Abstract: In an attempt to capture the complexity of the economic system many economists were led to the formulation of complex nonlinear rational expectations models that in many cases can not be solved analytically. In such cases, numerical methods need to be employed. In chapter one I review several numerical methods that have been used in the economic literature to solve nonlinear rational expectations models. I provide a classification of these methodologies and point out their strengths and weaknesses. I conclude by discussing several approaches used to measure accuracy of numerical methods. In the presence of uncertainty, the multistage stochastic optimization literature has advanced the idea of decomposing a multiperiod optimization problem into many subproblems, each corresponding to a scenario. Finding a solution to the original problem involves aggregating in some form the solutions to each scenario and hence its name, scenario aggregation. In chapter two, I study the viability of scenario aggregation methodology for solving rational expectation models. Specifically, I apply the scenario aggregation method to obtain a solution to a finite horizon life cycle model of consumption. I discuss the characteristics of the methodology and compare its solution to the analytical solution of the model. A growing literature in macroeconomics is tweaking the unbounded rationality assumption in an attempt to find alternative approaches to modeling the decision making process, that may explain observed facts better or easier. Following this line of research, in chapter three, I study the impact of bounded rationality on the level of precautionary savings in a finite horizon lifecycle model of consumption. I introduce bounded rationality by assuming that the consumer does not have either the resources or the sophistication to consider all possible future events and to optimize accordingly over a long horizon. Consequently, he focuses on choosing a consumption plan over a short span by considering a limited number of possible scenarios. While under these assumptions the level of precautionary saving in many cases is below the level that a rational expectations model would predict, there are also parameterizations of the model for which the reverse is true.
Review of Methods Used for Solving NonLinear Rational Expectations Models:
Limitations faced by most linear macroeconomic models coupled with the growing importance of rational expectations have led many economists, in an attempt to capture the complexity of the economic system, to turn to nonlinear rational expectation models. Since the majority of these models can not be solved analytically, researchers have to employ numerical methods in order to be able to compute a solution. Consequently, the use of numerical methods for solving nonlinear rational expectations models has been growing substantially in recent years. For the past decade, several strategies have been used to compute the solutions to nonlinear rational expectations models. The available numerical methods have several common features as well as differences, and depending on the criteria used, they may be grouped in various ways. Following is an adhoc categorization1 that will be used throughout this chapter. The first group of methods I consider has as a common feature the fact that the assumption of certainty equivalence is used at some point in the computation of the solutionThe second group of methods has as a common denominator the use of a discrete state space, or the discretization of an otherwise continuous space of the state variables. The methods falling into this category are often referred to as discrete statespace methods. They work well for models with a low number of state variables. The next set of methods is generically known as the class of perturbation methods. Since perturbation methods make heavy use of local approximations, in this presentation, I group them along with some other techniques that use local approximations under the heading of local approximations and perturbation methods. The fourth group, labeled here as projection methods consists of a collection of methodologies that approximate the true value of the conditional expectations of nonlinear functions with some finite parameterization and then evaluate the initially undetermined parameters. Several methods included in this group have recently become very popular in solving nonlinear rational expectations models containing a relatively small number of state variables. The layout of the chapter contains the presentation of a generic nonlinear rational expectations model followed by a description of the methods mentioned above. Throughout the chapter, special cases of the model described in section 2 are used to show how one can apply the methods discussed here.
Using Scenario Aggregation Method to Solve a Finite Horizon Life Cycle Model of Consumption:
Multistage optimization problems are a very common occurrence in the economic literature. While there exist other approaches to solving such problems, many economic models involving intertemporal optimizing agents assume that the representative agent chooses its actions as a result of solving some dynamic programming problem. Lately, an increasing number of researchers have investigated alternative approaches to modeling the representative agent, in an attempt to find one that may explain observed facts better or easier. Following the same line of research, I explore the suitability of scenario aggregation method as an alternative to describe the decision making process of an optimizing agent in economic models. The idea is that this methodology offers a different approach that might be more consistent with the observation that agents are more likely to behave like chess players, making decisions based only on a subset of all possible outcomes and using a relatively short horizon41. The advantage of scenario aggregation methodology is that, while it presents attractive features for use in models assuming bounded rationality, it can also be seen as an alternative numerical method that can be used for obtaining approximate solutions for rational expectation models. Therefore, I start by studying in this chapter the viability of the scenario aggregation method, as presented by Rockafellar and Wets (1991), to provide a good approximation for the optimal solution of a simple finite horizon lifecycle model of consumption with precautionary savings. In the next chapter, I will use scenario aggregation to model the decision making of the rationally bounded consumer. The layout of this chapter is as follows. First, I present the setup of a simple lifecycle consumption model with precautionary saving. Then, I introduce the notion of scenarios followed by a description of the aggregation method. Next, I introduce the progressive hedging algorithm followed by its application to a finite horizon lifecycle consumption model. Then, I present simulation results and conclude the chapter with final remarks.
