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The goal of this course is to give an introduction to some branches of mathematics which are extensively used in the modern economic theory, particularly in macro- and microeconomics. The course is obligatory and it is given at the first year in 1-st and 2-nd modules. The course includes 26 lectures and 12 seminars. Each of seven sections of the course is delivered by professor, Doctor of science who has extensive experience in corresponding branch of mathematics or in its applications in economics: E.G.Golshtein, V.Z. Belenkii, M.I. Levin, A.A. Shananin. In academic year 2000-2001, one of the sections is for the first time delivered by Prof. V.A.Bulavskii. The program of the course is coordinated by V.M. Polterovich. The students should take two exams: at the end of the first module (sections 1-3) and at the end of the second module (sections 4-7). Final grade of the course is computed as weighted mean in which the grade for seminars has weight 0.3 and the grade for exams (average of the two exam grades) has weight 0.7. The plan of the course is given below. After the title of each section, the name of the lecturer and the number of lectures in the section are indicated. References are at the end of the program in the common list. Introduction (V.M. Polterovich, 1) Mathematics in economic theory: The history and patterns. References: 1. Section 1. Convex sets and functions, elements of linear programming (E.G. Golshtein, 3) Linear spaces and subspaces, linear manifolds. Convex sets, interior, relative interior of convex sets, dimension of a convex set. Theorems on separating and supporting hyperplanes. Extreme points of convex sets. Convex polyhedra and polyhedral sets. Two equivalent definitions of convex (concave) functions, quasi-convex (quasi-concave) functions, properties of convex and quasi-convex functions. Subdifferential (superdifferential) of convex (concave) functions. Continuity, differentiability and subdifferentiability properties of convex functions. The basics of subdifferential calculus, associate functions and their properties. References: 1, 4, 6, 9, 10, 15. Section 2. Elements of nonlinear programming (E.G. Golshtein, 4) Problem of convex programming (CP), Lagrange function, dual problem. Saddle points of Lagrange function and pairs of dual problems of CP. Sleiter condition and its versions. Kuhn-Tucker theorem for CP problem. Necessary and sufficient conditions of optimality for CP problem in the differential form. Quasi-convex functions and optimization problems generated by them; necessary and sufficient conditions in the differential form. Examples of using the optimality conditions for finding the solution of the problem (the problem of efficient use of resources, utility maximization problems subject to the budget constraint). Disturbance functions of CP problems, the relationship between subdifferential of the disturbance function and the set of solutions to the dual problem, theorems of marginal values for CP problems and their economic interpretation. General problems of non-linear programming, local and global optimum, strict local optimum, regularity conditions for constraints, Kuhn-Tucker theorem for smooth non-linear programming problems (first-order necessary conditions of optimality). Second order necessary conditions of optimality, second order sufficient conditions of optimality. Differentiability of solutions and Lagrange multipliers of smooth non-linear programming problems with respect to their parameters, the relations with marginal values. Economic applications: The theory of demand (utility function and demand function, smoothness of demand function, Slutsky equation), decomposition of demand and supply; dynamic economic models. References: 3, 4, 6, 8, 9, 15, 19, 27, 28. Section 3. Differential equations and the theory of stability. (A.A.Shananin, 5) Ordinary differential equations. Existence and uniqueness of solutions. Gronwall’s lemma. Linear differential equations with fixed coefficients. Continuous dependence of solutions on parameters. Equations in variation. Frobenius’ theorem and integrability problem. Stability theorem. Concepts of stability by Dirichlet, Routh, Lyapunov. First Lyapunov’s method. Theorem of instability. Some tests for stability. Stable polynomials. Necessary condition of stability. Routh-Hurwitz criterion. Michailov-Nyquest criterion. Second Lyapunov’s method. Stability of price regulation processes. Limit cycles. The Hopf bifurcation. References: 11-14,22,26. Section 4. Maximum principle. (V.Z. Belenkii, 6) Various versions of the problem of optimal control. Problems in discrete and continuous time, finite and infinite horizons, various types of boundary conditions. Examples: speed of acting problem, model of planning with finite horizon (with terminal or integral, with discount, functionals), Ramsey’s model on infinite interval, model of optimal economic growth in discrete time. Two alternative approaches to analysis of problem of optimum control: “local” – Pontriagin’s maximum principle and “global” – Bellman’s dynamic programming. Notional tools of each of them. Maximum principle as necessary condition of optimality for a problem in finite interval with free right end (with proof). The cases of discrete and continuous time. Interpretation of dual variables. Cases in which maximum principle conditions are sufficient. Classic problem of calculus of variations in continuous time (CPCV). Euler equations as first-order extremum condition. Relationship between maximum principle and Kuhn-Tucker theorem in the discrete case. Maximum principle in other problems of optimal control: optimal speed of acting, case of mixed functional, problem with terminal constraints (transversality conditions), optimization over infinite period of time. References: 2,13,15,16,21,23,24,25. Section 5. Dynamic programming (V.Z. Belenkii, 3) Dynamic programming method for optimal control problem in discrete time with finite horizon. Payoff function and Bellman optimization principle. Recurrent relations. Strategy and transition mapping. Autonomous models. Terminal functional and “tail problem” in dynamic planning problem. Stationary version of the problem, Bellman equation. Strategy as the synthesis of optimal control. Dual interpretation of the stationary solution (as the solution to the model with infinite horizon and as the solution to the model with final horizon and objective terminal functional). The relationship between the derivatives of the payoff function and the dual variables in the maximum principle. The optimality principle for the continuous time problem. The optimality principle in the stochastic case. References: 2,10,13,16-18. Section 6. Pareto optimality (M.I. Levin, 2) Pareto optimality in strong and weak senses. Theorem of criteria convolution. Necessary and sufficient conditions of Pareto optimality. Pareto optimality and equilibrium states. Pareto optimality (efficiency) of trajectories in overlapping generations model. References: 5,7,10,19. Section 7. Fixed point theorems. (M.I. Levin, 2) Contracting mappings principle. Application: Existence of solutions to differential equations. Brouwer and Kakutani’s theorems. Geil-Nikado’s lemma. Applications: existence of Nash equilibrium, existence of competitive equilibrium. Birkgof-Tarskii theorem and its applications. References: 5-7,20.
References
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Economics
of Transition Elements
of the Economics History
of Economic
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