Kenneth Judd

Stanford University | SU · Hoover Institution on War, Revolution and Peace

We apply numerical dynamic programming techniques to solve discrete-time multi-asset dynamic portfolio optimization problems with proportional transaction costs and shorting/borrowing constraints. Examples include problems with multiple assets, and many trading periods in a finite horizon problem. We also solve dynamic stochastic problems, with a p...

We build a novel stochastic dynamic regional integrated assessment model (IAM) of the climate and economic system including a number of important climate science elements that are missing in most IAMs. These elements are spatial heat transport from the Equator to the Poles, sea level rise, permafrost thaw and tipping points. We study optimal polici...

We propose a general emulation method for constructing low-dimensional approximations of complex dynamic climate models. Our method uses artificially designed uncorrelated CO2 emissions scenarios, which are much better suited for the construction of an emulator than are conventional emissions scenarios. We apply our method to the climate model MAGI...

We build a novel stochastic dynamic regional integrated assessment model (IAM) of the climate and economic system including a number of important climate science elements that are missing in most IAMs. These elements are spatial heat transport from the Equator to the Poles, sea level rise, permafrost thaw and tipping points. We study optimal polici...

We propose a novel methodology for evaluating the accuracy of numerical solutions to dynamic economic models. It consists in constructing a lower bound on the size of approximation errors. A small lower bound on errors is a necessary condition for accuracy: If a lower error bound is unacceptably large, then the actual approximation errors are even...

This paper introduces a nonlinear certainty-equivalent approximation method for dynamic stochastic problems. We first introduce a novel, stable, and efficient method for computing the decision rules in deterministic dynamic economic problems. We use the results as nonlinear and global certainty-equivalent approximations for solutions to stochastic...

We develop a numerical method for computing all pure strategy subgame-perfect equilibrium values of dynamic strategic games with discrete states and actions. We define a monotone mapping that eliminates dominated strategies, and when applied iteratively, deliversan accurate approximation to the true equilibrium payoffs of the underlying game. Our a...

This chapter proposes a general approach for solving a broad class of difficult optimization problems using big data techniques. We provide a general description of this approach as well as some examples. This approach is ideally suited for solving nonconvex optimization problems, multiobjective programming problems, models with a large degree of h...

There is great uncertainty about future climate conditions and the appropriate policies for managing interactions between the climate and the economy. We develop a multidimensional computational model to examine how uncertainties and risks in the economic and climate systems affect the social cost of carbon (SCC)---that is, the present value of the...

Viability theory is the study of dynamical systems that asks what set of initial conditions will generate evolutions that obey the laws of motion of a system and some state constraints, for the length of the evolution. We apply viability theory to Judd's dynamic tax model [The welfare cost of factor taxation in a perfect-foresight model, Journal of...

Significance Most current cost−benefit analyses of climate change suggest global climate policy should be relatively weak. However, relatively few studies account for the market or nonmarket impacts of passing environmental tipping points that cause abrupt and irreversible damages. We use a stochastic dynamic model of the climate and economy to qua...

Perhaps the most 'dangerous'aspect of future climate change is the possibility that human activities will push parts of the climate system past tipping points, leading to irreversible impacts. The likelihood of such large-scale singular events is expected to increase with global warming, but is fundamentally uncertain. A key question is how should...

Dynamic programming is the essential tool in dynamic economic analysis. Problems such as portfolio allocation for individuals and optimal economic growth are typical examples. Numerical methods typically approximate the value function. Recent work has focused on making numerical methods more stable, and more efficient in its use of information. Thi...

We show how to enhance the performance of a Smolyak method for solving dynamic economic models. First, we propose a more efficient implementation of the Smolyak method for interpolation, namely, we show how to avoid costly evaluations of repeated basis functions in the conventional Smolyak formula. Second, we extend the Smolyak method to include an...

We propose a novel methodology for evaluating the accuracy of numerical solutions to dynamic economic models. Speci…cally, we construct a lower bound on the size of approximation errors. A small lower bound on errors is a necessary condition for accuracy: If a lower error bound is unacceptably large, then the actual approximation errors are even la...

A nonlinear programming formulation is introduced to solve infinite horizon dynamic programming problems. This extends the linear approach to dynamic programming by using ideas from approximation theory to avoid inefficient discretization. Our numerical results show that this nonlinear programming method is efficient and accurate.Institutional subs...

