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Starts by looking at the advances in macroeconomics that have been made since A. W. Phillips's 1958 Economica paper and his physical economic model (made of piping and valves) designed to teach students the complexities of macroeconomics. Two developments have contributed to these advances: first, macroeconomics is now based on the optimizing decision of individual agents who have formed expectations about an uncertain future, and emphasizes the use of Euler equations and dynamic programming to analyse the dynamic properties of economies; second, there has been a huge increase in the computational power available for studying model economies. The following two sections of the chapter are general discussions on the structure of (dynamic) economics models and the use of computational techniques. Lastly, an outline is given of the structure of the book.

There are many questions in economics for which heterogeneous‐agent dynamic models (i.e. models populated by agents that are different from each other) have to be used to provide answers. The first such model presented is one with infinitely lived agents subject to uninsurable idiosyncratic shocks to earnings; a very simple version of such a model is used to address the distribution of wealth under both a steady state and a non‐steady state situation. The second type of model presented is the overlapping generations model, which represents the situation where every year some agents die and new agents are born; such models are embodied in a neoclassical growth model structure with capital accumulation. Once again, both steady state and non‐steady state situations are considered. Finishes with a section on dynamic voting models; these are models that endogenously generate government policies as part of a Markov equilibrium, and are starting to be used to study positive policy issues such as redistributional policies. All the models are posed in such a way that they are susceptible to computation, and in all of them, the different agents are central in the sense that the question that the models are used to answer requires heterogeneity.

Introduces some of the methods and underlying ideas behind computational fluid dynamics—in particular, the use is discussed of finite‐difference methods for the simulation of dynamic economies. A standard stochastic dynamic programming model is considered of a macroeconomy. Finite‐difference methods are applied to this problem (model), resulting in a second‐order nonlinear partial differential equation that has some features in common with the governing equations of fluid dynamics; the idea is also introduced of ‘upwind’ or solution‐dependent differencing methods, and the stability of these is discussed through the analysis of model problems. An implicit solution to the nonlinear dynamic programming problem is then developed and tested, with the motivation of reducing the computer time required to solve it. Finally, the extension of the finite‐difference method to a two‐state dynamic programming problem is considered.

An extensive treatment is provided of methods that use log‐linear approximations to solve nonlinear dynamic discrete‐time stochastic models. These methods, based on their linear counterparts, have been extensively used in the macroeconomic literature, and Uhlig simplifies, integrates, and compares them. He shows how to log‐linearize the necessary equations characterizing the equilibrium without explicit differentiation, provides a general solution to a linearized system using the method of undetermined coefficients, allowing in particular for a vector of endogenous states, and provides a simulation‐free frequency‐domain method to calculate the model implications in its Hodrick–Prescott filtered version. These methods are easy to use if a numerical package such as MATLAB is available. Examples of the approach taken are presented in an appendix, which looks at Hansen's (1985) real business cycle model for the case of saddle‐point stability (using the Blanchard–Kahn approach) and the case where equilibria may be undetermined; the appendix also describes the MATLAB programs needed to carry out the calculations.

A number of numerical methods are discussed for solving dynamic stochastic general equilibrium models that fall within the common category of discrete state‐space methods. These methods can be applied in situations where the state space of the model in question is given by a finite set of discrete points; in these cases the methods provide an ‘exact’ solution to the model in question. However they are frequently applied in situations where the model's state space is continuous in which case the discrete state space can be viewed as an approximation to the continuous state space. Discrete state‐space methods are discussed in the context of two well‐known examples: a simple one‐asset version of Lucas's (1978) consumption‐based asset pricing model and the one‐sector neoclassical growth model. The discussion does not aim to exhaust the list of possible discrete state‐space methods as they are very numerous; rather it describes several examples that illustrate the basic principles involved. The main sections of the chapter describe the basic principles of numerical quadrature underlying most discrete state‐space methods, show how they can be applied in a very straightforward way to problems in which the state space consists entirely of exogenous state variables, and describe methods that can be used when there are endogenous state variables. The last section notes the several files associated with the chapter for use with MATLAB.

