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Modern business cycle theory and growth theory uses stochastic dynamic general equilibrium models. In order to solve these models, economists need to use many mathematical tools. This book presents various methods in order to compute the dynamics of general equilibrium models. In part I, the representative-agent stochastic growth model is solved with the help of value function iteration, linear and linear quadratic approximation methods, parameterised expectations and projection methods. In order to apply these methods, fundamentals from numerical analysis are reviewed in detail. In particular, the book discusses issues that are often neglected in existing work on computational methods, e.g. how to find a good initial value.

In part II, the authors discuss methods in order to solve heterogeneous-agent economies. In such economies, the distribution of the individual state variables is endogenous. This part of the book also serves as an introduction to the modern theory of distribution economics. Applications include the dynamics of the income distribution over the business cycle or the overlapping-generations model.

In an accompanying home page to this book, computer codes to all applications can be downloaded.



Representative Agent Models

1. Basic Models

This chapter introduces you to the framework of dynamic general equilibrium models. Our presentation serves two aims: first, we prepare the ground for the algorithms presented in subsequent chapters that use one out of two possible characterizations of a model's solution. Second, we develop standard tools in model building and model evaluation used throughout the book.
The most basic DGE model is the so called Ramsey model, where a single consumer-producer chooses an utility maximizing consumption profile. We begin with the deterministic, finite-horizon version of this model. The set of first-order conditions for this problem is a system of non-linear equations that can be solved with adequate software. Then, we consider the infinite-horizon version of this model. We characterize its solution along two lines: the Euler equations provide a set of difference equations that determine the optimal time path of consumption; dynamic programming delivers a policy function that relates the agent's choice of current consumption to his stock of capital. Both characterizations readily extend to the stochastic version of the infinite-horizon Ramsey model that we introduce in Section 1.3. In Section 1.4 we add productivity growth and labor supply to this model. We use this benchmark model in Section 1.5 to illustrate the problems of parameter choice and model evaluation. Section 1.6 concludes this chapter with a synopsis of the numerical solution techniques presented in Chapters 2 through 6 and introduces measures to evaluate the goodness of the approximate solutions.

2. Perturbation Methods

In the previous chapter we have seen that the solution of a DGE model with a representative agent is given by a set of policy functions that relate the agent's choice variables to the state variables that characterize the agent's economic environment. In this chapter we explore methods that use local information to obtain either a linear or a quadratic approximation of the agent's policy function.

3. Deterministic Extended Path

We know from Section 1.1 that the first-order conditions of the deterministic finite-horizon Ramsey model constitute a system of non-linear equations. The first section of this chapter employs a non-linear equations solver to obtain the approximate time profile of the optimal capital stock. We then extend this approach to the infinite-horizon deterministic Ramsey model of Section 1.2. At first sight this may seem impossible since this model has an infinite number of unknowns. However, we know from Section 1.2.4 that the optimal time profile of the capital stock converges monotonically to the stationary solution. We use this observation to reduce the system of first-order conditions to a finite number of equations. In Section 3.2 we turn to the stochastic Ramsey model (1.22). We use the property of this model to converge after a one-time productivity shock, to trace out a Rational expectations path for its variables. From this path we obtain the solution for the decision variables of the current period. This observation dates back to Fair and Tay l o r (1983) and was applied to the stochastic Ramsey model by Gagnon (1990) from whom we borrowed the label 'deterministic extended path'. More broadly speaking, the method is a forward iteration method, since we solve for current-period variables by determining a specific future path of the economy. We sketch the general structure of this approach at the end of Section 3.2 and close the chapter with two further applications in Section 3.3: our benchmark model of Example 1.5.1 and the small open economy model of Correia, Neves and Rebelo (1995). This latter model is less suited for the methods of Chapter 2, since it has no uniquely determined stationary solution. This feature, however, poses no problem for the deterministic extended path algorithm.
The mathematical tools that we employ in this chapter are algorithms that obtain approximate numeric solutions to systems of non-linear equations. We explain the mathematical background behind the most common routines in Section 11.5. It is the task of each researcher to prepare the system of non-linear equations to which her or his model gives rise so that a non-linear equations solver is able to obtain the solution. For this reason, this chapter is a collection of example applications that demonstrate the use of the deterministic extended path approach. Since the solution of non-linear equations with numerical methods is a delicate business, we hope nevertheless that it will be useful for the reader to go through the following pages.