This paper contains an analysis of incomplete market models with finitely but arbitrarily many heterogeneous agents. We discuss the mathematical foundation for equilibrium conditions which leads to two findings. First, we establish existence of recursive equilibria for small and large risks. Second, we develop a simple but general solution techniqu...

There is great uncertainty about the impact of anthropogenic carbon on future economic wellbeing. We use DSICE, a DSGE extension of the DICE2007 model of William Nordhaus, which incorporates beliefs about the uncertain economic impact of possible climate tipping events and uses empirically plausible parameterizations of Epstein-Zin preferences to r...

We apply numerical dynamic programming to multi-asset dynamic portfolio optimization problems with proportional transaction costs. Examples include problems with one safe asset plus two to six risky stocks, and seven to 360 trading periods in a finite horizon problem. These examples show that it is now tractable to solve such problems.Institutional...

We implement a dynamic programming algorithm on a computational grid consisting of loosely coupled processors, possibly including clusters and individual workstations. The grid changes dynamically during the computation, as processors enter and leave the pool of workstations. The algorithm is implemented using the Master-Worker library running on t...

Numerical dynamic programming algorithms typically use Lagrange data to approximate value functions over continuous states. Hermite data is easily obtained from solving the Bellman equation and can be used to approximate value functions. We illustrate this method with one-, three-, and six-dimensional examples. We find that value function iteration...

We introduce an algorithm for solving dynamic economic models that merges stochastic simulation and projection approaches: we use simulation to approximate the ergodic measure of the solution, we construct a fixed grid covering the support of the constructed ergodic measure, and we use projection techniques to accurately solve the model on that gri...

Numerical methods for dynamic programming often use value function iteration and interpolation. We present a novel shape-preserving rational spline approximation method that improves value function iteration in terms of both stability and accuracy compared to more common methods.Highlights► Dynamic programming methods require good approximation met...

Static and dynamic games are important tools for the analysis of strategic interactions among economic agents and have found many applications in economics. In many games equilibria can be described as solutions of polynomial equations. In this paper we describe state-of-the-art techniques for finding all solutions of polynomial systems of equation...

Continuous time is a superior representation of both the economic and climate systems that Integrated Assessment Models (IAM) aim to study. Moreover, continuous-time representations are simple to express. Continuous-time models are usually solved by discretizing time, but the quality of a solution is significantly affected by the details of the dis...

Numerical dynamic programming algorithms typically use Lagrange data to approximate value functions. This paper uses Hermite data obtained from the optimization step and applies Hermite interpolation to construct approximate value functions. Several examples show that Hermite interpolation significantly improves the accuracy of value function itera...

Dynamic programming is the essential tool in dynamic economic analysis. Problems such as portfolio allocation for individuals and optimal growth of national economies are typical examples. Numerical methods typically approximate the value function and use value function iteration to compute the value function for the optimal policy. Polynomial appr...

There are two important rules to patent races: minimal accomplishment necessary to receive the patent and the allocation of the innovation benefits. We study the optimal combination of these rules. A planner, who cannot distinguish between competing firms in a multistage innovation race, chooses the patent rules by maximizing either consumer or soc...

We use a dynamic stochastic general equilibrium model of integrated climate and economy (DSICE) to account for abrupt and irreversible climate change. We model a climate shock in the form of a stochastic tipping point. We investigate the impact of the tipping point externality on optimal mitigation policy.We conclude that the optimal mitigation pol...

This paper introduces a dynamic stochastic integrated model of climate and economy (DSICE), and a numerical dynamic programming algorithm for its solution. More specifically, we solve an example with annual time periods, a six hundred year horizon, and shocks to the economic and climate system. Our dynamic programming methods solve such models on a...

Estimating structural models is often viewed as computationally difficult, an impression partly due to a focus on the nested fixed-point (NFXP) approach. We propose a new constrained optimization approach for structural estimation. We show that our approach and the NFXP algorithm solve the same estimation problem, and yield the same estimates. Comp...

We study a two-person zero-sum game where players simultaneously choose sequences of actions, and the overall payoff is the average of a one-shot payoff over the joint sequence. We consider the maxmin value of the game played in pure strategies by boundedly rational players and model bounded rationality by introducing complexity limitations. First...

We introduce a technique called "precomputation of integrals" that makes it possible to compute conditional expectations in dynamic stochastic models in the initial stage of the solution procedure. This technique can be applied to any set of equations that contains conditional expectations, in particular, to the Bellman and Euler equations. After t...

