Introduction for Volume 3 of the Handbook of Computational Economics
Computational power continues to explode in terms of both hardware and algorithms. The previous volumes presented the state of the art in the past. Volume 1 of the Handbook of Computational Economics [Amman et al. (1996)] surveyed the growing literature on computational methods for solving standard economic models such as Arrow-Debreu-McKenzie general equilibrium models and rational expectations models, and dynamic optimization. Volume 2 (Tesfatsion and Judd, 2006) focused on the foundations and applications of Agent-based Computational Economics (ACE), a computationally intensive approach to economics that offers an alternative to standard modeling approaches in economics.
The increase in computational power over the past 20 years is measured in terms of orders of magnitude. The chapters in this volume give some examples of how these advances can be used to expand the breadth and quality of economic analyses. More specifically, they update the advances in algorithms that have improved econometric tools, solution methods for dynamic optimization and equilibrium models, and applications to public finance, macroeconomics, auctions, and finance. While much of the advance in methods is basically the incorporation of existing mathematical methods, many of these chapters show that economists are closing the gap between economic practice and the frontiers of computational mathematics. However, that frontier is progressing rapidly, implying that there is much more that can be done to expand the value of computational modeling in economics.
Some chapters also point to the opportunities arising from revolutions in computer architecture over the past 20 years. In the past, computational speed was increased by designing faster chips. The emphasis has switched to using massive parallelism to create more powerful computers. This is reflected in the development of high power and high throughput computing, as well as designing graphics processor units (GPU) capable of scientific computation.
The first chapter, “Learning About Learning in Dynamic Economic Models,” is by Kendrick, Amman, and Tucci. They summarize the long literature on dynamic learning and optimal control. These problems present challenges of both a theoretical and computational nature because decisions today affect not only the current payoff and the future state but also what is known about the system being controlled. This is an important part of any dynamic optimization problem, but is generally ignored due to the substantial difficulties. Kendrick, Amman, and Tucci summarize past research and present some suggestions for future work.
In their chapter, “On the Numerical Solution of Equilibria in Auction Models with Asymmetries within the Private-Values Paradigm,” Tim Hubbard and Harry Paarsch demonstrate the tight connections between theory, computation, and estimation in the auction literature. The empirical auction literature has been especially active in the past 20 years. Auction models present novel computational problems, and computational difficulties have often limited the range of models that can be estimated. Hubbard and Paarsch give an integrated presentation of auction theory and computational methods for private value auctions, describing past progress as well as current research which will substantially increase the range of models that can be efficiently and accurately solved.
Financial market research has been an intensive user of computational methods. The next two chapters cover two such areas. Asset pricing problems are the focus of “On Formulating and Solving Portfolio Decision and Asset Pricing Problems” by Chen, Cosimano, and Himonas. They discuss both the standard methods, such as log-linearization, as well as methods based on tools from functional analysis. The new tools, many of which were developed by the authors, are excellent examples of how quantitative asset market models can benefit from the use of modern computational and mathematical tools. Option pricing models are partial differential equations (or more general functional equations), and require the use of PDE methods. “Computational Methods for Derivatives with Early Exercise Features” by Chiarella, Kang, Meyer, and Ziogas surveys the literature related to complex derivatives that holders may exercise early.
Public economics is one area of economics that has used computational methods for close to 40 years. Nishiyama and Smetters summarize the current state of the art for solving substantive dynamic models in “Analyzing Fiscal Policies in a Heterogeneous-Agent Overlapping-Generations Economy.”
Macroeconomics research is becoming more computational, particularly as it moves away from the paradigm of solving the social planner’s problem in a simple representative agent model. The next two chapters outline the current state of the art for solving such models. Algan, Allais, den Haan, and Rendahl describe methods for solving models where the primary source of heterogeneity is idiosyncratic risk in “Solving and Simulating Models with Heterogeneous Agents and Aggregate Uncertainty.” “Numerical Methods for Large Scale Dynamic Economic Models” by Lilia Maliar and Serguei Maliar present methods for models where there are many asymmetric states, such as models with heterogeneity in tastes and technology, or when there are multiple shocks and constraints such as in a New Keynesian model with the zero lower bound on interest rates.
The Handbook continues with three papers on general software and hardware aspects of numerical analysis. Any numerical computation has error, and economists need to have confidence that numerical results are sufficiently accurate to support economic arguments. Peralta-Alva an Santos discuss this in “Analysis of Numerical Errors.”
The phrase “computational economics” refers to computers. This is obvious but we often ignore the fact that our choice of algorithms depends on the nature of the hardware that we use. Graphic processor units (GPUs) represent a new kind of hardware offering new challenges and opportunities. Eric Aldrich gives some economics examples of GPU computing in “GPU Computing in Economics.”
Cai and Judd describe new combinations of numerical ideas and the use of parallel hardware architecture for dynamic programming in “Advances in Numerical Dynamic Programming and New Applications.” These tools are expanding the range of multidimensional dynamic programming problems that economists can solve.
The final chapter ends the Handbook appropriately by giving us a peep into the future. Economic models often have multiple solutions, creating problems for both theorists and applied economists. Many economics problems can be represented as solutions to polynomial equations. Mathematicians have long known that there are methods for finding all solutions of systems of polynomial equations, but they doubted that these methods could be used for nontrivial problems such as problems in economics. Such was the case at the time that the chapters of Volume I of the Handbook of Computational Economics were written. There has been great progress in the field of computational commutative algebra in the past 20 years. The final chapter, “Computing All Solutions to Polynomial Equations in Economics” by Kubler, Renner, and Schmedders, introduces us to those advances and gives us a few hints as to the potential value they hold for economists.
The progress on both the hardware and algorithm dimensions has increased the power of computational machinery at a rate far faster than implied by Moore’s Law alone. While other fields of study have incorporated computational power into their mainstream research, there has been much slower progress in economics. It is the Editors’ aim and hope that these chapters will help economists see the vast potential for economics offered by modern computational methods.