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## Penn State 2011 – Numerical Methods by Ken Judd

### Introduction

This website contains materials related to my April, 2011, visit to Penn State.

The goal of these lectures is to build on the course that Paul and Mark have taught. I chose topics that I think will be of broad interest to macro and empirical IO students.

The schedule below is not final. I can adjust it in response to students’ interests.

**Schedule**

**Tuesday, April 12**

Introduction

I quickly review material in my book on basic topics, and describe some new methods I am using:

- Equations: Solving Equations
- Solving equations via reformulations as constrained optimization problems
- Approximation: Approximation
- L1 approximations – definitions, methods, and advantages over least squares
- Preserving shape

**Wednesday, April 13**

- Constraint qualification
- Quadrature:
- Gaussian quadrature
- Monomial rules
- Monte Carlo integration and methods
- Quasi-Monte Carlo methods.

**Thursday, April 14**

This lecture will integrate optimization, approximation, and quadrature methods to construct numerically stable and efficient methods for multidimensional dynamic programming. The initial applications will be life-cycle problems.

**Friday, April 15**

MPEC methods for structural estimation; quadrature methods for BLPI.

I will present my paper with Che-Lin Su, a paper by Valentina Michelangeli, and a paper with Ben Skrainka, all illustrating the value of using modern numerical methods in structural estimation.

**Monday, April 18:**

Solving high-dimensional models in macro: Judd Maliar Maliar

I will present recent papers I have written with Lilia and Serguei Maliar on solving rational expectations models.

#### **Tuesday, April 19**

Advanced Dynamic Programming and Structural Estimation

- Dynamic Programming: We will continue discussions of dynamic programming with a presentation of parallel methods, applications to portfolio problems, and the nonlinear programming approach to dynamic programming (DPNLP).
- Structural Estimation:We will then see applications to structural estimation (Michelangeli paper)

We finish with a discussion of Skrainka-Judd on quadrature rules in BLP

#### Wednesday, April 20

**Perturbation Methods**

I will introduce students to methods for computing Taylor series expansions in the neighborhood of a steady state of a dynamic model. I will focus on presenting the basic theoretical foundations for methods along with Mathematica software.

**Thursday, April 21**

Multidimensional optimal taxation and nonlinear pricing: Optimal Multidimensional Taxation

The last lecture will present recent work on solving optimal tax problems where individuals differ in several dimensions. I will illustrate the novel challenges in these problems (the failure of any useful constraint qualification plus the enormous number of constraints) as well as the insights from initial examples (basically, linear taxation is best). I will conclude with a discussion of why the standard Mirrlees-style approach is doomed, but can be replaced with a more tractable and realistic approach.

**Other Topics**

There are some more advanced topics that we could discuss in smaller sessions if there is enough interest. They are:

I am involved in a CGE project on climate change policies. I have recently solved dynamic stochastic models with productivity shocks and climate shocks (such as unexpected glacier melt).

Groebner Bases: Kubler and Schmedders hav

e introduced economists to powerful methods for solving systems of polynomial equations. I believe that these methods will be very useful in IO theory.

Bifurcation methods: Ordinary implicit function theorems do not apply to portfolio problems, but bifurcation methods do the job.

Solving dynamic games: Yeltekin-Judd is computing all Nash equilibria for dynamic games with state variables.

Hyperbolic discounting: Hyperbolic discounting is a popular topic and is just one example where one wants to compute time-consistent solutions, but there is no existence theory nor any sound computational methods. Both issues are addressed if one uses well-known nonlinear functional analysis.

Software engineering: Ben Skrainka has written up a nice set of slides on how to be a good coder. They are below – ShortSoftwareEngineering.

**Other links of interest**

Short Software Engineering**====================****Full Disclosure**

Graduate students from (at least) one Ivy League department are told that I am a monster. Here is an example why: http://sites.google.com/site/economicsandcomputation/.

### Additional Resources

- Approximation
- Doraszelski-Judd Paper and RAND’s rejection
- Dynamic Programming
- Judd Maliar Maliar
- Optimal Multidimensional Taxation: JuddSuOptimalTax.pdf and JuddSuTaxSlides.pdf
- Optimization: ConstrainedOptimization.pdf and ConstraintQualificaitonExamples.pdf
- Quadrature
- Solving Equations: EquationsNewIdeas.pdf
- Structural Estimation