Existing tax policies have many complex features that make distinctions across goods, factors, and financial structures that distort economic allocations. Their importance is ignored when tax policy is summarized by an “effective” tax rate. In fact, the losses due to asymmetries are often larger than the losses due to the level of taxation.

This paper describes the second model considered in the computational suite project that compares the performance of different numerical algorithms. It is a multi-country model in which countries face different productivity shocks. Solving such models is a challenging numerical problem unless the number of countries is small. The solutions are func...

Efficient, accurate, multi-dimensional, numerical integration has become an important tool for approximating the integrals which arise in modern economic models built on unobserved heterogeneity, incomplete information, and uncertainty. This paper demonstrates that polynomial-based rules out-perform number-theoretic quadrature (Monte Carlo) rules b...

Many bond portfolio managers argue that bond laddering tends to outperform other bond investment strategies because it reduces both market price risk and reinvestment risk for a bond portfolio in the presence of interest rate uncertainty. Despite the popularity of bond ladders as a strategy for managing investments in fixed-income securities, there...

In conventional stochastic simulation algorithms, Monte Carlo integration and curve fitting are merged together and implemented by means of regression. We perform a decomposition of the solution error and show that regression does a good job in curve fitting but a poor job in integration, which leads to low accuracy of solutions. We propose a gener...

This paper analyses what determines an individual investor's risk-sharing demand for options and, aggregating across investors, what the equilibrium demand for options. We find that agents trade options to achieve their desired skewness; specifically, we find that portfolio holdings boil down to a three-fund separation theorem that includes a so-ca...

We report here on the development of an open source software framework termed CIM-EARTH that is intended to aid decision-making in climate and energy policy. Numerical modeling in support of evaluating policies to address climate change is difficult not only because of inherent uncertainties but because of the differences in scale and modeling appr...

Authors: Dr. Kenneth L. Judd, Hoover Institution, and Prof. William A. Brock, University of Wisconsin Current climate models range from General Circulation Models (GCM's) with millions of degrees of freedom to models with few degrees of freedom. Simple Energy Balance Climate Models (EBCM's) help us understand the dynamics of GCM's. The same is true...

Both distributed systems and multicore systems are difficult programming environments. Although the expert programmer may be able to carefully tune these systems to achieve high performance, the non-expert may struggle. We argue that high level abstractions are an effective way of making parallel computing accessible to the non-expert. An abstracti...

We use the stochastic simulation algorithm, described in Judd, Maliar and Maliar (2009), and the cluster-grid algorithm, developed in Judd, Maliar and Maliar (2010a), to solve a collection of multi-country real business cycle models. The following ingredients help us reduce the cost in high-dimensional problems: an endogenous grid enclosing the erg...

Economics is the study of how scarce resources are allocated. Operations research studies how to accomplish goals in the least costly manner. These fields have much to offer each other in terms of challenging problems that need to be solved and the techniques to solve them. This was the case after World War II, partly because the individuals who we...

Estimation of games with multiple equilibria has received much attention in the recent econometrics literature. Unlike other estimation problems such as single-agent dynamic decision models or demand estimation, in which there is a unique solution in the underlying structural models, games usually admit multiple equilibria and the number of equilib...

We develop a projection method that can solve dynamic economic models with a large number of state variables. A distinctive feature of our method is that it operates on the ergodic set realized in equilibrium: we simulate a model, distinguish clusters on simulated series and use the clusters’ centers as a grid for projections. Making the grid endog...

Dynamic programming is the foundation of dynamic economic analysis and often requires numerical solution methods. Standard methods are either slow or unstable. These instabilities are avoided when one uses modern methods from numerical optimization and approximation. Furthermore, large dynamic programming problems can be solved by using modern para...

This paper describes the first model considered in the computational suite project that compares different numerical algorithms. It is an incomplete markets economy with a continuum of agents and an inequality (borrowing) constraint.

General equilibrium models have been used for decades to obtain insights into the economic implications of policies and decisions. Despite successes, however, these economic models have substantive limitations. Many of these limitations are due to computational and methodological constraints that can be overcome by leveraging recent advances in com...

This paper develops a numerical method for computing equilibria of dynamic games with state variables, and applies it to an oligopoly game with endogenous productive capacity. Our algorithm allows us to study the nature of cooperation and examine how the ability to collude is affected by state variables, such as current capacity. We study whether i...

