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## Penn State 2014 – Numerical Methods by Ken Judd

**Prerequisite** for these lectures: Watch the first six minutes of my presentation at Blue Waters:

### Lecture Topics and Schedule

There will be four main topics.

### 1: Death to Log-Linearization

The dominant method in dynamic economics is to compute the steady state of a deterministic dynamic system, compute a linear approximation to the stable manifold in the neighborhood of the steady state, and then use that as the law of motion for the equilibrium dynamics in a stochastic economy. The approximation to the stable manifold of the deterministic system is based on the Hartman-Grobman theorems for dynamic systems, whereas the use of that approximation in the stochastic problem assumes “certainty equivalence,” that is, that the decision rules for the deterministic problem are the same (or close to) the decision rules for the stochastic problem. Unfortunately, the quality of any linear approximation (before or after a nonlinear change of variables) breaks down as the state deviates from the steady state. Higher order perturbation methods will improve the approximation near the deterministic steady state but will surely create stability problems at states sufficiently far away.

I will describe linearization (and log-linearization) in a precise mathematical fashion, discuss optimal control methods for solving dynamic optimization problems with arbitrary initial states, describe a method for using solutions to the optimal control problems to compute the globally valid solutions for the deterministic problem, and the compare the performance of the global certainty equivalence solution to the local certainty equivalent solution that now dominates macroeconomic modeling.

### 2: Stabilizing Iterative Methods for Solving Equations

Nonlinear systems of equations are usually solved by some form of iteration — Gauss-Jacobi, Gauss-Seidel, SOR methods, etc. — that are intuitively based on tatonnement and best reply “stories.” Iterative methods often have convergence problems. Convergence often requires some appropriate reformulation of the iterative process. I will present some simple examples where the economics indicates the appropriate reformulation. There is no generally valid method. The goal here is to give you a conceptual foundation for finding a convergent iteration process.

### 3: Numerical Solution of Polynomial Systems of Equations

High quality (and low cost) software for solving polynomial systems has become available in the past year. More specifically, Bertini became open source a year ago, and Mathematica 10 has a much better NSolve command. I will present examples of where the polynomial approach can be applied, such as computing steady states of dynamic systems and Nash equilibrium of games.

### 4: Dynamic Stochastic Integration of Climate and the Economy

Yongyang Cai, Thomas Lontzek, and I have successfully solved a nine-dimensional dynamic programming problem that integrates a stochastic productivity process displaying long-run risk in consumption (one capital stock state variable, and two state variables modeling persistence in shocks to productivity growth) with a model of the climate, where the climate model combines the five-dimensional DICE climate model with a stochastic climate state that represents the impact of past temperature on current productivity (such as the current sea level which is determined by past temperature paths). The time period is one year.

We use this model to compute the stochastic optimal carbon taxes over the next few centuries. We find that the interaction of stochastic long-run growth and the climate implies a very large range of possible future carbon taxes, plausibly exceeding $1000/tC within this century. The vast range of possible future damages of carbon to the economy tells us that any rational greenhouse gas policy must incorporate the intrinsic risks of both the climate and economy.

This work also displays the vast untapped potential of computational science for economics. One of the examples I will discuss would have taken 20-30 years for a four-core laptop to solve (and then only if it had enough RAM). Fortunately for us, we found something better.