their Economic Applications. 2 nd Edition. Cambridge Press, Zang W.B. Differential Equations, Bifurcations, and Chaos in Economics. World Scientific Publishing Company, Puu T. Attractors, Bifurcations, and Chaos: Nonlinear Phenomena in Economics, 2 nd Edition, Springer, Rosser J.B. From Catastrophe to Chaos: A General Theory of Economic Discontinuities. 2 nd Edition, Springer, Stachurski D. Economic Dynamics: Theory and Computation. MIT Press, Lecture 5. Chaos theory and economic dynamics 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 dimension, 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 model. 1. Zang W.B. synergetic Economics. Time and Change in Nonlinear Economics. Springer, Shone R. Economic Dynamic. Phase Diagrams and Their Economic Applications. 2 nd Edition. Cambridge Press, Zang W.B. Differential Equations, Bifurcations, and Chaos in Economics. World Scientific Publishing Company, Puu T. Attractors, Bifurcations, and Chaos: Nonlinear Phenomena in Economics, 2 nd Edition, Springer, Rosser J.B. From Catastrophe to Chaos: A General Theory of Economic Discontinuities. 2 nd Edition, Springer, Stachurski D. Economic Dynamics: Theory and Computation. MIT Press, Lecture 6. Probability distribution function of stock market instruments. Correlation in the stock market Empirical distributions of stock returns and number of shares. Stable distribution. Characteristic function. Properties of stable distribution: infinitely divisible, leptokurtotic, closure under convolution. A generalized central limit theorem.

8 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 transactions Mantegna R.N., Stanley H.E. An Introduction to Econophysics, Correlation and Complexity in Finance. Cambridge University Press, Malliaris A.G., Brock W.A. Stochastic Methods in Economics and Finance. North Holland, 2. Rachev S.T., Kim Y.S. Financial Models with Levy processes and Volatility Clustering. Wiley, 3. Schoutens W. Levy Processes in Finance: Pricing Financial Derivatives. Wiley, Bottasso A. Stable Levy Processes in Finance: Economics. Lambert, Challet D., Marsili M. Minority Games: Interacting Agents in Financial Markets. Oxford Press, Lecture 7. Random walk in financial market models 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 Mantegna R.N., Stanley H.E. An Introduction to Econophysics, Correlation and Complexity in Finance. Cambridge University Press, Malliaris A.G., Brock W.A. Stochastic Methods in Economics and Finance. North Holland, 2. Rachev S.T., Kim Y.S. Financial Models with Levy processes and Volatility Clustering. Wiley, 3. Schoutens W. Levy Processes in Finance: Pricing Financial Derivatives. Wiley, Bottasso A. Stable Levy Processes in Finance: Economics. Lambert, Challet D., Marsili M. Minority Games: Interacting Agents in Financial Markets. Oxford Press, Lecture 8. Minority games Price dynamics. Formulation of the minority game. Thermal minority game. Minority game without information. Grand-canonical minority game. Analytic approach.

9 Practice 8. Speculative trading Challet D., Marsili M. Minority Games: Interacting Agents in Financial Markets. Oxford Press, Malliaris A.G., Brock W.A. Stochastic Methods in Economics and Finance. North Holland, 2. Rachev S.T., Kim Y.S. Financial Models with Levy processes and Volatility Clustering. Wiley, 3. Schoutens W. Levy Processes in Finance: Pricing Financial Derivatives. Wiley, Bottasso A. Stable Levy Processes in Finance: Economics. Lambert, 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 dπ = βf(u) β(1 ξ)π dt for π(0) = π. Show that for β > 0 and 0 < ξ < 1 the equilibrium π is asymptotically stable. Problem 2. Given the following parameters for the Solow growth model a = 4, α = 0.25, s = 0.1, δ = 0.4, n = 0.03 (i) to plot the graph of k(t) (ii) plot the function k = sak (n + δ)k (iii) Linearise k about the equilibrium in (ii) and establish whether it is stable or unstable. Problem 3. For the following Holling Tanner predatory prey model x = x 1 x 6 6xy 8 + 8x y = 0.2y 1 0.4y x (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 X(t) = x + D(t), where x > 0 and [D(t), t 0] is a Brownian motion with positive drift parameter μ and variance parameter σ 2. At time point t=0 a speculator acquires an American call option on this share with finite expiry date τ. Assume that x + μt > 3σ t, 0 t τ. (i) Why does the assumption make sense?

10 (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 equilibrium. 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, orbital 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. 6. Open and closed systems. Dynamic and static equilibrium.

11 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 stability. Structural instability of Lotka-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 catastrophe, butterfly catastrophe. 15. Potential functions of two active variables: hyperbolic umbilic catastrophe, elliptice umbilic catastrophe, 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 dimension, 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, leptokurtotic, 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 convergence. 27. Berry-Essen theorem 1. Berry-Essen theorem 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.

12 36. Minority game without information. 37. Grand-canonical minority game. Example of Problem Consider the following Walrasian price and quantity adjustment model φ (Y) = Y D(p) = 0.025p p 6p + 40 p = 0.75Dp, φ(y) Y Y = 2[p φ (Y)] (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, 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, Required Reading Shone R. Economic Dynamic. Phase Diagrams and Their Economic Applications. 2 nd Edition. Cambridge Press, 2002 Zhang W.B. Differential Equations, Bifurcations, and Chaos in Economics. World Scientific Publishing Company, Malliaris A.G., Brock W.A. Stochastic Methods in Economics and Finance. North Holland, Rachev S.T., Kim Y.S. Financial Models with Levy processes and Volatility Clustering. Wiley, Supplementary Reading Puu T. Attractors, Bifurcations, and Chaos: Nonlinear Phenomena in Economics, 2 nd Edition, Springer, Rosser J.B. From Catastrophe to Chaos: A General Theory of Economic Discontinuities. 2 nd Edition, Springer, Stachurski D. Economic Dynamics: Theory and Computation. MIT Press, Schoutens W. Levy Processes in Finance: Pricing Financial Derivatives. Wiley, Bottasso A. Stable Levy Processes in Finance: Economics. Lambert, Challet D., Marsili M. Minority Games: Interacting Agents in Financial Markets. Oxford Press, 2013 Ian Jacques. Mathematics for Economics and Business. Pearson, 7 edition, Kenneth Shaw. Mathematical Modeling in Business and Economics: A Data-Driven Approach. Business Expert Press, September, 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, Software Mathematica v. 9.0

13 9.6 Distance Learning HSE Electronic Library access: Books24*7, Scopus, EBSCOHost, Science Direct, Web of Knowledge HSE Learning Management System 10 Technical Provision Computer, projector (for lectures or practice), computer class