We develop numerically stable stochastic simulation approaches for solving dynamic economic models. We rely on standard simulation procedures to simultaneously compute an ergodic distribution of state variables, its support and the associated decision rules. We differ from existing methods, however, in how we use simulation data to approximate deci...

Many bond portfolio managers argue that bond laddering tends to outperform other bond investment strategies because it reduces both market price risk and reinvestment risk for a bond portfolio in the presence of interest rate uncertainty. Despite the popularity of bond ladders as a strategy for managing investments in fixed-income securities, there...

This article was submitted without an abstract, please refer to the full-text PDF file.

This article was submitted without an abstract, please refer to the full-text PDF file.

Both distributed systems and multicore computers are difficult programming environments. Although the expert programmer may be able to tune distributed and multicore computers to achieve high performance, the non-expert may struggle to achieve a program that even functions correctly. We argue that high level abstractions are an effective way of mak...

We present a general method for computing the entire set of equilibrium payoffs for dynamic games with state variables, with and without complete information. We extend our earlier methods (Judd, Yeltekin, Conklin (Econometrica, 2003)) to games with physical states variables, and apply the procedure to a dynamic oligopoly game with costly capacity...

Seasonal products have an effective inventory deadline, a time by which the inventory must be ready to distribute. The deadline creates an incentive to start early with production. However, opportunities to gather information that might change production ...

Maximum likelihood estimation of structural models is often viewed as computationally difficult. This impression is due to a focus on the Nested Fixed-Point approach. We present a direct optimization approach to the general problem and show that it is significantly faster than the NFXP approach when applied to the canonical Zurcher bus repair model...

John Deere & Company (Deere), one of the world's leading producers of machinery, manufactures products composed of various features, within which a customer may select one of a number of possible options. On any given Deere product line, there may ...

Discrete-time stochastic games with a finite number of states have been widely applied to study the strategic interactions among forward-looking players in dynamic environments. The model as written down by Ericson & Pakes (1995), Pakes & McGuire (1994, 2001) (hereafter, EP, PM1, and PM2), the subsequent literature (e.g., Gowrisankaran 1999, Fersht...

Prof. Mirowski has written a provocative discussion of new ways to model markets that focus on their algorithmic aspects. Many of his substantive points about the weaknesses of standard theory are widely accepted; in particular, we have little idea of how prices are formed, we agree that economic agents are not infinitely intelligent, and it is cle...

Computer technology presents economists with new tools, but also raises novel methodological issues. This essay discusses the challenges faced by computational researchers, and proposes some solutions.

Beginning with Mirrlees, the optimal taxation literature has generally focused on economies where individuals are differentiated only by their productivity. In this paper we examine models where individuals are differentiated by two or more characteristics. We examine cases where individuals differ in productivity, elasticity of labor supply, and t...

In a comment, Peter Bossaerts and William R. Zame [2006. Finance Research Letters. This issue] claim that the main result of our paper [Judd, K.L., Kubler, F., Schmedders, K., 2003. The Journal of Finance 58, 2203–2217], namely the no-trade theorem for the dynamic Lucas infinite horizon economy with heterogeneous agents, is an artifact of the assum...

Beginning with Mirrlees, the optimal taxation literature has generally focused on economies where individuals are differentiated by only their productivity. In this pa- per we examine models with discrete types where individuals are differentiated by two or more characteristics. For example, we examine the case where individuals also have different...

Maximum likelihood estimation of structural models is regarded as computationally difficult by many who want to apply the Nested Fixed-Point approach. We present a direct optimization approach to the problem and show that it is significantly faster than the NFXP approach when applied to the canonical Zurcher bus repair model. The NFXP approach is i...

Economic analysis often leads to multidimensional numerical problems. The {\em Curse of Dimensionality\/} often leads researchers to adopt methods designed for very high-dimension problems, but inefficient for problems of intermediate dimension. However, a little mathematics can greatly help dealing with the {\em Curse\/}. We will survey methods fr...

"Social-ecological systems are complex adaptive systems where social and biophysical agents are interacting at multiple temporal and spatial scales. The main challenge for the study of governance of social-ecological systems is improving our understanding of the conditions under which cooperative solutions are sustained, how social actors can make...

In this paper we analyze a credit economy � la Kiyotaki and Moore [1997. Credit cycles. Journal of Political Economy 105, 211-248] enriched with learning dynamics, where both borrowers and lenders need to form expectations about the future price of the collateral. We find that under homogeneous learning, the MSV REE for this economy is E-stable and...

