Description
Bounded rationality is the idea that in decision-making, rationality of individuals is limited by the information they have, the cognitive limitations of their minds, and the finite amount of time they have to make a decision.
CASE STUDY ON DECISION MAKING UNDER
UNCERTAINTY AND BOUNDED RATIONALITY
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 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.
ii
Table of Contents
Dedication ..................................................................................................................... ii
Table of Contents......................................................................................................... iii
Chapter I. Review of Methods Used for Solving Non-Linear Rational Expectations
Models........................................................................................................................... 1
I.1. Introduction ........................................................................................................ 1
I.2. Generic Model .................................................................................................... 3
I.3. Using Certainty Equivalence; The Extended Path Method ................................ 6
I.3.1. Example ....................................................................................................... 7
I.3.2. Notes on Certainty Equivalence Methods ................................................. 10
I.4. Local Approximation and Perturbation Methods ............................................. 11
I.4.1. Regular and General Perturbation Methods .............................................. 11
I.4.2. Example ..................................................................................................... 13
I.4.3. Flavors of Perturbation Methods ............................................................... 15
I.4.4. Alternative Local Approximation Methods............................................... 16
I.4.5. Notes on Local Approximation Methods .................................................. 18
I.5. Discrete State-Space Methods .......................................................................... 19
I.5.1. Example. Discrete State-Space Approximation Using Value-Function
Iteration ............................................................................................................... 20
I.5.2. Fredholm Equations and Numerical Quadratures ..................................... 21
I.5.3. Example. Using Quadrature Approximations ........................................... 24
I.5.4. Notes on Discrete State-Space Methods.................................................... 26
I.6. Projection Methods........................................................................................... 27
I.6.1. The Concept of Projection Methods.......................................................... 28
I.6.2. Parameterized Expectations....................................................................... 39
I.6.3. Notes on Projection Methods .................................................................... 42
I.7. Comparing Numerical Methods: Accuracy and Computational Burden.......... 44
I.8. Concluding Remarks ........................................................................................ 47
Chapter II. Using Scenario Aggregation Method to Solve a Finite Horizon Life Cycle
Model of Consumption ............................................................................................... 49
II.1. Introduction ..................................................................................................... 49
II.2. A Simple Life-Cycle Model with Precautionary Saving ................................ 50
II.3. The Concept of Scenarios ............................................................................... 52
II.3.1. The Problem ............................................................................................. 52
II.3.2. Scenarios and the Event Tree ................................................................... 53
II.4. Scenario Aggregation...................................................................................... 57
II.5. The Progressive Hedging Algorithm............................................................... 60
II.5.1. Description of the Progressive Hedging Algorithm................................. 61
II.6. Using Scenario Aggregation to Solve a Finite Horizon Life Cycle Model .... 63
II.6.1. The Algorithm .......................................................................................... 65
II.6.2. Simulation Results.................................................................................... 68
II.6.3. The Role of the Penalty Parameter........................................................... 72
II.6.4. More simulations...................................................................................... 74
II.7. Final Remarks ................................................................................................. 76
iii
Chapter III. Impact of Bounded Rationality on the Magnitude of Precautionary
Saving ......................................................................................................................... 77
III.1. Introduction.................................................................................................... 77
III.2. Empirical Results on Precautionary Saving................................................... 80
III.3. The Model...................................................................................................... 82
III.3.1. Rule 1 ...................................................................................................... 87
III.3.2. Rule 2 ...................................................................................................... 96
III.3.3. Rule 3 .................................................................................................... 104
III.4. Final Remarks .............................................................................................. 110
Appendices................................................................................................................ 112
Appendix A. Technical notes to chapter 2............................................................ 112
Appendix A1. Definitions for Scenarios, Equivalence Classes and Associated
Probabilities ...................................................................................................... 112
Appendix A2. Description of the Scenario Aggregation Theory ..................... 115
Appendix A3. Solution to a Scenario Subproblem........................................... 118
Appendix B. Technical notes to chapter 3 ............................................................ 121
Appendix B1. Analytical Solution for a Scenario with Deterministic Interest
Rate ................................................................................................................... 121
Appendix B2. Details on the Assumptions in Rule 1 ....................................... 124
Appendix B3. Details on the Assumptions in Rule 2 ....................................... 125
iv
Chapter I. Review of Methods Used for Solving Non-Linear
Rational Expectations Models
I.1. Introduction
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 categorization
1
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
solution.
1
This classification draws on Binder et al. (2000), Burnside (1999.), Marcet et al. (1999),
McGrattan (1999), Novales et al. (1999), Uhlig (1999) and Judd (1992, 1998).
1
The 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
2
.
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
3
.
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.
2
Examples include Baxter et al. (1990), Christiano (1990a, 1990b), Coleman (1990),
Tauchen (1990) and Taylor and Uhlig (1990), Tauchen and Hussey (1991), Deaton and
Laroque (1992), and Rust (1996)
3
This approach is used, for example, by Binder et al. (2000), Christiano and Fisher
(2000), Judd (1992) and Miranda and Rui (1997).
2
I.2. Generic Model
I start by presenting a generic model in discrete time that will be used along the
way to exemplify the application of some of the methods discussed in this chapter. I
assume that the problem consists of maximizing the expected present discounted value of
an objective function:
max E
?
¿ |
tt
(u
t
) O
0
? ·
subject to
u
t
?
?t =0
x
t
=
h
(x
t
÷
1
,u
t
, y
t
)
f (x
t
, x
t
÷
1
) > 0
?
?
(1.2.1)
(1.2.2)
(1.2.3)
where u
t
and x
t
denote the values of the control and state variables u and x
respectively, at the beginning of period t . y
t
is a vector of forcing variables, | e (0,1) a
constant discount factor while t represents the objective function. I further assume that
t
(
?
) is twice continuously differentiable, strictly increasing, and strictly concave with
respect to u
t
.
E
(? O
0
) denotes the mathematical expectations operator, conditional on
the information set at the beginning of period 0, O
0
. At any point in time, t , the
information set is given by O
t =
{u
t
,u
t
÷
1
,...; x
t
, x
t
÷
1
,...; y
t
, y
t
÷
1
,...}
4
. Finally, y
t
is assumed
to be generated by a first-order process
y
t
=
q
( y
t
÷
1
, z
t
) , (1.2.4)
4
The elements of the information set point to the fact that variables become known at the
beginning of the period. During the chapter this assumption may change to allow for an
easier setup of the problem.
3
where the elements of z
t
are distributed independently and identically across t and are
drawn from a distribution with a finite number of parameters.
The preceding generic optimization problem covers various examples of models
in economics, including the life-cycle model of consumption under uncertainty with or
without liquidity constraints, stochastic growth model with or without irreversible
investment and certain versions of asset pricing models. The present specification does
not cover models that have more than one control variable. However, some of the
techniques presented in this chapter could be used to solve such models.
If the underlying assumptions are such that the Bellman principle holds, one can
use the Bellman equation method to solve the dynamic programming problem. The
Bellman equation for the problem described by (1.2.1) - (1.2.2) is given by
V
( x
t
, y
t
) = muax
t
(u
t
) + | E ?
V
(
h
( x
t
,u
t
+
1
, y
t
+
1
), y
t
+
1
) | O
t
?
t
{
?
?
}
(1.2.5)
where
V
(
?
) is the value function. An alternative way to solve the model is to use the
Euler equation method. If u can be expressed as a function of x , i.e. u
t
=
g
(x
t
, x
t
÷
1
, y
t
) ,
the Euler equation for period t for the same problem is:
t'
u
?
g
( x
t
, x
t
÷
1
, y
t
)? g
'
x ( x
t
, x
t
÷
1
, y
t
) +
? ?
t
+ | E t'
u
?
g
( x
t
, x
t
+
1
, y
t
+
1
)? g
'
x
t
( x
t
, x
t
+
1
, y
t
+
1
) | O
t
= 0
{
? ?
}
(1.2.6)
So far, it has been assumed that the inequality constraint was not binding. If one
considers the possibility of constraint (1.2.3) being binding, then one must employ either
the Kuhn-Tucker method or the penalty function method. In the case of the former, the
Euler equation for period t becomes:
t'
u
?
g
( x
t
, x
t
÷
1
, y
t
)? g
'
x ( x
t
, x
t
÷
1
, y
t
) + ·
t
f
x
'
( x
t
, x
t
÷
1
) + ·
t
+
1
f
x
'
( x
t
, x
t
+
1
) +
? ?
t t t
(1.2.7)
+ | E t ?
g
( x
t
, x
t
+
1
, y
t
+
1
)? g
{
'
u
?
4
?
'
x
t
( x
t
, x
t
+
1
, y
t
+
1
) | O
t
} = 0
where ·
t
and ·
t
+
1
are Lagrange multipliers. The additional Kuhn-Tucker conditions are
given by:
·
t
> 0,
f
( x
t
, x
t
÷
1
) > 0, ·
t f
( x
t
, x
t
÷
1
) = 0 (1.2.8)
Alternatively, one can use penalty methods to account for the inequality constraint. One
approach is to modify the objective function by introducing a penalty term
5
. Then the
new objective function becomes:
E
e
9 |
t
t
t
(u
t
) + µ min
f
(x
t
, x
t
÷
1
), 0
?
O
0
÷
c
•
3
©
t =
0
e _
( )
?
?
?
?
where · is the penalty parameter. Consequently, the Bellman equation is given by:
V
( x
t
, y
t
) = muax
t
(u
t
) + · min (
f
( x
t
, x
t
÷
1
),
0
)3 + | E ?