Some practical issues are discussed that relate to the use of the parameterized expectations approach (PEA) for solving nonlinear stochastic dynamic models with rational expectations. This approach has been applied widely as it turns out to be a convenient algorithm, especially when there are large numbers of state variables and stochastic shocks in key conditional expectations terms. The first main section of the chapter provides a detailed discussion of some practical issues associated with the algorithm, and of its application. This is done using a set of examples—the Lucas asset pricing model, the simple stochastic growth model, and four variations of the latter, each selected to demonstrate a different issue. The next section describes a FORTRAN program used for implementing the algorithm, and the following one shows how it is applied to and adapted for each example previously presented.

Many problems in economics require the solution to a functional equation as an intermediate step, and typically, decision functions are sought that satisfy a set of Euler conditions or a value function that satisfies Bellman's equation. However, in many cases, analytical solutions cannot be derived for these functions, and numerical methods are needed instead. Shows how to apply weighted residual and finite‐element methods to this type of problem by illustrating their application to various examples. The first type of problem involves a simple differential equation because the coefficients to be computed satisfy a linear system of equations, and no computer is needed for the solution. Weighted residual and finite‐element methods are then applied to a deterministic growth model and a stochastic growth model—two standard models used in economics; in these examples, the coefficients to be computed satisfy nonlinear systems of equations, which, fortunately, are exploitably sparse if they are derived from a finite‐element method.

This is a brief introduction to dynamic programming and the method of using linear quadratic (LQ) approximations to the return function; the method is an approximation because it computes the solution to a quadratic expansion of the utility function about the steady state or the stable growth path of model economies. The main purpose of the chapter is to review the theoretical basis for the LQ approximation and to illustrate its use with a detailed example (social planning). The author demonstrates that, using the LQ approximation approach and the certainty equivalence principle, solving for the value function is a relatively easy task. The different sections of the chapter describe the standard neoclassical growth model, present a social planner problem that can be used to solve the model, give a recursive formulation of the social planner's problem, and describe an LQ approximation to this problem. Exercises are included throughout, and an appendix presents a MATLAB program to illustrate the LQ method.

Discusses the main issues involved in practical applications of solution methods that have been proposed for rational expectations models, based on eigenvalue–eigenvector decompositions. It starts by reviewing how a numerical solution can be derived for the standard deterministic Cass–Koopmans–Brock–Mirman economy, pointing out the relevance of stability conditions. Next the general structure used to solve linear rational expectations models, and its extension to nonlinear models, is summarized. The solution method is then applied to Hansen's (1985) model of indivisible labour, and comparisons with other solution approaches are discussed. It is then shown how the eigenvalue–eigenvector decomposition can help to separately identify variables of a similar nature (as is the case when physical capital and inventories are inputs in an aggregate production technology), and how the solution method can be adapted to deal with endogenous growth models.

A core topic of current economic research (and policy debate) is the evaluation of social security systems and their possible reforms. Shows how models of social security can be computed in economies where agents have uncertain lifespans and earnings profiles. In particular, it shows how to solve stationary equilibria and (within a linear quadratic formulation) how to solve transitional equilibria, such as the transition following a reform of the system. The two main sections of the chapter present: a model of social security with heterogeneous agents, which is related to several recent large‐scale general equilibrium, overlapping generations models; and a linear quadratic model of social security. These are both versions of an overlapping generations model with incomplete markets, and both assume that private annuity markets are missing, but they differ in their preference structures and certain other respects.

The Lucas and Stokey (1983) economy without capital is used to exhibit features of the Lucas and Stokey model of optimal taxation, and show how they compare with Barro's (1979) tax‐smoothing model. Computation of optimal fiscal policies for Lucas and Stokey's model requires repeated evaluations of the present value of the government's surplus, an object formally equivalent to an asset price. The functional equation for an asset price is typically difficult to solve. A linear quadratic version of Lucas and Stokey's model is specified, which makes both asset pricing computations and optimal fiscal policy calculations easy. Martingale returns on government debt are discussed, and examples and extensions of Lucas and Stokey's model given. Two appendices describe and discuss: the key steps for two basic kinds of stochastic process (a stochastic first‐order linear difference equation and a Markov chain), and time consistency and the structure of debt. Lastly, details are given of the appropriate MATLAB programs.