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2024 | Book

Basic Models Read chapter Read first chapter

Dynamic General Equilibrium Modeling

Computational Methods and Applications

Authors: Burkhard Heer, Alfred Maußner

Publisher: Springer International Publishing

Book Series : Springer Texts in Business and Economics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

Contemporary macroeconomics is built upon microeconomic principles, with its most recent advance featuring dynamic stochastic general equilibrium models. The textbook by Heer and Maußner acquaints readers with the essential computational techniques required to tackle these models and employ them for quantitative analysis. This third edition maintains the structure of the second, dividing the content into three separate parts dedicated to representative agent models, heterogeneous agent models, and numerical methods. At the same time, every chapter has been revised and two entirely new chapters have been added.

The updated content reflects the latest advances in both numerical methods and their applications in macroeconomics, spanning areas like business-cycle analysis, economic growth theory, distributional economics, monetary and fiscal policy. The two new chapters delve into advanced techniques, including higher-order perturbation, weighted residual methods, and solutions to high-dimensional nonlinear problems. In addition, the authors present further insights from macroeconomic theory, complemented by practical applications like the Smolyak algorithm, Gorman aggregation, rare disaster models and dynamic Laffer curves. Lastly, the new edition places special emphasis on practical implementation across various programming languages; accordingly, its accompanying web page offers examples of computer code for languages such as MATLAB®, GAUSS, Fortran, Julia and Python.

Table of Contents

Frontmatter

Representative Agent Models

Frontmatter
Chapter 1. Basic Models
Abstract
This chapter presents the basic Ramsey model, which is fundamental for modern macroeconomic research and paves the way for the algorithms presented in subsequent chapters. The solution is characterized along two lines: 1) The Euler equations provide a set of nonlinear difference equations that determine the optimal time path of consumption. 2) Dynamic programming seeks a policy function that relates the agent’s choice of current consumption to his or her stock of capital. In the benchmark business cycle model with endogenous labor, stochastic productivity, and growth, Heer and Maußner study the problems of parameter choice and model evaluation. The chapter concludes with a synopsis of the numerical solution techniques presented in Chapters 2 through 7 and introduces measures for evaluating the approximate solutions’ goodness of fit.
Burkhard Heer, Alfred Maußner
Chapter 2. Perturbation Methods: Framework and Tools
Abstract
Perturbation methods are the tool most frequently employed to solve dynamic general equilibrium models. Their popularity is due to the fact that they are easy to apply. Toolkit programmers can keep all the algorithm details hidden away from users. Due to their widespread use, Heer and Maußner examine perturbation methods in detail, over three chapters. The present chapter introduces readers to the perturbation approach and related issues, develops a general framework, and reviews the required tools.
Burkhard Heer, Alfred Maußner
Chapter 3. Perturbation Methods: Solutions
Abstract
This chapter presents examples of the perturbation solutions of the canonical dynamic stochastic general equilibrium (DSGE) model introduced in Chap. 2. Heer and Maußner provide formulas for computing the matrices of the first-, second-, and third-order approximations of the policy functions, respectively, and document their use in the computer code. At the book’s accompanying website, readers can download toolboxes written in both GAUSS and MATLAB® that implement the perturbation solution. Descriptions of these toolboxes and their uses are provided at the end of the chapter.
Burkhard Heer, Alfred Maußner
Chapter 4. Perturbation Methods: Model Evaluation and Applications
Abstract
This chapter introduces readers to two standard tools for evaluating the solution of a dynamic stochastic general equilibrium (DSGE) model: second moments and impulse response functions. Drawing on these diagnostic tools, Heer and Maußner consider three applications: the benchmark business cycle model, a time-to-build variant of this model, and a New Key-nesian model with consumption habits, investment adjustment costs, and sticky nominal prices and wages for the study of monetary policy.
Burkhard Heer, Alfred Maußner
Chapter 5. Weighted Residuals Methods
Abstract
This chapter introduces readers to global methods, which, unlike local perturbation methods, draw on information from various points in the state space of a dynamic stochastic general equilibrium (DSGE) model. The methods presented here are interchangeably referred to as weighted residuals methods and projection methods. Applications cover the deterministic growth model, the benchmark business cycle model, a model with labor market frictions, and a disaster risk model driven by shocks that trigger infrequent but severe recessions.
Burkhard Heer, Alfred Maußner
Chapter 6. Simulation-Based Methods
Abstract
This chapter presents methods that combine stochastic simulation with other numerical tools to find approximate solutions on the model’s ergodic set. In the first part, Heer and Maußner describe the extended path method, which is based on the repeated solution of a large system of nonlinear equations, and apply it to compute the benchmark business cycle model and the model of a small open economy. In the second part, they consider approaches that combine stochastic simulation with the methods of function approximation. The computation of the residuals and the ultimate solution can involve a wide variety of methods, as illustrated with the help of the benchmark business cycle model. The chapter concludes with a detailed description of a computationally more involved monetary model.
Burkhard Heer, Alfred Maußner
Chapter 7. Discrete State Space Value Function Iteration
Abstract
The methods presented in the previous chapters use a system of stochastic difference equations that governs the time path of an economy to find approximations of the policy functions that determine the endogenous variables given the current state of the economy. In this chapter, we switch the perspective from the Euler equations approach to the dynamic programming approach
Burkhard Heer, Alfred Maußner