V
(
h
( x
t
,u
t
+
1
, y
t
+
1
), y
t
+
1
) | O
t
?
t
{
?
?
}
(1.2.9)
Let u
*
t =
d
( x
t
, y
t
) denote the solution of the problem. When an analytical solution
for
d
(
?
) can not be computed, numerical techniques need to be used. Three main
approaches have been used in the literature to solve the problem (1.2.1) - (1.2.4) and to
obtain an approximation of the solution. First approach consists of modifying the
specification of the problem (1.2.1) - (1.2.2) so that it becomes easier to solve, as is the
case with the linear quadratic approximation
6
. Second approach is to employ methods
that seek to approximate the value and policy functions by using the Bellman equation
7
.
5
6
This approach is used by McGrattan (1990).
This approach has been used, among others, by Christiano (1990b) and McGrattan
(1990).
7
Examples of this approach are: Christiano (1990a), Rust (1997), Santos and Vigo
(1998), Tauchen (1990).
5
Finally, the third approach focuses on approximating certain terms appearing in the Euler
equation such as decision functions or expectations
8
.
These approaches have shaped the design of numerical algorithms used in solving
dynamic non-linear rational expectation models. In the next few sections, I will present
several of the numerical methods employed by researchers in their attempt to solve
functional equations such as the Euler and Bellman equations (1.2.5) - (1.2.9) presented
above.
I.3. Using Certainty Equivalence; The Extended Path Method
Certainty equivalence has been used especially for its convenience since it may
allow researchers to compute an analytical solution for their models. It has also been used
to compute the steady state of a model as a prerequisite for applying some linearization or
log-linearization around its equilibrium state
9
or to provide a starting point for more
complex algorithms
10
. One methodology that received a lot of attention in the literature is
the extended path method developed by Fair and Taylor (1983). Solving a model such as
(1.2.1) - (1.2.3) usually leads to a functional equation such as a Bellman or an Euler
equation.
8
Examples of this approach are Binder et al. (2000), Christiano and Fisher (2000), Judd
(1992), Marcet (1994), Mc-Grattan (1996).
9
This is the case in the linear quadratic approach where the law of motion is linearized
and the objective function is replaced by a quadratic approximation around the
deterministic steady state.
10
Certainty equivalence has also been used to provide starting values or temporary values
in algorithms used to solve models leading to nonlinear stochastic equations as in early
work by Chow (1973, 1976), Bitros and Kelejian (1976) and Prucha and Nadiri (1984).
6
Let
F x
t
, x
t
÷
1
,u
t
,u
t
÷
1
, y
t
, y
t
÷
1
, E
t
t
'
?
h
( x
t
, x
t
+
1
, y
t
+
1
)? h '
x
t
, E
t
t ?
h
( x
t
, x
t
+
1
, y
t
+
1
)? = 0
(
{
? ?
}
? ?
)
(1.3.1)
denote such a functional equation for period t . As before, x
t
is the state variable, u
t
is
the control variable, y
t
is a vector of forcing variables,
t
(
?
) is the objective function, t'
is the derivative of t with respect to the control variable, and E
t
is the conditional
expectations operator based on information available through period t . F is a function
that may be nonlinear in variables and expectations. For numerous models if the
expectations terms appearing in F were known, (1.3.1) could be easily solved. Since that
is not the case, the approach of the extended path method is to first set current and future
values of the forcing variables to their expected values. This is equivalent to assuming
that all future values of z
t
in equation (1.2.4) are zero. Then equation (1.3.1) becomes:
F x
t
, x
t
÷
1
,u
t
,u
t
÷
1
, y
t
, y
t
÷
1
,t
'
?
h
( x
t
, x
t
+
1
, E
t
y
t
+
1
)? h '
x
t
,t ?
h
( x
t
, x
t
+
1
, E
t
y
t
+
1
)? ,... = 0
(
? ? ? ?
) (1.3.2)
Then, the idea is to expand the horizon and iterate over solution paths. Let us consider an
example to see how this method can be applied.
I.3.1. Example
11
Consider the following problem where the social planner or a representative agent
maximizes an objective function
max E
e
9 |
tt
(u
t
) O
0
÷ •
u
t
c
©t=0
?
?
(1.3.3)
11
The application of the extended path method in this example draws to some extent on
the model presented in Gagnon (1990).
7
subject to
x
t
=
h
(x
t
÷
1
,u
t
, y
t
) (1.3.4)
where y
t
is a Gaussian AR (
1
) process with the law of motion y
t
= µ y
t
÷
1
+ z
t
where z
t
is i.i.d. N 0,o
2
. It is further assumed that u can be expressed as a function of x , i.e.
( )
u
t
=
g
(x
t
, x
t
÷
1
, y
t
) . Then the Euler equation for period t is:
0 = t ' ?
g
( x
t
, x
t
÷
1
, y
t
)? ? g '
x
t
( x
t
, x
t
÷
1
, y
t
)
? ?
+ | E t ' ?
g
( x
t
+
1
, x
t
, y
t
+
1
)? ? g '
x
t
( x
t
+
1
, x
t
, y
t
+
1
) O
t
{
? ?
}
(1.3.5)
If the expectation term were known in equation (1.3.5), it would be easy to find a
solution. The idea of the extended path method is to expand the horizon and then iterate
over solution paths. As in Fair and Taylor (1983), I consider the horizon t,...,t + k +1 and
assume that x
t
÷
1
and y
t
÷
1
are given and that z
t
+
s
= 0 for s = 1,..., k +1 . Following is an
algorithm that would implement the extended path methodology. The first step is to
choose initial values for x
t
+
s
and y
t
+
s
for s = 1,..., k +1 and denote them by ˆ
t
+
s
and x
yˆ
t
+
s
. Then, for period t , the Euler equation becomes:
0 = t ' ?
g
( x
t
, x
t
÷
1
, y
t
)? ? g '
x
t
( x
t
, x
t
÷
1
, y
t
)
? ?
(1.3.6)
+ |t ' ?
g
(ˆ
t
+
1
, x
t
, ˆ
t
+
1
)? ? g '
x
t
( x
t
+
1
, x
t
, ˆ
t
+
1
)
?x y?
ˆ y
Similarly, for period t + s , the Euler equation is given by:
0 = t ' ?
g
( x
t
+
s
, x
t
+s÷
1
, y
t
+
s
)? ? g '
x
t+
s
( x
t
+
s
, x
t
+s÷
1
, y
t
+
s
)
? ?
(1.3.7)
+ |t ' ?
g
(ˆ
t
+s+
1
, x
t
+
s
, ˆ
t
+s+
1
)? ? g '
x
t+
s
(ˆ
t
+
1
, x
t
+
s
, ˆ
t
+s+
1
)
In addition,
?x
y
?
x y
y
t
+
s
= µ y
t
+s÷
1
+ z
t
+s (1.3.8)
u
t
+
s
=
g
(x
t
+
s
, x
t
+s÷
1
, y
t
+
s
) (1.3.9)
8
Therefore, for period t + s , equations (1.3.7) - (1.3.9) define a system where x
t
+s÷
1
, y
t
+s÷
1
,
ˆ
t
+s+
1
, ˆ
t
+s+
1
are known so one can determine the unknowns x
t
+
s
, y
t
+
s
and u
t
+
s
. Let x
t
j+
s
,
x y
y
t
j+
s
and u
t
j+
s
denote the solutions of the system for s = 0,..., k +1 , where j represents the
iteration for a fixed horizon, in this case t,...,t + k +1. If the solutions x
t
j+s
{}
k +1
k +1
s =0
, y
t
j+s
{}
k +1
s =0
and u
t
j+s
{}
s =0
obtained in iteration j are not satisfactory then proceed with the next
iteration
where
{ˆ
t
j++s
1
}
s
=
1
=
{x
t
j+
s
}
s
=
1
,
{ˆ
t
j++s
1
}
s
=
1
=
{y
t
j+
s
}
s
=
1
. Notice that the horizon remains
k +1 k +1 k +1 k +1
x y
the same for iteration j +1. The iterations will continue until a satisfactory solution is
obtained. At this point, the methodology calls for the extension of the horizon without
modifying the starting period. Fair and Taylor extend the horizon by a number of periods
that is limited to the number of endogenous variables. This is in essence an ad-hoc rule.
In the present example, the horizon is extended by 2 periods, that is, t,...,t + k + 3. The
same steps are followed for the new horizon with the exception of the end criterion,
which should consist of a comparison between the last obtained solution, using the
t,...,t + k + 3 horizon, and the solution provided using the previous horizon, t,...,t + k +1.
The expansion of the horizon continues until a satisfactory solution is obtained. At that
point, the procedure will start over with a new starting period and a new horizon. In our
example the next starting period should be t +1 and the initial horizon t +1,...,t + k + 2 .
One of the less mentioned caveats of this method is that no general convergence
proofs for the algorithm are available. In addition, the method relies on the certainty
equivalence assumption even though the model is nonlinear. Since expectations of
functions are treated as functions of the expectations in future periods in equation (1.3.2),
9
the solution is only approximate unless function F is linear. This assumption is similar
to the one used in the case of linear-quadratic approximation to rational expectations
models that has been proposed, for example, by Kydland and Prescott (1982).
In the spirit of Fair and Taylor, Fuhrer and Bleakley (1996), following an
algorithm from an unpublished paper by Anderson and Moore (1986), sketch a
methodology for finding the solution for nonlinear dynamic rational expectations models.
