## Policy Responses to Climate Change in a Dynamic Stochastic World

### Lars Hansen, University of Chicago

### Usage Details

Kenneth Judd, Yongyang Cai, Lars Hansen, Simon Scheidegger, Gregor Reich, Taylor Canann, Philipp Renner, Carlos RangelIntegrated Assessment Models (IAM) aim to analyze the impact and effectiveness of alternative policy responses to climate change. Current analyses rely on very simple models, ignoring heterogeneity in the economy as well as uncertainties in economic conditions and the climate. The standard claim is that computational limitations make it impossible to do better.

Our recent work on Blue Waters clearly showed otherwise. Our DSICE framework uses modern numerical methods along with the computational power of Blue Waters to look at far more realistic models that incorporate uncertainty.

In the next year, we will apply DSICE and its Blue Waters implementation to explore how climate change policies affect different countries, different sectors, and different generations. As in most IAM analyses, our earlier work assumed that there was one decision maker, the "social planner," who makes all decisions. The real world is instead comprised of many actors pursuing different goals, and implementing a variety of policies, such as taxes, to pursue other policy objectives. Our Blue Waters work will examine the complexities of such environments.

Mathematically, we aim to solve the decision rules of all actors, such decision rules being functions of the many current state variables. Economics and physics tell us how the states evolve and how today’s decisions are determined by today’s objectives and tomorrow’s decision rules. Formally, the decision rules are functions in a Banach space, and the decision rules at time t are determined by the laws of motion between time t and time t+1, and the decision rules at time t+1. This map is smooth in standard Banach space topologies, and we use flexible methods to approximate each decision rule at each time t. One way to think about this is we are solving a difference equation in a Banach space.

Our current work solves dynamic programming problems, which are the discrete-time analogs of Hamilton-Jacobi-Bellman (HJB) partial differential equations. The multi-actor problems we will solve are similar to the HJB equations in dynamic games, and our computational approach is essentially spectral in the space dimensions and a finite-difference approach in time. The computational challenge lies in handling the large number of state variables in the combined multi-actor economic and climate systems. We are currently solving six- to twenty-dimensional discrete-time analogs to HJB equations, which is made possible by our flexible (and somewhat novel) approaches to solving high-dimensional smooth functions.

Our problems can exploit Blue Waters because the computations mapping the time t+1 functions to the time t functions can be parallelized. Furthermore, the extensive use of nonlinear optimization implies that the number of messages is small relative to the numerical operations and each message is "small" (a few megabytes). For example, we used 84,000 cores in Blue Waters to solve one DP problem having 372 billions of such tasks, and it was solved in eight wall clock hours, which otherwise would use about 77 years without parallelism. Scaling tests show that our examples scale linearly as we go from 30 nodes to 5,000 nodes; that is, running time is cut in half when we have twice as many nodes.