Decision Making under Uncertainty and Bounded Rationality

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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 non-linear 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 life-cycle 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 Non-Linear 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 non-linear 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 ad-hoc 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 state-space
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 non-linear 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 life-cycle 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 life-cycle
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 non-linear 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 life-cycle 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.

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