I.3.2. Notes on Certainty Equivalence Methods
All the methods that use certainty equivalence either as a main step or as a
preliminary step in finding a solution, incur an approximation error due to the assumption
of perfect foresight. The magnitude of this error depends on the degree of nonlinearity of
the model being solved. Fair (2003), while acknowledging its limitations, argues that the
use of certainty equivalence may provide good approximations for many
macroeconometric models.
In the case of the extended path algorithm, the error propagates through each level
of iteration and therefore it forces the use of strong convergence criteria. Due to this fact,
the extended path algorithm tends to be computationally intensive. Other methodologies
that only use certainty equivalence as a preliminary step as in the case of linearization
methods or linear quadratic approaches are not subject to the same computational burden.
In conclusion, while there are cases where certainty equivalence may be used to
obtain good approximations, one needs to be careful when using this methodology since
there are no guarantees when it comes to accuracy.
10
I.4.Local Approximation and Perturbation Methods
Economic modeling problems have used a variety of approximation methods in
the absence of a closed form solution. One of the most used approximation methods,
coming in different flavors, is the local approximation. In particular, the first order
approximation has been extensively used in economic modeling. Formally, a function
a(x) is a first order approximation of b(x) around x
0
if a(x
0
) = b(x
0
) and the
derivatives at x
0
are the same, a '(x
0
) = b '(x
0
) . In certain instances, first order
approximations may not be enough so one would have to compute higher order
approximations. Perturbation methods often use high order local approximation and
therefore rely heavily on two very well own theorems, Taylor's theorem and implicit
function theorem.
I.4.1. Regular and General Perturbation Methods
Perturbation methods are formally addressed by Judd (1998). In this section,
following Judd's framework, I try to highlight the basic idea of regular perturbation
methods. I start by assuming that the Euler equation of the model under consideration is
given by:
F
( u,
c
) = 0 (1.4.1)
where
u
(
c
) is the policy I want to solve for and c is a parameter. Further on, I assume
that a solution to (1.4.1) exists, that F is differentiable,
u
(
c
) is a smooth function and
u
(
0
) can be easily determined or is known. Differentiating equation (1.4.1) leads to:
F
u
(
u
(
c
),
c
)u
'
(
c
) + F
c
(
u
(
c
),
c
) = 0 (1.4.2)
11
Making c = 0 in equation (1.4.2) allows one to compute u
'
(
0
) :
F
c
(
u
(
0
),
0
)
u
'
(
0
) = ÷
F
u
(
u
(
0
),
0
)
(1.4.3)
The necessary condition for the computation of u
'
(
0
) is that F
u
(
u
(
0
),
0
) = 0 . Assuming
that indeed F
u
(
u
(
0
),
0
) = 0 , it means that now u
'
(
0
) is known and one can compute the
first order Taylor expansion, of
u
(
c
) around c = 0 :
F
c
(
u
(
0
),
0
)
u
(
c
) ?
u
(
0
) ÷
F
u
(
u
(
0
),
0
)
c
(1.4.4)
This is a linear approximation of
u
(
c
) around c = 0 . In order to be able to compute
higher order approximations of
u
(
c
) one needs to know at least the value of u
''
(
0
) . That
can be found by differentiating (1.4.2):
u
''
(
0
) = ÷
F
uu
(
u
(
0
),
0
)(u
'
(
0
))2 + 2F
u
c
(
u
(
0
),
0
)u
'
(
0
) + F
cc
(
u
(
0
),
0
)
F
u
(
u
(
0
),
0
)
(1.4.5)
The necessary condition for the computation of u
''
(
0
) is, once again, that
F
u
(
u
(
0
),
0
) = 0 . In addition, second order derivatives shall exist. Then the second order
approximation of
u
(
c
) around c = 0 is given by:
u
(
c
) ?
u
(
0
) ÷
F
c
(
u
(
0
),
0
)
c÷
1
c
2
F
uu
(
u
(
0
),
0
)(u
'
(
0
)) + 2F
u
c
(
u
(
0
),
0
)u
'
(
0
) + F
cc
(
u
(
0
),
0
) 2
F
u
(
u
(
0
),
0
)
2
F
u
(
u
(
0
),
0
)
In general, higher order approximations of
u
(
c
) can be computed if higher
derivatives of
F
(u,
c
) with respect with u exist and if F
u
(
u
(
0
),
0
) = 0 . The advantage
of regular perturbation methods based on an implicit function formulation is that one
12
directly computes the Taylor expansions in terms of whatever variables one wants to use,
and that expansion is the best possible asymptotically.
I.4.2. Example
Consider the following optimization problem
max E
e
9 |
tt
(u
t
) | O
0
÷ •
subject to
u
t
c
©t=0
x
t
=
h
(x
t
÷
1
,u
t
÷
1
, y
t
)
?
?
(1.4.6)
(1.4.7)
with y
t
= y
t
÷
1
+ c z
t
, where u
t
is the control variable, x
t
is the state variable, c is a scalar
parameter and z
t
is a stochastic variable drawn from a distribution with zero mean and
unit variance. x
t
, u
t
, c and z
t
are all scalars. The Bellman equation is given by:
V (x
t
) = muax
t
(u
t
) + | E ?V (
h
( x
t
,u
t
+
1
,c z
t
+
1
)) | O
t
?
t
{
?
?
}
(1.4.8)
Then the first order condition is:
0 = t
u
(u
t
) + | E ?V
'
(
h
( x
t
,u
t
,c z
t
+
1
)) h
u
( x
t
,u
t
,c z
t
+
1
)?
?
Differentiating the Bellman equation with respect to x
t
, one obtains:
?
(1.4.9)
V
'
(x
t
) = | E ?V
'
(
h
( x
t
,u
t
,c z
t
+
1
)) h
x
( x
t
,u
t
,c z
t
+
1
)?
? ?
(1.4.10)
Let the control law
U
( x,
c
) be the solution of this problem. Then the above equation
becomes:
V
'
(x) = | E
?
V
'
h ( x,
U
( x,
c
),c
z
) h
x
?
?
( )
?
The idea is to first solve for steady state in the deterministic case, which here is
equivalent to c = 0 , and then find a Taylor expansion for
U
( x,
c
) around c = 0 .
13
Assuming that there exists a steady state defined by (x
*
,u
*
) such that x
*
= h x
*
,u
*
, one
( )
can use the following system to obtain steady state solutions:
x
*
= h x
*
, u
*
( )
(1.4.11)
0 = t
u
u
*
+ |V ' h x
*
,u
*
h
u
x
*
,u
*
() (( )) ( ) (1.4.12)
V ' x
*
= |V ' h x
*
,u
*
h
x
x
*
,u
*
() (( )) ( ) (1.4.13)
V x
*
= t u
*
+ | V x
*
() () () (1.4.14)
Further assuming local uniqueness and stability for the steady state, equations (1.4.11)-
(1.4.14) provide the solutions for the four steady state quantities x
*
, u
*
, V x
*
, and ()
V ' x
*
. Given that the time subscript for all variables is the same, I drop it for the ()
moment. Going back to equations (1.4.9) - (1.4.10), in the deterministic case, that is, for
c = 0 , one obtains:
0 = t
u
(
U
(
x
)) + |V ' ?h ( x,
U
(
x
))? h
u
( x,
U
(
x
))
? ?
(1.4.15)
V
'
(x) = |V
'
?
h
( x,
U
(
x
))? h
x
( x,
U
(
x
))
? ?
(1.4.16)
Differentiating (1.4.15) and (1.4.16) with respect to x yields
0 = t
uu
U
'
x + |V
"
(
h
)(h
x
+ h
u
U
'
x ) h
u
+ |V
'
(
h
)(h
ux
+ h
uu
U
'
x ) (1.4.17)
V " = |V
"
(
h
)(h
x
+ h
u
U
'
x ) h
x
+ |V
'
(
h
)(h
xx
+ h
xu
U
'
x ) (1.4.18)
Therefore, the steady state version of the system (1.4.17) - (1.4.18) is given by:
0 = t
uu
x
*
,u
*
U
'
x x
*
+ |V " x
*
h
x
x
*
,u
*
( ) () () ( ) e
(1.4.19)
+h
u
x ,u U
x
x
?
h
u x ,u + |V '
x
e
h
ux x ,u + h
uu
x ,u U
x
x
?
(
* *
) () (
'
*
?
* *
) () (
*
* *
) (
* *
) ()
V " x
*
= |V " x
*
h
x
x
*
,u
*
+h
u
x
*
,u
*
U
'
x x
*
? h
x
x
*
,u
*
'
*
?
() () ( ) ( ) () ( )
e ?
(1.4.20)
+|V ' x
*
h
xx
x
*
,u
*
+ h
xu
x
*
,u
*
U
'
x x
*
?
()
e
( ) ( ) ()
?
These equations define a quadratic system for the unknowns V "(x ) and
U
(
x
) .
* ' *
x
14
Going back to the stochastic case, the first order condition with respect to u is given by:
0 = t
u
(
U
( x,
c
)) + | E V '
h
( x,
U
( x,
c
),c z
t
+
1
) h
u
( x,
U
( x,
c
),c z
t
+
1
) | O
t
(1.4.21)
{( ) }
Taking the derivative of the Bellman equation with respect to x yields:
V
'
(x) = | E V
'
h (x,
U
(x,
c
),c z
t
+
1
) h
x
(x,
U
(x,
c
),c z
t
+
1
) | O
t
{( ) } (1.4.22)
In order to obtain a local approximation of the control law around c = 0 , its derivatives
with respect to c must exist and be known. To find these values one needs to
differentiate equations (1.4.21) - (1.4.22) with respect to c , make c = 0 and solve the
resulting system for the values of the derivatives of U with respect to c when c = 0 ,
i.e., for U
c
'
( x
*
,
0
) . Once that value is found, one can compute a Taylor expansion for
U
( x,
c
)
around
( x
*
,
0
) .