4. Discrete State Space Methods

In this chapter we explore methods that replace the original model by a model whose state space consists of a finite number of discrete points. In this case, the value function is a finite dimensional object. For instance, if the state space is one-dimensional and has elements X = {x1,x2, …,x n }, the value function is just a vector of n elements where each element gives the value attained by the optimal policy if the initial state of the system is x j X. We can start with an arbitrary vector of values representing our initial guess of the value function and then obtain a new vector by solving the maximization problem on the rhs of the Bellman equation. This procedure will converge to the true value function of this discrete valued problem. Though simple in principle, this approach has a serious drawback. It suffers from the curse of dimensionality. On a one-dimensional state space, the maximization step is simple. We just need to search for the maximal element among n. Yet, the value function of an m-dimensional problem with n different points in each dimension is an array of n m different elements and the computation time needed to search this array may be prohibitively high.

5. Parameterized Expectations

We know from Chapter 1 that there are two ways to characterize the solution of a Ramsey problem or, more generally, of a recursive dynamic general equilibrium (DGE) model: (1) in terms of a policy function that relates the model's decision or control variables to the model's state variables or (2) in terms of a system of stochastic difference equations that determines the time paths of the model's endogenous variables. The method presented in this chapter rests on yet a third solution concept. In the Rational expectations equilibrium of a recursive DGE model agents' conditional expectations are time invariant functions of the model's state variables. The parameterized expectations approach (PEA) applies methods from function approximation (see Section 11.2) to these unknown functions. In particular, it uses simple functions instead of the true but unknown expectations and employs Monte Carlo techniques to determine their parameters.
The PEA has several advantages vis-a-vis both the value function iteration approach and the extended path algorithm. In contrast to the former, it does not suffer as easily from the curse of dimensionality and, therefore, can be applied to models with many endogenous state variables. Unlike the latter, it deals easily with binding constraints. Our applications in Section 5.3 illustrate these issues.

6. Projection Methods

The parameterized expectations approach (PEA) considered in the previous chapter solves DGE models by approximating the agents' conditional expectations and determines the best approximation via Monte-Carlo simulations. In this chapter, we also employ methods from function approximation. Yet, these methods are not limited to functions that determine the agents' conditional expectations, nor do they necessarily resort to simulation techniques to find a good approximation. These methods, known as projection or weighted residual methods, may, thus, be viewed as generalizations of the PEA along certain dimensions. 1) The functions that we approximate do not need to be the conditional expectations that characterize the first-order conditions of the agents in our model. Instead, we may approximate the agent's policy function, or the value function of the problem at hand. 2) We use different criteria to determine the goodness of the fit between the true but unknown function and its polynomial representation. These criteria prevent the problem that we encountered in the previous chapter, namely, that it may be difficult to increase precision by using a higher degree polynomial. 3) Some of these criteria require numerical integration. The Monte-Carlo simulation is just one way to do this. Other techniques exist and often are preferable.
This chapter is structured as follows. First, the general idea of projection methods is presented. Second, we consider the various steps that constitute this class of methods in more detail. It will become obvious that we need several numerical tools to implement a particular method. Among them are numerical integration and optimization as well as finding the zeros of a set of non-linear equations. Third, we apply projection methods to the deterministic and the simple stochastic growth model and compare our results to those of Chapter 2 and Chapter 4. As an additional application, we study the equity premium puzzle, i.e. the (arguably) missing explanation for the observation that the average return on equities has been so much higher than the one on bonds over the last century. For this reason, we consider asset pricing within the stochastic growth model.

Heterogeneous Agent Models

7. Computation of Stationary Distributions

This chapter introduces you to the modeling and computation of heterogeneous-agent economies. In this kind of problem, we have to compute the distribution of the individual state variable(s). While we focus on the computation of the stationary equilibrium in this chapter, you will learn how to compute the dynamics of such an economy in the next chapter.
The representative agent framework has become the standard tool for modern macroeconomics. It is based on the intertemporal calculus of the household that maximizes lifetime utility. Furthermore, the household behaves rationally. As a consequence, it is a natural framework for the welfare analysis of policy actions. However, it has also been subject to the criticism whether the results for the economy with a representative household carry over to one with heterogenous agents. In the real economy, agents are different with regard to many characteristics including their abilities, their education, their age, their marital status, their number of children, their wealth holdings, to name but a few. As a consequence it is difficult to define a representative agent. Simple aggregation may sometimes not be possible or lead to wrong implications. For example, if the savings of the households are a convex function of income and, therefore, the savings rate increases with higher income, the definition of the representative household as the one with the average income or median income may result in a consideration of a savings rate that is too low.1 In addition, we are unable to study many important policy and welfare questions that analyze the redistribution of income among agents like, for example, through the reform of the social security and pensions system or by the choice of a flat versus a progressive schedule of the income tax.

