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## Penn State – Numerical Methods by Ken Judd

**Lecture Topics and Schedule**

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

#### April 1: 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.

#### April 2, 3: Methods for Dynamic Programming Problems With One Continuous Dimension

Many economics problems are formulated as dynamic programming problems with one continuous dimension, such as wealth in a life-cycle model or capital in a growth model. These problems may have other states that are discrete (wage level, productivity). The main computational challenge is to approximate the value function in the continuous dimension. Basic value function iteration with polynomial or spline interpolation may be unstable. These lectures will outline stability, accuracy, and efficiency issues in solving dynamic programming problems in general, and then present ideas and methods that work for problems with one continuous dimension.

I will ask students to write programs to solve these problems. I will meet with students the following week to discuss how their code worked.

#### April 6: Methods For Multidimensional Dynamic Programming Methods

We present methods for solving problems with many continuous dimensions and discuss some examples. One example will be portfolio management in the presence of transaction costs, and determining the social value of allowing trade in options.

#### April 7, 8 : Nonlinear Complementarity Problems

Many problems in economics have solutions with binding constraints. For example, a firm may choose zero output in an oligopoly model, or a country may choose to not produce a traded good. Most models avoid this possibility by making extreme assumptions, such as a zero marginal cost of production at zero output, so that equilibrium is the solution to a system of nonlinear equations. The more general situation is modeled by nonlinear complementarity problems, a generalization of equations. I will describe nonlinear complementarity problems, how to describe them, and give some basic examples of how to solve them. I will also show how to use nonlinear complementarity methods to solve optimal tax problems.