If the model requires the addition of an inequality constraint such as (1.2.3) which
could be the representation of a liquidity constraint or a gross investment constraint, the
Bellman equation (1.4.8) becomes:
V (x
t
) = muax
t
(u
t
) + · min (
f
( x
t
, x
t
÷
1
),
0
)3 + | E ?
V
(
h
( x
t
,u
t
,c z
t
)) | O
t
? (1.4.23)
t
{
?
?}
where · is the penalty parameter.
I.4.3. Flavors of Perturbation Methods
Economic modeling problems have used a variety of approximation methods that
may be characterized as perturbation methods. The most common use of perturbation
methods is the method of linearization around the steady state. Such linearization
provides a description on how a dynamical system evolves near its steady state. It has
often been used to compute the reaction of a system to shocks. While the first-order
15
perturbation method exactly corresponds to the solution obtained by standard
linearization of first-order conditions, one well known drawback of such a solution,
especially in the case of asset pricing models, is that it does not take advantage of any
piece of information contained in the distribution of the shocks. Collard and Juillard
(2001) use higher order perturbation methods and apply a fixed-point algorithm, which
they call "bias reduction procedure", to capture the fact that the policy function depends
on the variance of the underlying shocks. Similarly, Schmitt-Grohé and Uribe (2004)
derive a second-order approximation to the policy function of a general class of dynamic,
discrete-time, rational expectations models using a perturbation method that incorporates
a scale parameter for the standard deviations of the exogenous shocks as an argument of
the policy function.
I.4.4. Alternative Local Approximation Methods
There are also certain local approximations techniques used in the literature that
may look like perturbation methods when in fact they are not. One frequently used
approach is to find the deterministic steady state and then to replace the original nonlinear
problem with a linear-quadratic problem that is similar to the original problem. The
linear-quadratic problem can then be solved using standard methods. This method differs
from the perturbation method in that the idea here is to replace the nonlinear problem
with a linear-quadratic problem, whereas the perturbation approach focuses on computing
derivatives of the nonlinear problem. Let me consider again the problem defined by
equations (1.2.1) - (1.2.2). The idea is to approximate the original problem by a
16
combination of a quadratic objective and a linear constraint, which would take the
following form:
max E ?
¿
( ? · |
t
Q +Wu + Ru
2
| O ?
u
t
?t =0
t
t
) 0
?
?
(1.4.24)
s.t. x
t
= Ax
t
÷
1
+ Bu
t
+ Cy
t
+ D (1.4.25)
where Q, R, W , A, B, C and D are scalars.
In order to obtain the new specification, the first step is to compute the steady
state for the deterministic problem (which means z
t
= 0 in equation (1.2.4)). Therefore,
one has to formulate the Lagrangian:
·
L
=
¿ |t {
t
(u
t
) ÷ ì
t
?x
t
÷
h
( x
t
÷
1
,u
t
, y
0
)
?
}
t =0
? ?
(1.4.26)
The first order conditions for (1.4.26) is a system of 3 equations with unknowns
x,u and ì . The solution of the system represents the steady
state,
( x
*
,u
*
, ì
*
) . The next
step is to take the second order Taylor expansion for
t
(u
t
) and first order Taylor
expansion for
h
(x
t
÷
1
,u
t
, y
t
)
around
( x
*
,u
*
, y
0
) . Thus,
t
(
u
) =
t
(
u
) + t
'
(
u
) (u ÷
u
) + t
"
(
u
) (
*2
t * * t * *
u ÷
u
) t
2
(1.4.27)
h
( x
t
÷
1
,u
t
, y
t
) =
h
( x
*
,u
*
, y
0
) + h
'
x ( x
*
,u
*
, y
0
)( x
t
÷
1
÷ x
*
) +
(1.4.28)
+ h
u
'
( x
*
,u
*
, y
0
)(u
t
÷ u
*
) + h
'
y ( x
*
,u
*
, y
0
)( y
t
÷ y
0
)
These expansions allow one to identify the parameters Q, R, W , A, B, C and D .
Specifically,
*2
Q =
t
( u
*
) ÷ t
'
( u
*
) u
*
+ t
"
(
u
*
)
u
2
t
"
(
u
*
)
(1.4.29)
W = t
'
( u
*
) ÷ t
"
( u
*
)
u
*
17
R=
2
A = h
'
x ( x
*
,u
*
, y
0
) B = h
u
'
( x
*
,u
*
, y
0
) C = h
'
y ( x
*
,u
*
, y
0
)
(1.4.30)
D =
h
( x
*
,u
*
, y
0
) ÷ h
'
x ( x
*
,u
*
, y
0
) x
*
÷ h
u
'
( x
*
,u
*
, y
0
)u
*
÷ h
'
y ( x
*
,u
*
, y
0
) y
0
Once the parameters have been identified, the problem can be written in the form
described by (1.4.24) and (1.4.25) which has a quadratic objective function and linear
constraints
12
.
If the model needs to account for an additional inequality constraint such as
(1.2.3), the Lagrangian (1.4.26) becomes
L
=
¿ |t {
t
(u
t
) ÷ ì
t
?x
t
÷
h
( x
t
÷
1
,u
t
, y
0
)? + ·
t f
( x
t
, x
t
÷
1
)
} ·
t =0
? ?
(1.4.31)
and the additional Kuhn-Tucker conditions have to be taken into account.
I.4.5. Notes on Local Approximation Methods
The perturbation methods provide a good alternative for dealing with the major
drawback of the method of linearization around steady state, that is, its lack of accuracy
in the case of high volatility of shocks or high curvature of the objective function. While
the first order perturbation method coincides with the standard linearization, the higher
order perturbation methods offer a much higher accuracy
13
.
Some of the local approximation implementations such as the linear-quadratic
method
14
do fairly well when it comes to modeling movements of quantities, but not as
12
There are some other variations of this approach used in the literature such as
Christiano (1990b).
13
See Collard and Juillard (2001) for a study on the accuracy of perturbation methods in
the case of an asset-pricing model.
14
Dotsey and Mao (1992), Christiano (1990b) and McGrattan (1990) have documented
the quality of some implementations of the macroeconomic linear-quadratic approach.
18
well with asset prices. The reason behind this result is that approximation of quantity
movements depends only on linear-quadratic terms whereas asset-pricing movements are
more likely to involve higher-order terms.
I.5. Discrete State-Space Methods
15
These methods can be applied in several situations. In the case where the state
space of the model is given by a finite set of discrete points these methods may provide
an "exact" solution
16
. In addition, these methods are frequently applied by discretizing an
otherwise continuous state space. The use of discrete state-space methods in models with
a continuous state space is based on the result
17
that the fixed point of a discretized
dynamic programming problem may converge point wise to its continuous equivalent
18
.
The discrete state-space methods sometimes prove to be a useful alternative to
linearization and log-linear approximations to the first order necessary conditions,
especially for certain model specifications.
15
16
This section draws heavily on Burnside (1999) and on Tauchen and Hussey (1991)
This may be the case in models without endogenous state variables, especially when
there is only one state variable that follows a simple finite state process. Examples are
Mehra and Prescott (1985) and Cecchetti, Lam and Mark (1993).
17
As documented in Burnside (1999), Atkinson (1976) and Baker (1977) present
convergence results related to the use of discrete state spaces to solve integral equations.
Results concerning pointwise and absolute convergence of solutions to asset pricing
models obtained using discrete state spaces are presented in Tauchen and Hussey (1991)
and Burnside (1993).
18
The procedure employed by discrete state-space methods in models with a continuous
state space is sometimes referred to as 'brute force discretization'.
19
I.5.1. Example. Discrete State-Space Approximation Using Value-Function Iteration
As before, I consider the following maximization problem:
max E
?
¿ |
tt
(u
t
) | O
0
? ·
subject to
u
t
?
?t =0
x
t
+
1
=
h
(x
t
,u
t
, y
t
)
?
?
(1.5.1)
(1.5.2)
where y
t
is a realization from an n -state Markov chain, u
t
is the control variable and x
t
is the state variable. Let Y
=
{Y
1
, Y
2
,..., Y
n
} be the set of all possible realizations for y
t
.
In order to be able to apply the above mentioned methodology one has to establish a grid
for the state variable. Let the ordered set X
=
{X
1
, X
2
,..., X
k
} be the grid for x
t
.
Assuming that the control variable u
t
can be explicitly determined from equation (1.5.2)
as a function of x
t
, x
t
+
1
and y
t
, then the dynamic programming problem can be
expressed as:
V (x
t
, y
t
) = max
t
( x
t
, x
t
+
1
, y
t
) + | E ?
V
( x
t
+
1
, y
t
+
1
) | O
t
?
x eXt+1
{
?
?
}
(1.5.3)
Let H(x
t
, y
t
) be the Cartesian product of Y and X , that is, the set of all possible
m = n ? k pairs (x
i
, y
j
) . Formally, H(x
t
, y
t
) = (x
i
, y
j
) | x
i
ŒX ¸ ÷
k
and y
j
ŒY ¸ ÷
n
.
{ }
Hence H(x
t
, y
t
) ¸ ÷
k
· ÷
n
= ÷
m
. If equation (1.5.3) is discretized using the grid given
by H(x
t
, y
t
) one can think of function
V
(
?