8. Dynamics of the Distribution function

This chapter presents methods in order to compute the dynamics of an economy that is populated by heterogenous agents. In the first section, we show that this amounts to compute the law of motion for the distribution function F(ϵ, α) of wealth among agents. In the second section, we concentrate on an economy without aggregate uncertainty. The initial distribution is not stationary. For example, this might be the case after a change in policy, e.g. after a change in the income tax schedule, or during a demographic transition, as many modern industrialized countries experience it right now. Given this initial distribution, we compute the transition to the new stationary equilibrium. With the methods developed in this section we are able to answer questions as to how the concentration of wealth evolves following a change in capital taxation or how the income distribution evolves following a change in the unemployment compensation system. In the third section, we consider a model with aggregate risk. There are many ways to introduce aggregate risk, but we will focus on a simple case. We distinguish good and bad times which we identify with the boom and recession during the business cycle. In good times, employment probabilities increase and productivity rises. The opposite holds during a recession. As one application, we study the income and wealth distribution dynamics over the business cycle in the final section of this chapter. We will need to find an approximation to the law of motion F′ = G(F) and introduce you to the method developed by Krusell and Smith (1998).

9. Deterministic Overlapping Generations Models

In this chapter, we introduce an additional source of heterogeneity. Agents do not only differ with regard to their individual productivity or their wealth, but also with regard to their age. First, you will learn how to compute a simple overlapping generations model (OLG model) where each generation can be represented by a homogeneous household. Subsequently, we study the dynamics displayed by the typical Auerbach-Kotlikoff model. We will pay particular attention to the updating of the transition path for the aggregate variables.
The previous two chapters concentrated on the computation of models based on the Ramsey model. In this chapter, we will analyze overlapping generations models. The central difference between the OLG model and the Ramsey model is that there is a continuous turnover of the population. The lifetime is finite and in every period, a new generation is born and the oldest generation dies. In such models, many cohorts coexist at any time. In the pioneering work on OLG models by Samuelson (1958) and Diamond (1965), the number of coexisting cohorts only amounts to two, the young and working generation on the one hand and the old and retired generation on the other hand. In these early studies of simple OLG models, Samuelson (1958) and Diamond (1965) focused on the analysis of theoretical problems, i.e. if there is a role for money and what are the effects of national debt, respectively.

10. Stochastic Overlapping Generations Models

In this chapter, we introduce both idiosyncratic and aggregate uncertainty into the OLG model. The methods that we will apply for the computation of these models are already familiar to you from previous chapters and will only be modified in order to allow for the more complex age structure of OLG models. In particular, we will apply the log-linearization from Chapter 2, the algorithm for the computation of the stationary distribution from Chapter 7, and the Algorithm by Krusell and Smith (1998) from Chapter 8 for the solution of the non-stochastic steady state and the business cycle dynamics of the OLG model.
In the following, we will first introduce individual stochastic productivity in the standard OLG model, and, then, aggregate stochastic productivity. In the first section, agents have different productivity types. Different from the traditional Auerbach-Kotlikoff models, agents are subject to idiosyncratic shocks and may change their productivity types randomly. As a consequence, the direct computation of policies and transition paths is no longer feasible. As an interesting application, we are trying to explain the empirically observed wealth heterogeneity. In the second section, we introduce aggregate uncertainty and study the business cycle dynamics of the OLG model.


11. Numerical Methods

In this section we provide some elementary and some more advanced, but very useful concepts and techniques from linear algebra. Most of the elementary material gathered here is found in any undergraduate textbook on linear algebra as, e.g., Lang (1987). For the more advanced subjects Bronson (1989) as well as Golub and Van Loan (1996) are good references. In addition, many texts on econometrics review matrix algebra, as, e.g., Greene (2003), Appendix A, or Judge et al. (1982), Appendix A.

12. Various Other Tools

Dynamic models are either formulated in terms of difference or differential equations. Here we review a few basic definitions and facts about difference equations.


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