TOPICS IN ECONOMIC THEORY
Instructor: A. Banerji
Course Outline This course will provide an introduction to
deterministic and stochastic dynamics in discrete time and some economic applications.
the Python programming language and applications to the dynamics covered in the course.
Since the course is introductory, much of the time will be devoted to developing textbook-level theory and techniques. The treatment will be rigorous, so the theory will be developed slowly. We will be further slowed down by the implementation of dynamic programming problems on the computer. Therefore, we will be able to discuss mainly dynamic optimization, leaving little time to discuss equilibrium models beyond baby ones. This course design therefore will give a lower current payoff in order to have a higher continuation value, in dynamic programming parlance. Thus it makes more sense for you to take this course if you wish to continue your study of economics and can postpone the natural desire to use all of the stuff immediately.
Prerequisites: You would have learned (i) basic math econ, (ii) probability and (iii) game theory in the compulsory courses. The knowledge from these courses (mostly, from (i) and (ii)) is presumed, but will be required less than the sophistication that you have developed doing them. No prior programming knowledge is necessary; what is presumed is the desire to debug code that does not work, and to write code that can be used in multiple applications.
Books:
[1] John Stachurski (2009): Economic Dynamics – Theory and Computation.
[2] Thomas Sargent and John Stachurski (ongoing): Quantitative Economics.
(Access the ‘book’ at its website, quant-econ.net. You can always download a pdf
version for your computer/tablet/phone. You can access the codes at github:
https://github.com/QuantEcon/QuantEcon.py. Github is a social network for collaborative projects that uses the version control software git.)
[3] Nancy Stokey, Robert Lucas with Ed Prescott (1989): Recursive Methods in Economic Dynamics.
[4] Lars Ljungqvist and Thomas Sargent (2012): Recursive Macroeconomic Theory (3rd edition).
[5] Efe Ok: Real Analysis with Economic Applications (the chapters on Metric Spaces).
[6] Richard Bass (2013): Real Analysis for Graduate Students.
[7] Marek Capinski and Peter Kopp: Measure, Integral and Probability.
[8] Krishna Athreya and Soumendra Lahiri (2006): Measure Theory and Probability Theory.
[9] Sidney Resnick (2005): Adventures in Stochastic Processes.
[10] Allen Downey (2013): Think Python.
[11] Hans Peter Langtangen (2012): A Primer on Scientific Programming with Python.
[12] Kenneth Judd (1998): Numerical Methods in Economics.
These are all great books. We will use mainly [1] – [4], apart from a set of lecture notes, as indicated against each topic below (required readings are starred).
There are numerous applications of stochastic dynamics to game theory, something that is not evident from the books above, as we won’t have time to study much of these. If you’ve not done any programming earlier, please read the first 5 or 6 chapters of [10]. [11] is a nice, easy to understand book on scientific programming in Python: its discussion of object-oriented programming is pretty good. [5] and [6] are very good for math buffs. [9] is very nice if you want to read
further on stochastic processes. Finally, our go-to reference for numerical methods will be [12].
The way to do this course is to work through the material using pencil, paper and writing and running codes on your machine.
Internal Assessment will be based on coding and other exercises that I will hand out; this may involve writing code to run illustrations from a theory paper. We can discuss the structure of the final exam later.
Topics:
1. Metric Spaces; Contraction Mapping Theorem; Correspondences; Theorem of the Maximum; Blackwell’s conditions for a Contraction.
Readings: Lecture Notes*, [3]* (Chapter 3).
2. Deterministic Dynamic Programming.
Readings: [3]* (Chapters 4,5), [4] (Chapters 3,4).
3. Introduction to Programming in Python:
Setting up a Python environment – Anaconda, Jupyter notebook. Python basics.
Introduction to Classes in Python (OOP). Python Libraries: NumPy, SciPy, Matplotlib.
Readings: Lecture Jupyter Notebooks*, [2]* (Chapter 1, sections 1.1-1.9), [11]*(Chapter 7).
4. Intoduction to Dynamics:
Deterministic dynamical systems; Finite State Markov Chains: Dobrushin coefficient, stability, Markov Chains as Stochastic Recursive Sequences.
Applications (with coding in Python) – introduction to optimization and dynamic programming (deterministic, and stochastic with discrete state space), shortest paths, Schelling’s Segregation Model, the Law of Large Numbers.
Readings: Lecture Slides*, [1]* (Chapter 4),[2], [3](Chapters 6, 11.1), [4](Chapters 3-5).
5. Discrete Dynamic Programming:
Readings: Lecture Notes*, [1]*(Chapter 5.1*) 6. Infinite State Space:
Basics; dynamic programming applied to a simple optimal stochastic growth model, Linear-Quadratic Problems with applications to Rational Expectations equilibrium.
Readings: Lecture Notes*, [1]*(Chapter 6.1, 6.2*), [2]* (Chapter 2.9, 2.12, 2.14*), [4]*(Chapters 5*, 7).
7. Measure, Integration and Probability:
Classes of Sets, Measures, Probability Measures, Measurable Functions, Integration, Monotone Convergence Theorem, Fatou’s Lemma, Dominated Convergence Theorem, Product Measures, Fubini’s Theorem, Conditional Expectation.
Readings: Lecture Notes*, [3](Chapter 6), [6](Chapters1-15, 21), [7], [8].
8. Markov Processes:
Transition functions, probability measures on spaces of sequences, iterated integrals, stochastic difference equations.
Readings: [3]* (especially Chapter 8.1*)
9. Stochastic Dynamic Programming:
Principle of Optimality, bounded returns, constant returns to scale, unbounded returns, stochastic Euler equations, policy functions and transition functions. Applications: Optimal growth, Industry investment under uncertainty, Search models.
Readings: [3]*(Chapters 9, 10, especially 9.2*).