) as a point in ÷
m
. Similarly, the expression
t
(x
t
, x
t
+
1
, y
t
) + |
E
(
V
(x
t
+
1
, y
t
+
1
) | O
t
) can be thought of as a mapping M from ÷
m
into
÷
m
. In this context
V
(
?
) is a fixed point for M , that is, V =
M
(
V
) . One of the methods
commonly used to solve for the fixed point in these situations is the value function
iteration.
20
In order to solve the maximization problem one can use various algorithms. The
algorithm I am going to present follows, to some degree, Christiano (1990a). Let
S
j
(X
p
, Y
q
) be the value of x
t
+
1
that maximizes
M
(V
j
) for given values of x
t
and y
t
,
( x
t
, y
t
)
=
(X
p
,Y
q
) c H . Formally,
S
t
j+
1
(X
p
, Y
q
) = arg max
t
(X
p
, x
t
+
1
, Y
q
) + | E ?V
j
( x
t
+
1
, y
t
+
1
) | O
t
?
x
t
+
1
eX
{
?
?
}
(1.5.4)
where j represents the iteration. The idea is to go through all the possible values for x
t
+
1
,
that is, the set X , and find the value that maximizes the right hand side of (1.5.4). That
will become the value assigned to S
j
(X
p
, Y
q
) . Then the procedure will be repeated for a
different value of the pair x
t
and y
t
belonging to set H(x
t
, y
t
) and, finally, a global
maximum will be found. The exposition of the algorithm so far implies an exhaustive
search of the grid. The speed of the algorithm can be improved by choosing a starting
point for the search in every iteration and continue the search only until the first decrease
in the value function is encountered
19
. The decision rule for u
t
can then be derived by
substituting S
t
+
1
for x
t
+
1
in the law of motion.
I.5.2. Fredholm Equations and Numerical Quadratures
Let me consider the model specified by (1.2.1) - (1.2.2). Then the Bellman
equation is given by:
V
( x
t
, y
t
) = muax
t
(u
t
) + | E ?
V
( x
t
+
1
, y
t
+
1
) | O
t
?
t
{
?
?
}
(1.5.5)
19
T his change in the algorithm, as presented by Christiano (1990a), is valid only when
the value function is globally concave.
21
If y
t
follows a process such as (1.2.4), one can rewrite the conditional expectation and
consequently the whole equation (1.5.5) as:
V
( x
t
, y
t
) = muax
t
(u
t
) +
|
}
V
( x
t
+
1
, y
t
+
1
)
q
( y
t
+
1
| y
t
) dy
t
+1{t
}
(1.5.6)
In the above equation, the term needing approximation is the integral
}
V
( x t +1
, y
t
+
1
)
q
( y
t
+
1
| y
t
) dy
t
+1
If V (x
t
+
1
, y
t
+
1
) is continuous in y
t
+
1
for every x , the integral can be replaced by an N-
point quadrature approximation. An N-point quadrature method is based on the notion
that one can find some points y
i
,
N
and some weights w
i
,
N
in order to obtain the following
approximation
N
¿
V
( x t +1
i=1
, y
i
.
N
) w
i
,
N
~
}
Y
V
( x
t
+
1
, y
t
+
1
)
q
( y
t
+
1
| y
t
) dy
t
+1
(1.5.7)
where the points y
i
,
N
eY ,i = 1,K, N , are chosen according to some rule, while the weight
given to each point, w
i
,
N
, relates to the density function
q
(
y
) in the neighborhood of
those points. In general, a quadrature method requires a rule for choosing the points, y
i
,
N
,
and a rule for choosing the weights, w
i
,
N
. The abscissa y
i
,
N
and weights w
i
,
N
depend only
on the density
q
(
y
) , and not directly on the function V .
Quadrature methods differ in their choice of nodes and weights. Possible choices
are Newton-Cotes, Gauss, Gauss-Legendre and Gauss-Hermite approximations. For a
classical N-point Gauss rule along the real line, the abscissa y
i
,
N
and weights w
i
,
N
are
determined by forcing the rule to be exact for all polynomials of degree less than or equal
to 2N ÷1.
22
For most rational expectation models, integral equations are a very common
occurrence both in Bellman equations such as (1.5.6), as well as in Euler equations. One
of the most common forms of integral equations mentioned in the literature is the
Fredholm equation
20
. Therefore, in this section I will present an algorithm similar to the
one used by Tauchen and Hussey (1991) for solving such equation.
Now let me assume for a moment that the Euler equation of the model is given by
a Fredholm equation of the second kind:
v( y
t
)
=
v¢ ( y
t
+
1
, y
t
)v( y
t
+
1
)q( y
t
+
1
y
t
)dy
t
+
1
+¸ ( y
t
) (1.5.8)
where y
t
is an n-dimensional vector of variables, E
t
is the conditional expectations
operator based on information available through period t , and
¢
( y
t
, y
t
+
1
) and
¸
( y
t
) are
functions of y
t
and y
t
+
1
that depend upon the specific structure of the economic model,
and where
v
( y
t
) is the solution function of the model. The
process
{y
t
} is characterized
by a conditional density, q( y
t
+
1
y
t
) .
Following Tauchen and Hussey (1991), let the T[?] operator define the integral
term in equation (1.5.8). Then (1.5.8) can be written as:
v = T [v ] + ¸ (1.5.9)
Under regularity conditions, the operator [I ÷ T ]
÷
1
exists, where I denotes the identity
operator, and the exact solution is:
v = [I ÷ T ]
÷
1¸ (1.5.10)
20
One example where this form of integral equation appears is a version of the asset
pricing model. See Tauchen and Hussey (1991) and Burnside (1999) for more details.
23
An approximate solution is obtained using T
N
in place of T , where T
N
is an
approximation of T using quadrature methods for large N . Then [I ÷ T
N
] can be
inverted.
v
N
= [I ÷ T
N
]
÷
1¸ (1.5.11)
In some cases, the function ¸ is of the form ¸ = T[¸
0
] and then the approximate solution
is taken as [I ÷ T
N
]
÷
1
T
N
[¸
0
] .
I.5.3. Example. Using Quadrature Approximations
This is an example of discrete state-space approximation using quadrature
approximations and value-function iterations. I consider a similar model to the one
described in section I.5.1 with the difference that y
t
is a Gaussian
AR
(
1
) process as
opposed to a Markov chain. Again, the representative agent solves the following
optimization problem
max
E
?
¿ |
tt
(u
t
) O
0
? ·
subject to
u
t
?
?t =0
x
t
+
1
=
h
(x
t
,u
t
, y
t
)
?
?
(1.5.12)
(1.5.13)
where y
t
is a Gaussian
AR
(
1
) process with the law of motion y
t
= µ y
t
÷
1
+ z
t
where z
t
is i.i.d. N 0,o
2
. I assume that u
t
can be expressed as a function of x , i.e.
( )
u
t
=
g
(x
t
, x
t
+
1
, y
t
) . Then the Bellman equation for the dynamic programming is given by
V
( x
t
, y
t
) = mxax
t
(
g
( x
t
, x
t
+
1
, y
t
)) + |
E
{
V
( x
t
+
1
, y
t
+
1
) O
t
+
1
} t+1
(1.5.14)
24
Writing the expectation term explicitly, equation (1.5.14) becomes:
V
( x
t
, y
t
) = mxax
t
(
g
( x
t
, x
t
+
1
, y
t
)) +
|
}
V
( x
t
+
1
, y
+
1
)
f
( y
t
+
1
y
t
) dy
t
+1t+1
(1.5.15)
where
y
t
+
1
= µ y
t
+ z
t
+1 (1.5.16)
To convert the dynamic programming problem in (1.5.15) to one involving
discrete state spaces one needs first to approximate the law of motion of y
t
using a
discrete state-space process. That is, redefine y
t
to be a process which lies in a set
Y = y
i
,N
{}
N
with y
i
,
N
= o a
i
,
N
, where
{
a
}
N
is the set of quadrature points
i=1
i,N
i=1
corresponding to an N-point rule for a standard normal distribution
21
. Let the probability
that y
t
+
1
= y
i
,
N
conditional on y
t
= y
j
,
N
be given by
f y
i
,
N
y
j
,
N
w
i
,N
p
ji
=
(
f y
i
,
N
0 (
) )
s
j
(1.5.17)
where
N
s
j
=
9
f y
i
,
N
y
j
,N (
)
w
(1.5.18)
i=1
f y
i
,
N
0 ( )
i,N
and {w
i
,
N
}N=
1
are the quadrature weights as described in section I.5.2.. With this i
approximation, the Bellman equation can be written as:
V
( x
t
, y
t
) =
mxax ?
?
t g x , x , y + |N
V
( x ,
y
)
p
?
t+
1
?
(
( t t+1
j
) ¿1t+1iji ?? )
i=
(1.5.19)
given y
t
= y
j
, j = 1,..., N .
21
This is in fact the approach used by Tauchen and Hussey (1991) and Burnside (1999),
among others.
25
The next step is to replace the state space by a discrete domain X from which the
solution is chosen. There is no universal recipe for choosing a discrete domain and
therefore it is usually done on a priori knowledge of possible values of the state
variable
22
. The maximization problem can now be solved by value function iteration as
presented in section I.5.1..
I.5.4. Notes on Discrete State-Space Methods
Discrete state-space methods tend to work well for models with a low number of
state variables. As the number of variables increases, this approach becomes numerically
intractable, suffering from what the literature usually refers to as the curse of
dimensionality. In addition, as pointed out in Baxter et al. (1990), when the method is
used to solve continuous models there are two sources of approximation error. One is due
to forcing a discrete grid on continuous state variables and second from using a discrete
approximation of the true distribution of the underlying shocks. There are also instances
where the use of discrete state-space methods is entirely inappropriate since the
discretization process transforms an infinite state space into a finite one and in the
process is changing the information structure. This may not be an issue in most models,
but it definitely has an impact in models with partially revealing rational expectations
equilibria
23
.