Impact of Bounded Rationality on the Magnitude of Precautionary Saving:
It is fair to say that nowadays the assumption of rational expectations has become routine in most economic models. Recently, however, there has been an increasing number of papers, such as Gali et al. (2004), Allen and Carroll (2001), Krusell and Smith (1996), that have modeled consumers using assumptions that depart from the standard rational expectations paradigm. Although they are not explicitly identified as modeling bounded rationality, these assumptions clearly take a bite from the unbounded rationality, which is the standard endowment of the representative agent. The practice of imposing limits on the rationality of agents in economic models is part of the attempts made in the literature to circumvent some of the limitations associated with the rational expectations assumption. Aware of its shortcomings, even some of the most ardent supporters58 of the rational expectations paradigm have been looking for possible alterations of the standard set of assumptions. As a result, a growing literature in macroeconomics is tweaking the unbounded rationality assumption resulting in alternative approaches that are usually presented under the umbrella of bounded rationalityOne may ask why is there a need to even consider bounded rationality. First, individual rationality tests led various researchers to “hypothesize that subjects make systematic errors by using ... rules of thumb which fail to accommodate the full logic of a decision” (J. Conlisk, 1996). Secondly, some models assuming rational expectations fail to explain observed facts, or their results may not match empirical evidence. Since most of the time models include other hypotheses besides the unbounded rationality assumption, the inability of such models to explain certain observed facts could not be blamed solely on rational expectations. Yet, it is worth investigating whether bounded rationality plays an important role in such cases. Finally, as Allen and Carroll (2001) point out, even when results of models assuming rational expectations match the data, it is still worth asking the question of how can an average individual find the solution to complex optimization problems that until recently economists could not solve. To summarize, the main idea behind this literature is to investigate what happens if one changes the assumption that agents being modeled have a deeper understanding of the economy than researchers do, as most rational expectations theories assume. Therefore, instead of using rational expectations, it is assumed that economic agents make decisions behaving in a rational manner but being constrained by the availability of data and their ability to process the available information. While the vast literature on bounded rationality continues to grow, there is yet to be found an agreed upon approach to modeling rationally bounded economic agents. Among the myriad of methods being used, one can identify decision theory, simulationbased models, artificial intelligence based methodologies such as neural networks and genetic algorithms, evolutionary models drawing their roots from biology, behavioral models, learning models and so on. Since there is no standard approach to modeling bounded rationality, most of the current research focuses on investigating the importance of imposing limits on rationality, as well as on choosing the methods to be used in a particular context. When modeling consumers, the method of choice so far seems to be the assumption that they follow some rules of thumb59. Instead of imposing some rules of thumb, my approach in modeling bounded rationality focuses on the decision making process. I borrow the idea of scenario aggregation from the multistage optimization literature and I adapt it to fit, what I believe to be, a reasonable description of the decision making process for a representative consumer. Besides the decision making process per se, I also add a few other elements of bounded rationality that have to do with the ability to gather and process information. In the previous chapter, the method of scenario aggregation was introduced as an alternative method for solving nonlinear rational expectation models. Even though it performs well in certain circumstances, the real advantage of the scenario aggregation lays in a different area. Its structure presents itself as a natural way to describe the process through which a rationally bounded agent, faced with uncertainty, makes his decision. In this chapter, I consider several versions of a lifecycle consumption model with the purpose of investigating how the magnitude of precautionary saving changes with the underlying assumptions on the (bounded) rationality of the consumer.
