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# Computational Economics, 17 February – 29 May 2020

### Dr. Kenneth L. Judd

judd@stanford.edu

Office: Hoover Institution, Herbert Hoover Memorial Building, Room 334

## Objective

This course, taught by Ken Judd, introduces computational approaches for solving economic models. It focuses on a broad range of numerical methods and then applies them to economic problems. We formulate economic problems in computationally tractable forms and use numerical analysis techniques to solve them. We will study examples of computational techniques in the current economic literature as well as discuss areas of economic analysis where numerical analysis may be useful in future research of dynamic economic problems. The substantive applications will cover a wide range of problems including public finance, macroeconomics, game theory, mechanism design, econometrics, finance, and industrial organization.

## Lectures

The lectures will be given 18:00 CET (9:00am PST). They will be streamed using Zoom. They will also be recorded and put on the web. Exact details will be posted later. Ken Judd will deliver the lectures except where noted otherwise below.

## Overview

We begin with an overview of the necessary numerical analysis and approximation theory. First, we review standard numerical analysis – interpolation and approximation techniques, numerical optimization, numerical solutions to systems of linear and nonlinear equations, numerical integration, and basic solution techniques for ordinary and partial differential equations. We also discuss perturbation methods useful in economics.

Second, there will be an emphasis on applications related to current research areas. We will discuss methods for solving dynamic programming problems, as well as dynamic stochastic equilibrium models. We will solve for optimal incentive mechanisms using numerical optimization.

The course aims to acquaint students with the range of techniques that have been useful in economic analysis as well as expose students to techniques that have potential use in economic applications. I want to develop a basic understanding of numerical methods, demonstrate their use in economic examples, show how existing techniques fit into the broader numerical literature, and point the students to potentially useful numerical techniques.

## Prerequisites

It is best if students have a good undergraduate background in mathematics (such as calculus, linear algebra, eigenvalues and eigenvectors) and are familiar with static optimization theory (such as Lagrangians and Kuhn-Tucker conditions) as well as basic dynamic optimization theory. I will also presume some experience with graduate level economics, however any student who is interested in the course and has taken either Masters or PhD courses in econometrics, microeconomics, and macroeconomics will be able to follow the course.

## Computing Languages

Students will need to know use some computational language. My recommendation is that you use Python, Matlab, or R. Students are welcome to use FORTRAN, C, or C++. Statistical software such as TSP, Eviews, RATS, and SAS are not acceptable. The US Federal Reserve uses Eviews but I expect more of you. Don’t even ask about Excel. If you do not know any suitable language, I recommend that you learn Matlab or Python. Both are adequate for this course and useful in research. Matlab is the most used language in economics. Python is a more complex programming language but much better in terms of long-run value and flexibility. See the QuantEcon project of Sargent and Stachurski.

In operations research, the popular languages are GAMS and AMPL. They are is easy to learn and allows you to access the best numerical software through the NEOS website. NEOS time is FREE, reflecting the “Wisconsin idea”! If there is interest, I will have a couple of optional tutorial sessions on AMPL (sorry, I don’t know GAMS).

I will use Mathematica in my lectures and some classnotes. It allows one to use symbolic methods, has good graphics and is overall more flexible than Matlab. It is free for UZH students. All students will be asked to download the free CDF player; it will allow you to run my examples on your laptops.

## Textbook

The text is Numerical Methods in Economics by Judd, published by MIT Press. I will be particularly appreciative of any comments you may have on the text (typos, lack of clarity, etc.) since I am beginning to prepare the second edition.

I am working on a second edition. As we progress, I will (hopefully) have drafts on the course website of chapters for the second edition.

## Course Outline and Schedule

Date | Topic | Video | Details |
---|---|---|---|

Feb. 18 | Introduction | Video | Chapter 1: Computational power. Computational math as an economics problem: managing scarce resources. Economics VERSUS computational mathematics. |

Feb. 20 | Computer arithmetic | Video | Chapter 2. General ideas of computational errors, and rates of convergence. Finite precision arithmetic and finite – difference derivatives. |

Feb. 25 | Linear algebra and equations | Chapter 3. LU, QR, and Cholesky decomposition, condition numbers, Gauss-Jacobi and Gauss-Seidel methods. | |

Feb. 27 | Unconstrained optimization | Chapters 4 and 5. Search methods, bisection, gradient descent, Newton’s method, derivative free optimization (DFO). Applications to consumer demand and life-cycle problems, and maximum likelihood estimation. | |

March 3 | Nonlinear equations | Video | Chapters 4 and 5. Bisection, Newton’s method, BFGS and DFP updates, and Powell hybrid. Applications to general equilibrium and Nash equilibrium. |

March 5 | Constrained optimization: theory and methods | Video | Chapters 4 and 5. Linear and nonlinear optimization. KKT conditions, augmented lagrangian, SQP and interior point methods. |

March 10 | Constrained optimization: applications | Video | Introduction to multiobjective optimization. Applications to consumer demand and incentive problems |

March 12 | Structural estimation I | Video | Basic ideas. MPEC versus NFXP |

March 17 | Finite-difference ODEs | Video | |

March 19 | Version control using Git | (Robert Erbe and Gregor Reich) | |

March 24 | Automatic Differentiation | (Philipp Mueller); includes CasADi Tutorial | |

March 26 | Homotopy | (Philipp Mueller and Karl Schmedders) Chapter 5. Applications will include general equilibrium. | |

March 30 | Approximation I | Video | Chapter 6. Interpolation, regression, orthogonal polynomials, splines, least squares, LAD and Lasso fits. |

April 2 | Numerical quadrature, MC, qMC | Chapter 7, 8, and 9. Integration methods for single- and multiple-dimensional integrals. Monte Carlo simulation methods. Applications to portfolio choice and dynamic problems. | |

April 7 | Dynamic optimization, equilibrium, NLCEQ | Video | |

April 9, 14, and 16 | No lectures. Easter break. | ||

April 21 | Dynamic programming – discrete state | Video | Chapter 12. Value function iteration, policy iteration, acceleration methods. |

April 23 | Structural estimation II | Video | Su-Judd and Skrainka-Judd papers. |

April 28 | Dynamic programming – continuous state | Video | Chapter 12. Solutions to deterministic and stochastic dynamic programming problems using approximation, integration, and optimization methods. Applications to savings-consumption problems, climate change policy, and portfolio problems. |

April 30 | Projection methods I | Video | Chapter 10, 11, and 17. Methods for solving ordinary differential equations as well as the more complex equations arising in dynamic economic models. |

May 5 | Projection methods II | Video | Chapter 10, 11, and 17. Methods for solving ordinary differential equations as well as the more complex equations arising in dynamic economic models. |

May 7 | Perturbation methods | Video | Chapter 13, 14, and 15. Taylor series approximations to find numerical solutions of equations, linearizing around a steady state, simple bifurcation methods. |

May 12 | Approximation II | Video | Neural nets, radial basis functions, machine learning |

May 14 | Dynamic games | Video | |

May 19 | Multiobjective optimization | Video | |

May 20 | Structural estimation III | (Note that this is a Wednesday. Ascension Day on May 21 is a holiday in Zurich.) Philipp Mueller and Gregor Reich. | |

May 26 | DSICE: High-Power Computing Applied to Climate Policy Issues | Video | |

May 28 | Concluding remarks | Video | Future of computational economics |

## Contact

If you have any questions, feel free to contact Ken.