22
23
See Tauchen (1990) for an example.
See Judd (1998) pp. 578-581 for an example.
26
I.6. Projection Methods
24
As opposed to the previously presented numerical methods, the techniques that
are going to be presented in this section have a high degree of generality. Projection
methods appear to be applicable to solving a wide variety of economic problems. In fact,
projection methods can be described as general numerical methods that make use of
global approximation techniques
25
to solve equations involving unknown functions.
The idea is to replace the quantity that needs to be approximated by parameterized
functions with arbitrary coefficients that are to be determined later on
26
, or to represent
the approximate solution to the functional equation as a linear combination of known
basis functions whose coefficients need to be determined
27
. In either case, there are
coefficients to be computed in order to obtain the approximate solution. These
coefficients are found by minimizing some form of a residual function.
Further on, a step by step description of the general projection method is
presented, followed by a discussion of the parameterized expectations approach.
24
I borrow this terminology from Judd (1992, 1998). These methods are also called
weighted residual methods by some authors (for example Rust (1996), McGrattan (1999),
Binder et al. (2000)). In fact, one can argue that weighted residual methods are just a
subset of the projection methods with a given norm and inner product.
25
In some cases local approximations are used on subsets of the original domain and then
they are pieced together to give a global approximation. One such case is the finite
element method.
26
See Marcet and Marshall (1994a), Marcet and Lorenzoni (1999), Wright and Williams
(1982a, 1982b, 1984) and Miranda and Helmberger (1988)
27
See McGrattan (1999)
27
I.6.1. The Concept of Projection Methods
Suppose that the functional equation can be described by:
F (d ) = 0 (1.6.1)
where F is a continuous map, F : C
1
÷ C
2
with C
1
and C
2
complete normed function
spaces and d : D c ?
k
÷ ?
m
is the solution to the optimization problem. More generally,
d is a list of functions that enter in the equations that define the equilibrium of a model,
such as decision rules, value functions, and conditional expectations functions, while the
F operator expresses equilibrium conditions such as Euler equations or Bellman
equations.
I.6.1.1. Defining the Problem
The problem is to find d : D c ?
k
÷ ?
m
that satisfies equation (1.6.1). This
translates into finding an approximation
d
ˆ(x;u ) which depends on a finite-dimensional
vector of parameters u
=
|u
1
,u
2
,K,u
n
| such that F
dˆ
( x;
u
) is as close as possible to
( )
zero.
I.6.1.1.1. Example
28
Consider the following finite horizon problem where the social planner or a
representative agent maximizes
E
e
9 |
tt
(u
t
) O
0
÷ T
subject to
c
©t=0
?
?
(1.6.2)
28
x
t
=
h
(x
t
÷
1
,u
t
, y
t
)
The example in section I.6.1 draws heavily on Binder et al. (2000)
28
(1.6.3)
with x
0
and x
T
given. y
t
is an
AR
(
1
) process with the law of motion
y
t
= µ y
t
÷
1
+ z
t
(1.6.4)
and z
t
are i.i.d. with z
t
~ N 0,o 2
y
. I assume that u can be expressed as a function of
( )
x , i.e. u
t
=
g
(x
t
÷
1
, x
t
, y
t
) . Then the Euler equation for period T ÷ 1 is given by
0 = t '
g
(x
T
÷
2
, x
T
÷
1
, y
T
÷
1
) ? g '
x
T÷
1
(x
T
÷
2
, x
T
÷
1
, y
T
÷
1
)
( )
(1.6.5)
+ | E t '
g
(x
T
÷
1
, x
T
, y
T
) ? g '
x
T÷
1
(x
T
÷
1
, x
T
, y
T
) O
T
÷1
{( ) }
Let the optimal decision rule for x
T
÷
1
be given by x
*
÷
1
= d
T
÷
1
(x
T
÷
2
, y
T
÷
1
) where T
d
(
?
) is a smooth function. The projection methodology consists of approximating
d
(
?
)
by d
ˆ
(?,
u
) , where u represents an unknown parameter matrix. The unknown parameters
are computed such that the Euler equation also holds for d
ˆ
(?,
u
) .
Further on in this section I present the necessary steps one needs to take when
applying the projection methods, drawing heavily on the formalization provided by Judd
(1998)
29
. As I mentioned above, the methodology consists of finding an approximation
dˆ(x;u ) such that F d
ˆ
( x;
u
) is as close as possible to zero. It becomes obvious that
( )
there are a few issues that need to be addressed: what form of approximation to choose
for dˆ(x;u ) ; does the operator F need to be approximated; what does one understand by,
or in other words, what is the formal representation of "as close as possible to zero".
29
Judd provides a five step check list for applying the projection methods.
29
I.6.1.2. Finding a Functional Form
The first step comes quite naturally from the need to address the question on how
to represent d (x;u ) . In general d
ˆ
is defined as a finite linear combination of basis
functions, ?
i
(x), i = 0,K, n :
n
dˆ(x;u ) = ?
0
(x)
+
¿u
i
?
i
(x) (1.6.6)
i=1
Therefore, the first step consists of choosing a basis over C
1
.
Functions ?
i
(x), i = 0,K, n are typically simple functions. Standard examples of
basis functions include simple polynomials (such as ?
0
(x) = 1, ?
i
(x) = x
i
), orthogonal
polynomials (for example, Chebyshev polynomials), and piecewise linear functions.
Choosing a basis is not a straightforward task. For example, ordinary polynomials are
sometimes adequate in simple cases where they may provide a good solution with only a
few terms. However, since they are not orthogonal on R
+
and they are all monotonically
increasing and positive for x ΠR
+
, for x big enough, they are almost indistinguishable
and hence they tend to reduce numerical accuracy
30
. Consequently, orthogonal bases are
usually preferred to avoid the shortcomings just mentioned.
One of the more popular orthogonal bases is formed by Chebyshev polynomials.
They constitute a set of orthogonal polynomials with respect to the weight function
30
In order to solve for the unknown coefficients u
i
one needs to solve linear systems of
equations. The accuracy of these solutions depends on the properties of the matrices
involved in the computation, i.e. linear independence of rows and columns. Due to the
properties already mentioned, regular polynomials tend to lead to ill-conditioned
matrices.
30
e(x) = 1 1÷ x
2
, that is,
}
1
p
i
(x) p
j
(x)e(x)dx = 0 for all i = j . Chebyshev polynomials
÷1
are defined on the closed
interval
|÷1,
1
| and can be computed recursively as follows:
p
i
(x) = 2xp
i
÷
1
(x) ÷ p
i
÷
2
(x), i = 2, 3, 4,K (1.6.7)
with p
0
(x) = 1 and p
1
(x) = x or, non-recursively, as:
p
i
(x) = cos (i arccos (
x
)) (1.6.8)
Another set of possible basis functions that can be used to construct a piecewise
linear representation for d
ˆ
is given by:
? x ÷ x
i
÷
1
if x
e
| x ,
x
|
? x
i
÷ x
i
÷1
?
i÷1 i
?
i
(x) = ? x
i
+
1
÷ x if x
e
|x
i
, x
i
+
1
|
? i+1 i
?x ÷ x
(1.6.9)
?
?
?
0 elsewhere
The points x
i
, i = 1,K, n that divide the domain D c ? need not be equally spaced. If,
for example, it is known that the function to be approximated has large gradients or kinks
in certain places then the subdivisions can be smaller and clustered in those regions. On
the other hand, in areas where the function is near-linear the subdivisions can be larger
and hence fewer.
Once the basis is chosen, the next step is to choose how many terms and
consequently how many parameters the functional form will have. In general, if the
choice of the basis is good, the higher the number of terms the better the approximations.
However, due to the fact that the more terms are chosen the more parameters have to be
computed, one should choose the smallest number of terms, n , that yields an acceptable
31
approximation. One possible approach is to begin with a small n and then increase its
value until some approximation threshold is reached.
I.6.1.2.1. Example
Going back to the model defined by equations (1.6.2) and (1.6.3) the next step is
choosing a base. I assume that Chebyshev polynomials are used in constructing the
functional form for dˆ
T
÷
1
(?,
u
) . Then:
dˆ
T
÷
1
(x
T
÷
2
, y
T
÷
1
;u
T
÷
1
) =
n
x
,T ÷
1
n
y
,T ÷1
9 9u
T ÷1,sq
p
s
÷
1
(x
T
÷
1
) p
q
÷
1
( y
T
÷
1
)
s=1 q=1
% %
(1.6.10)
where u
T
÷1,sq is
the
(s,
q
) element of u
T
÷
1
, p
l
(?) is the l -th order Chebyshev polynomial
as defined in (1.6.7) - (1.6.8), while n
x
,T ÷
1
and n
y
,T ÷
1
are the maximum order of the
Chebyshev polynomials assumed for x
T
÷
1
and y
T
÷
1
respectively. In order to restrict the
% %
domain of the polynomials to the unit interval the following transformation is applied:
%T ÷
1
= 2 x
T
ma÷1x ÷ x
T
m÷i1
n
÷ 1
min
x
%
x
T
÷
1
÷ x
T
÷1
min
%
(1.6.11)
y
T
÷
1
= 2 y
T
ma÷1x ÷ y
T
m÷i1
n
÷1 (1.6.12)
y
T
÷
1
÷ y
T
÷1
I.6.1.3. Choosing a Residual Function
In many cases, computing F (dˆ) may require the use of numerical approximations
such as when F (d ) involves integration of d . In those cases, the F operator has to be
approximated. In addition, once the methodology for approximating d and F has been
32
established, one needs to choose a residual function. Therefore, the third step consists of
defining the residual function and an approximation criterion. Let
R(x;u ) } F (dˆ(?,u ))(x) ˆ
(1.6.13)
be the residual function. At this point, a decision has to be made on how an acceptable
approximation is defined. That is accomplished by choosing an approximation criterion.