Heterogenous Agent Models

Frontmatter
Chapter 8. Computation of Stationary Distributions
Abstract
This chapter introduces readers to the modeling, computation of heterogeneous-agent economies. In this kind of problem, one has to compute the distribution of the individual state variable(s). In the first part, Heer, Maußner present avery simple heterogeneous-agent model with aspecial form of preferences — Gorman preferences — that allows to use the representative household’s policy function for the computation of the aggregate dynamics. The second part considers models in which the distribution of wealth, income affects the dynamics of the aggregate economy. Applications cover the puzzle of the low risk-free interest rate, the distributional effects of switching from an income tax to aconsumption tax. The chapter concludes with ashort survey of recent literature on the theory of income distribution.
Burkhard Heer, Alfred Maußner
Chapter 9. Dynamics of the Distribution Function
Abstract
This chapter presents methods for computing the dynamics of an economy that is populated by heterogeneous agents. The law of motion for the distribution of individual wealth is first computed in a model with individual uncertainty and subsequently in an economy with both individual and aggregate risk. To solve for the dynamics of the stochastic heterogeneous-agent neoclassical growth model, the algorithm developed by Krusell and Smith is described in detail. As two prominent applications from the literature, Heer and Maußner consider the welfare effects of business-cycle fluctuations and the cyclical dynamics of the income shares.
Burkhard Heer, Alfred Maußner
Chapter 10. Overlapping Generations Models with Perfect Foresight
Abstract
This chapter studies overlapping generations (OLG) models as pioneered by Auerbach and Kotlikoff. Here, agents differ not only with regard to their individual productivity or wealth but also with regard to their age. Heer and Maußner pay particular attention to the computation of the steady state in a large-scale OLG model with more than one hundred endogenous variables — the curse of dimensionality — and to updating the transition path for the aggregate variables. Two applications in this chapter include the study of Laffer curves and the consequences of the demographic transition for the US economy.
Burkhard Heer, Alfred Maußner
Chapter 11. OLG Models with Uncertainty
Abstract
This chapter introduces both idiosyncratic and aggregate uncertainty into overlapping generations (OLG) models. The methods used for the computation of these models are already familiar from previous chapters on the (stochastic) neoclassical growth model and only need to be modified to allow for the more complex age structure of OLG models. Heer and Maußner first introduce individual stochastic productivity into the standard OLG model and apply the model to explain the empirically observed income and wealth distribution. In the second part, aggregate uncertainty is introduced into the OLG model and its business-cycle properties are compared to those of the standard real-business cycle model.
Burkhard Heer, Alfred Maußner

Numerical Methods

Frontmatter
Chapter 12. Linear Algebra
Abstract
This chapter covers some elementary and some relatively advanced but very useful concepts and techniques from linear algebra. The latter comprise various matrix factorizations employed in Chapters 4 and 6
Burkhard Heer, Alfred Maußner
Chapter 13. Function Approximation
Abstract
There are numerous instances in which we need to approximate functions of one or several variables. Examples include the perturbation methods considered in Chaps. 2–4, the discrete state space methods of Chap. 7, and the weighted residuals methods of Chap. 5. This chapter gathers the most useful tools for this purpose. First, we clarify the notion of function approximation in the next section. We distinguish between local and global methods.
Burkhard Heer, Alfred Maußner
Chapter 14. Differentiation and Integration
Abstract
Numerical derivatives and integrals are basic components of virtually every algorithm discussed in this book. The first part of this chapter presents the standard methods used to numerically approximate first-order and second-order partial derivatives from finite differences. The second part covers numerical integration, with a special emphasis on the approximation of expectations.
Burkhard Heer, Alfred Maußner
Chapter 15. Nonlinear Equations and Optimization
Abstract
This chapter presents numerical methods used to locate the roots of systems of nonlinear equations or find the extrema of real-valued functions defined on a subset of Euclidean n-space. It demonstrates the close link between the two kinds of problem, since the solution of a root-finding problem can be transformed into a nonlinear minimization problem. With one exception (genetic search), the authors consider algorithms that are iterative, proceeding from a given initial point in successive steps. First, they develop stopping criteria for the iterative algorithms. Next, they consider standard methods used to solve nonlinear equations including the (modified) Newton-Raphson and Gauss-Seidel methods, before describing numerical optimization methods such as the golden section search, Newton’s method, and quasi Newton methods.
Burkhard Heer, Alfred Maußner
Chapter 16. Difference Equations and Stochastic Processes
Abstract
Stochastic processes are an integral part of all models examined in Parts I and II of this book. Linear difference equations, the building blocks of (vector) autoregressive stochastic processes, are reviewed in the first part of the chapter. Subsequently, Heer and Maußner formally introduce readers to stochastic processes in discrete time, before analyzing Markov chains and techniques used to approximate continuously valued autoregressive processes via finite-state Markov chains. The final part of the chapter considers time series filters, which can help to extract stochastic trends from nonstationary stochastic processes.
Burkhard Heer, Alfred Maußner
Backmatter
Metadata
Title
Dynamic General Equilibrium Modeling
Authors
Burkhard Heer
Alfred Maußner
Copyright Year
2024
Electronic ISBN
978-3-031-51681-8
Print ISBN
978-3-031-51680-1
DOI
https://doi.org/10.1007/978-3-031-51681-8