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Introduction and course description This course will cover selected topics at the frontier of research focusing on control theory, differential games, and complementarity and supermodularity in economic theory. The prerequisites are the macro courses as well as some familiarity with continuous-time and discrete-time dynamic optimization. Because of this, the students are encouraged to look at these references: Stokey, N. and R. E. Lucas, Jr. (1989), ”Recursive methods in Economic Dynamics”, Harvard University Press. Beavis B. and I. Dobbs (1990), ”Optimization and Stability Theory for Economic Analysis”, Cambridge University Press. Gandolfo, G. (1995), ”Economic Dynamics”, 3rd. ed., Springer-Verlag, Berlin.
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Part I - Control Theory and Dynamic Games The main references for the first part of the course are: Dockner, E. J., Van Long, N., Sorger, G. and S. Jørgensen (2000), ”Differential Games in Economics and Management Science”, Cambridge University Press. Basar, T. and G.J. Olsder (1999), ”Dynamic Noncooperative Game Theory”, SIAM, Academic Press, 2nd edition. [1]. Introduction: Basic concepts of game theory (a) Axioms of game theory (b) Game theoretic models (c) Dynamic games
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[2]. Control theoretic methods (a) The Hamilton-Jacobi-Bellman equation (b) Pontryagin’s maximum principle (c) Information, commitment, and strategies (d) Infinite time horizon (e) Conditions for non-smooth problems
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[3]. Markovian equilibria with simultaneous play (a) The Nash equilibrium (b) Equilibrium conditions (c) Time consistency and subgame perfectness
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[4]. Differential games with hierarchical play (a) Cournot and Stackelberg equilibria in one-shot games (i) The one-shot Cournot duopoly game (ii) The one-shot Stackelberg duopoly game (iii) Introduction to time inconsistency of Stackelberg equilibria (b) Open-loop Stackelberg equilibria (c) Non-degenerateMarkovian Stackelberg equilibria
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[5]. Differential games with special structures (a) Linear quadratic games (b) Linear state games (c) Exponential games
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[6]. Stochastic differential games (a) Piecewise deterministic games (i) A piecewise deterministic control model (ii) Stationary Markovian Nash equilibria (iii) Piecewise open-loopNash equilibria (b) Differential games with white noise (i) A control problem with white noise (ii) Markovian Nash equilibria
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[7]. Capital accumulation games (a) The structure of capital accumulation games (b) Qualitative properties of equilibrium strategies (c) Stability properties of equilibria (d) Games without adjustment costs (e) Knowledge as a public good
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[8]. Differential games in resources and environmental economics (a) Non-renewable resources (i) The cooperative case (ii) The non-cooperative case (b) Non-renewable resources: some variations (i) The doomsday problem (ii) Stock-dependent utility (c) Renewable resources (i) A fishery model (ii) A fishery model with capacity constraints
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Part II - Complementarity and Supermodularity in Economic Theory The main references for the second part of the course are: Cooper W. R. (1999), ”Coordination Games”, Cambridge University Press Topkis (1998), ”Supermodularity and Complementarity”, Princeton University Press [1]. Pareto-Edgeworth Complementarity (a) Individual decision problems (b) Cardinal complementarity (c) The need of ordinal complementarity
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[2]. Sensitivity solutions of optimal solutions (a) Monotone comparative statics on the real line (b) Monotone comparative statics on the plane (c) Complementarity on posets which are not the product of chains (d) Monotone comparative statics on lattices
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[3]. Supermodular and Quasisupermodular Games (a) The best reply of a quasisupermodular game (b) Tarski-Zhou fixed point theorem (c) The structure of the Nash set (d) Extensions and applications