One choice is to compute the sum of squared residuals, R(?;u ) } R(?;u ), R(?;u ) and
then determine u such that R(?;u ) is minimized. An alternative would be to choose a
collection of n test functions in C
2
, p
i
: D C R
m
, i = 1,..., n , and for each guess of u to
compute the n projections, P
i
(?) } R(?;u ), p
i
(?)
31
.
It is obvious that this step creates the
projections that will be used to determine the value of the unknown coefficients, u .
Another popular choice in the literature is the weighted residual criterion defined as
32
:
v ¢ (x)R(x;u )dx = 0,
iD
i = 1,K, n
(1.6.14)
where ¢
i
(x), i = 1,K, n are weight functions. Alternatively, the set of equations (1.6.14)
can be written as
v
D
e (x)R(x;u )dx = 0
n
(1.6.15)
where D is the domain for function d , e (x)
=
9e ¢
i
(x) and (1.6.15) must hold for i
i=1
any non-zero weights e
i
, i = 1,K, n . Therefore, the method sets a weighted integral of
R(x;u ) to zero as the criterion for determining u .
31
The choice of the criterion gives the method its name. That is why in the literature the
method appears both under the name "projection method" and "weighted residual
method".
32
See McGrattan (1999).
33
I.6.1.3.1. Example
Going back to the example, recall that Chebyshev polynomials were used in
constructing the functional form for dˆ
T
÷
1
(?,
u
) :
dˆ
T
÷
1
(x
T
÷
2
, y
T
÷
1
;u
T
÷
1
) =
n
x
,T ÷
1
n
y
,T ÷1
9 9u
T ÷1,sq
p
s
÷
1
( %T÷
1
) p
q
÷
1
( %T÷
1
)
s=1 q=1
x y
As mentioned above, the Euler equation (1.6.5) needs to hold for d
ˆ
(?,
u
) . Therefore, its
right hand side is a prime candidate for defining the residuals function. Let v
T
÷
1
= _
T
÷
2
ˆ
.
x
·
y
T÷1 ˜
With this notation, the residual function is given by:
. +
R
T
÷
1
?v
T
÷
1
; dˆ
T
÷
1
(v
T
÷
1
;u
T
÷
1
)? =
? ?
g v
T
÷
1
, dˆ
T
÷
1
(v
T
÷
1
;u
T
÷
1
), y
T
÷
1
? g '
x
T÷
1
v
T
÷
1
, dˆ
T
÷
1
(v
T
÷
1
;u
T
÷
1
), y
T
÷1
( ) ( ) (1.6.16)
+
(t
'
)÷1 E
?
|t ' g dˆ
T
÷
1
(v
T
÷
1
;u
T
÷
1
), x
T
, y
T
? g '
x
T÷
1
dˆ
T
÷
1
(v
T
÷
1
;u
T
÷
1
), x
T
, y
T
?
{
?
( ) (
)
}
?
Then the criterion for computing u%
T
÷
1
is given by the weighted residual integral equation:
v
R
T
÷
1
v
T
÷
1
; dˆ
T
÷
1
(v
T
÷
1
;uˆT ÷
1
)
?
W (v
T
÷
1
)dv
T
÷
1
= 0
v
T
÷1
e ?
(1.6.17)
where W is a weighting function. In the next section it will become clear why the choice
of W is important in the computation of u%
T
÷
1
.
I.6.1.4. Methods Used for Estimating the Parameters
Evidently, the next step is to find u ΠR
n
that minimizes the chosen criterion. In
order to determine the coefficients u
1
,K,u
n
several methods can be used, depending on
the criterion chosen.
34
If the projection criterion is chosen, finding the n components of u means
solving the n equations
R
(x,
u
), p
i
= 0 for some specified collection of test functions,
p
i
. The choice of the test functions p
i
defines the implementation of the projection
method. In the least squares implementation the projection directions are given by the
gradients of the residual function. Therefore, the problem is reduced to solving the
c
R
( x ,
u
)
nonlinear set of equations generated by
R
(x,
u
),
cu
i
= 0 i = 1,..., n .
One alternative is to choose the first n elements of the basis u , that is,
?
i
(x) i = 1,..., n , as the weight functions, ¢
i
(x), i = 1,K, n . In other words, n elements of
the basis used to approximate dˆ(x;u ) are also used as test functions to define the
projection direction, ¢
i
(x) = ¢
i
(x), i = 1,K, n . This technique is known as the Galerkin
method. As a result of this choice, the Galerkin method forces the residual to be
orthogonal to each of the basis functions. Therefore u is chosen to solve the following
set of equations:
P
i
(
u
) =
R
(x,
u
),¢
i
(
x
) = 0 i = 1,..., n (1.6.18)
As long as the basis functions are chosen from a complete set of functions, system
(1.6.18) provides the exact solution, given that enough terms are included. If the basis
consists of monomials, the method is also known as the method of moments. Then u is
the solution to the system:
P
i
(
u
) =
R
(x,
u
), x
i
÷
1
= 0 i = 1,..., n (1.6.19)
The collocation method chooses u so that the functional equation holds exactly at
n fixed points, x
i
, called the collocation points. That is, u is the solution to:
35
R(x
i
;u ) = 0, i = 1,..., n (1.6.20)
where
{x
i
}ni=
1
are n fixed points from D . It is easy to see that this is a special case of the
projection approach, since R(x;u ),o (x ÷ x
i
) = R(x
i
;u ) , where o (x ÷ x
i
) is the Dirac
function at x
i
. If the collocation points x
i
are chosen as the n roots of the n
th
orthogonal
polynomial basis element and the basis elements are orthogonal with respect to the inner
product, the method is called orthogonal collocation. The Chebyshev polynomial basis is
a very popular choice for an orthogonal collocation method.
I.6.1.4.1. Example
Going back to the example, it was established that the criterion for computing
u%
T
÷
1
is given by the following integral equation:
v
R
T
÷
1
v
T
÷
1
; dˆ
T
÷
1
(v
T
÷
1
;uˆT ÷
1
)?W (v
T
÷
1
)dv
T
÷
1
= 0
v
T
÷1
e ?
As discussed in this section, given this criterion, the collocation method is a
sensible choice for computing u%
T
÷
1
. Then the choice for the weighting functions, as used
in Binder et al. (2000), is the n
x
,T ÷
1
, n
y
,T ÷
1
Dirac delta functions o x
T
÷
1
÷ x
i
T ÷
1
, y
T
÷
1
÷ y
i
T ÷
1
,
( )
where x
iT
÷
1
and y
iT
÷
1
are chosen such that x
iT
÷
1
and y
iT
÷
1
are the n
x
,T ÷
1
and n
y
,T ÷
1
zeros of
% %
the Chebyshev polynomials forming the basis of the approximation dˆ
T
÷
1
(v
T
÷
1
;u
T
÷
1
) . The
zeros for the Chebyshev polynomials are given by
?
?
co
s
(2
i
÷
1
)
t
?
?
v%T÷
1
=
? i
2n
x
,T ÷1 ? (1.6.21)
? (2i ÷1)t ?
? cos 2n ?
? y,T ÷1 ?
36
Then the integral equation can be reduced to:
R
T
÷
1
v
ij
÷
1
; dˆ
T
ij÷
1
= 0
for all
(
T
)
(1.6.22)
v
ij
÷
1
= x
i
T ÷
1
, y
T
j÷
1
, i = 1, 2,..., n
x
,T ÷
1
,
and
T
( )
dˆ
T
ij÷
1
= dˆ
T
÷
1
v
ij
÷
1
;uˆT ÷1
j = 1, 2,..., n
y
,T ÷1
(1.6.23)
(
T
)
(1.6.24)
The discrete orthogonality of Chebyshev polynomials implies that:
n
x
,T ÷
1
n
y
,T ÷1
9 9 e
p
(x
%
)
p
(y
%
)??e
p
(x
%
)
p
(y
%
)?? =
0
i=1 j =1 w÷1
i
T ÷1
p÷1
j
T ÷1
s÷1
i
T ÷1
q÷1
j
T ÷1
(1.6.25)
for w t s and /or p t q , and
n
x
,T ÷
1
n
y
,T ÷1
9 9 e
p
(x
%
)
p
(y
%
)??e
p
(x
%
)
p
(y
%
)?? =
c
(n
i=1 j =1 w÷1
i
T ÷1
p÷1
j
T ÷1
s÷1
i
T ÷1
q÷1
j
T ÷1
sq
x,T ÷1
, n
y
,T ÷
1
(1.6.26) )
for w = s and p = q , with
?n
x
,T ÷
1
n
y
,T÷
1
, w = s = p = q = 1
?
? ?w = s = 1 and p = q = 1,
c
sq
(n
x
,T ÷
1
, n
y
,T÷
1
) = ?x,T÷1y,T÷
1
) / 2, ?or
?
(n n
?
(1.6.27)
?
?
??w
= s =
1 and
p =
q
=
1,
?
(n
x
,T ÷
1
n
y
,T÷
1
) / 4, w = s = 1 and p = q = 1 ?