The two-fund separation theorem from static porfolio analysis generalizes to dynamic Lucas-style asset model only when a consol is presemt. If all bonds have finite maturity and do not span the consol, then equilibrium will devitate, often significantly, from two-fund separation even with the classical preference assumptions. Furthermore, equilibri...

We explore alternative approaches to numerical solutions oflarge rational-expectations models. We discuss and compare severalcurrent alternatives, focusing on the trade-offs in accuracy, space,and speed. The models range from representative-agent models withmany goods and capital stocks, to models of heterogeneous agents withcomplete or incomplete...

We study a two-person zero-sum game where players simultaneously choose sequences of actions, and the overall payoff is the average of a one-shot payoff over the joint sequence. We consider the maxmin value of the game played in pure strategies by boundedly rational players and model bounded rationality by introducing complexity limitations. First...

We study computational aspects of moral-hazard problems. We consider deterministic contracts as well as contracts with action and/or compensation lotteries, and formulate each case as a mathematical program with equilibrium constraints (MPEC). We investigate and compare solution properties of the MPEC approach with the linear programming (LP) appro...

Numerical methods are increasingly used in graduate student research. I will discuss the problems of how to teach the necessary skills and the challenges of incorporating such teaching into a graduate program

We explore methods for solving continuous-time, finite-state dynamic games, and document the advantages of such models over comparable discrete-time models. In particular, continuous-time models are more flexible since they avoid lumpiness in time, and computational methods for continuous-time models avoid a curse of dimensionality that arises in d...

We quantitatively explore how asset market structure affects risk-sharing, welfare, and trading volume in stylized rational expectations models. We examine five market structures: perfect asset trading, only bonds and equity, only equity, only bonds, and autarchy. We find a variety of results. First, trade in a single asset, either bond or stock, a...

We present a general method for computing the set of supergame equilibria in infinitely repeated games with perfect monitoring and public randomization. We present a three-stage algorithm that constructs a convex set containing the set of equilibrium values, constructs another convex set contained in the set of equilibrium values, and produces stra...

A modern market-based economy is an example of a complex adaptive system, consisting of a decentralized collection of autonomous adaptive agents interacting over time in various market contexts. These massively parallel local interactions give rise to global regularities such as trade networks, socially accepted monies, market protocols, business c...

We present asymptotically valid analyses of a simple optimal growth model with hyperbolic discounting. We use the implicit function theorem for Banach spaces to show that for small deviations from exponential discounting there is a unique solution in the Banach space of con-sumption functions with bounded derivatives. The proof is constructive and...

We examine a multistage model of an R&D race where players have multiple projects. We also develop perturbation methods for general dynamic games that can be expressed as analytic operators in a Banach space. We apply these perturbation methods to solve races with a small prize. We compute second-order asymptotically valid solutions for equilibrium...

Trading volume of infinitely lived securities, such as equity, is generically zero in Lucas asset pricing models with heterogeneous agents. More generally, the end-of-period portfolio of all securities is constant over time and states in the generic economy. General equilibrium restrictions rule out trading of equity after an initial period. This r...

The Scarf algorithm was the first practical, almost surely convergent method for computing general equilibria of competitive models. The current focus of much computational research is computing equilibrium of dynamic stochastic models. While many of these models are examples of Arrow-Debreu equilibria, Scarf’s algorithm and subsequent homotopy met...

This paper examines a dynamic, stochastic endowment economy with two agents and two financial securities. Markets are incomplete and agents can have heterogeneous tastes. We develop a new computational method to solve the dynamic general equilibrium model. We allow for various forms of portfolio constraints, transaction costs, and portfolio penalti...

The parametric path method applies projection methods to compute the equilibrium time path of economic variables in infinite-horizon dynamic models. We exploit the special structure of equilibrium paths common in such models to construct a low-dimensional set of candidate solutions, and then use efficient integration and equation solution methods t...

We describe a general Taylor series method for computing asymp-totically valid approximations to deterministic and stochastic rational expectations models near the deterministic steady state. Contrary to conventional wisdom, the higher-order terms are conceptually no more difficult to compute than the conven-tional deterministic linear approximatio...

In this paper we examine the impact of membership in Preferential Trade Agreements (PTAs) on trade between PTA members. Rather than considering the impact of PTA membership on the volume of trade we consider the impact of membership on the structure of trade. For a large sample of countries over the period 1962-2000 we find that membership in a PTA...