Then u is given by:
n
x
,T ÷
1
n
y
,T ÷1
1
uˆT ÷1,sq =
9 9
p
(x
%
)
p
(y
%
)?e
g
(v
( )
s÷1
i
T ÷1
q÷1
j
T ÷1
T ÷1
, dˆ
T
ij÷
1
, y
T
÷
1
? g '
x
T÷
2
v
T
÷
1
, dˆ
T
ij÷
1
,
y
T
÷1
c
sq
n
x
,T ÷
1
, n
y
,T ÷1
i=1 j =1
) ( )
+
(t%
'
)÷1 E |t ' g dˆ
T
ij÷
1
, x
T
, y
T
? g '
x
T÷
1
dˆ
T
ij÷
1
, x
T
, y
T
? v
T
÷1 ?
({
e
( ) ( )
?
}
)
?
?
(1.6.28)
for s = 1, 2,..., n
x
,T ÷
1
, q = 1, 2,..., n
y
,T ÷
1
.
37
The conditional expectation from the above equation needs to be computed
numerically. In order to compute the integral one can use some of the quadrature methods
such as the Gauss quadrature presented in section I.5.2. All that remains is to solve
equation (1.6.28) for u
T
÷1,sq , s = 1, 2,..., n
x
,T ÷
1
, q = 1, 2,..., n
y
,T ÷
1
. Once dˆ
T
÷
1
v
T
÷
1
;uˆT ÷1 (
)
is
computed, one can proceed recursively backwards to period T ÷ 2 . Note that
x
*
÷
1
= dˆ
T
÷
1
v
T
÷
1
;uˆT ÷
1
will be used in the definition of R
T
÷
2
v
ij
÷
2
; dˆ
T
ij÷
2
. The computation
T
( ) (
T
)
of uˆT ÷
2
can now follow the same logic as the computation of uˆT ÷
1
.
So far the flavors of the projection methodology have been categorized either with
respect to the choice of the approximation criterion or with respect to the method
employed for estimating the parameters. The choice of basis functions for the
representation in (1.6.6) can be used to further divide projection methods into two
categories: spectral methods and finite-element methods. Spectral methods use basis
functions that are smooth and non-zero on most of the domain of x such as Chebyshev
polynomials and the same functions are used on all regions of the state space. Finite-
element methods use basis functions that are equal to zero on most of the domain and
non-zero on only a few subdivisions of the domain of x (these are in general piecewise
linear functions such as those defined in (1.6.9)) and they provide different
approximations in different regions of the state space. For problems with many state
variables, there are typically many coefficients to compute and it implies the inversion of
a large, dense matrix. With the finite-element method, however, the same matrix is sparse
and its structure can typically be exploited. For the above-mentioned reasons McGrattan
38
(1996, 1999) argues that a finite-element method is better suited to problems in which the
solution is nonlinear or kinked in certain regions.
I.6.2. Parameterized Expectations
While Marcet (1988) is largely credited in the literature with the introduction of
the parameterized expectations approach, Christiano and Fisher (2000) point out that the
underlying idea of parameterized expectations seems to have surfaced earlier in the work
of Wright and Williams (1982a, 1982b, 1984), and then in the work of Miranda and
Helmberger (1988). Marcet (1988)
33
implemented a variation of that idea and the
approach finally caught on with the publication of Den Haan and Marcet (1990).
In this section, I will concentrate on what Christiano and Fisher (2000) call the
conventional parameterized expectations approach due to Marcet (1988). While one may
argue that this methodology does not belong under the label of projection methods, I
believe that it can be viewed as a special case of projection methods by virtue of its use of
parameterized functions to approximate an unknown quantity, of an implicit choice of a
residual function and an approximation criterion similar to projection methods. In
addition, the techniques used to estimate the parameters are also common to projection
methods. The assumption is that the functional equation has the following form:
g E
t
?
|
(q
t
+
1
,q
t
)? ,q
t
÷
1
,q
t
, z
t
= 0
(
? ?
)
(1.6.29)
where q
t
includes all the endogenous and exogenous variables and z
t
is a vector of
exogenous shocks. As it has been repeatedly asserted in this chapter, the reason why
33
For more information of this variant of the parameterized expectations approach, see
the references cited in Marcet and Marshall (1994b).
39
many dynamic models are difficult to solve is that conditional expectations often appear
in the equilibrium conditions. The assumption under which this methodology operates is
that conditional expectations are a time-invariant function c of some state variables:
c (u
t
) = E
t
||(q
t
+
1
,q
t )
| (1.6.30)
where E
t
||(q
t
+
1
,q
t )
| = E ?|(q
t
+
1
,q
t
) u
t
? is the conditional expectation based on the
? ?
available information at time t , u
t
e R
l
where u
t
is a subset
of
(q
t
÷
1
, z
t
) . As Marcet and
Lorenzoni (1999) point out, a key property of c is that under rational expectations, if
agents use c to form their decisions, the series generated is such that c is precisely the
best predictor of the future variables inside the conditional expectations. So, if c were
known, one could easily simulate the model and check whether this is actually the
conditional expectation.
The basic approach of Marcet and Marshall (1994a) is to substitute the
conditional expectations in equation (1.6.29) by parameterized functions of the state
variables with arbitrary coefficients. Then (1.6.29) is used to generate simulations for u
t
consistent with the parameterized expectations. With these simulations, one can iterate on
the parameterized expectations until they are consistent with the solution they generate.
In this fashion, the process of estimating the parameters is reduced to a fixed-point
problem.
I.6.2.1. Example
Consider again the model specified by (1.6.2) - (1.6.3) with the Euler equation for
period t given by:
40
0 = t '
g
(x
t
, x
t
÷
1
, y
t
) ? g '
x
t
(x
t
, x
t
÷
1
, y
t
)
( )
(1.6.31)
+ | E t '
g
(x
t
+
1
, x
t
, y
t
+
1
) ? g '
x
t
(x
t
+
1
, x
t
, y
t
+
1
) O
t
{( ) }
The idea is to substitute
E
t
t '
g
(x
t
+
1
, x
t
, y
t
+
1
) ? g '
x
t
(x
t
+
1
, x
t
, y
t
+
1
)
{( ) }
by a parameterized function
¢
( x
t
÷
1
, y
t
;
u
) where u is a vector of parameters. For
simplicity, let the function ¢ be given by:
¢
t
( x
t
÷
1
, y
t
;u
1
,u
2
) = u
1
x
t
÷
1
+u
2
y
t
(1.6.32)
The next step is to generate a
series
{z
t
}Tt=
1
as draws from a Gaussian distribution and to
choose starting values for the elements of u , u
i
0
, i = 1, 2 . Then, for uˆi = u
i
0
and assuming
that the initial values for x
t
and y
t
, that is, x
÷
1
and y
0
are given, one can use the
following system
t
'
(
g
(x
t
, x
t
÷
1
, y
t
))? g '
x
(x
t
, x
t
÷
1
, y
t
) +uˆ1x
t
÷
1
+uˆ2 y
t
= 0 for t = 0,...,T ÷1 t
x
t
=
h
(x
t
÷
1
,u
t
, y
t
) for t = 0,...,T , with x
÷
1
given (1.6.33)
y
t
= µ y
t
÷
1
+ z
t
for t = 1,...,T , with y
0
given
to generate
series
{ˆ
tt
j
}
t
=
0
,
{ˆ
t
j
}
t
=
1
and
{u
t
j
}
t
=
0
where j represents the iteration. In order
T T
x y ˆT
to estimate the parameters u , proponents of this methodology run a regression of
?
tj
uˆj = t '
g
(ˆ
t
j
, ˆ
t
j÷
1
, ˆ
t
j
) ? g '
x
t
(ˆ
t
j
, ˆ
t
j÷
1
, ˆ
t
j
)
() ( xx y ) xx y (1.6.34)
on ¢
t
. Formally, the regression can be written as:
?
tj
uˆj = a
1
ˆ
t
j÷
1
+ a
2
ˆ
tj
+ ç
t
() x y
where ç
t
is the error term. The estimates for a
1
and a
2
provide a new set of values for u
for the next iteration. With those values new series will be generated for {ˆ
t
j+
1
}
t
=
0
and T
x
41
{u
ˆ
} j
+1 T
y
T
t t =0
. In this particular case, there is no need to generate new series
for
{ˆ
t
j+
1
}
t
=
1
if the
same vector of shocks {z
t
}Tt=1 is used. In addition, note that a
1
and a
2
are in fact
functions of uˆ . Specifically, for iteration j , the vector of parameters a is a function of
uˆ
j
, a = G uˆ
j
. Hence the final step is to find the fixed point u =
G
(
u
) . One approach ()
suggested by Marcet and Lorenzoni (1999) is to compute the values of uˆ for iteration
j +1 using the following expression uˆj+
1 =
(1÷
b
)uˆj + bG uˆ
j
where b > 0 . The iteration ()
process should stop when uˆ
j
and G uˆ
j
are sufficiently close. ()
I.6.3. Notes on Projection Methods
As Judd (1992) points out, the advantage of the projection method framework is
that one can easily generate several different implementations by choosing among
different basis, residual functions or methods for estimating the parameters. Obviously,
the many choices also imply some trade-offs among speed, accuracy, and reliability. For
example, the orthogonal collocation method tends to be faster than the Galerkin method,
while the Galerkin method tends to offer more accuracy
34
.
The generality of the projection techniques can also be seen from the fact that
even methods that discretize the state space can be thought of as projection methods that
are using step function bases.
While throughout this section I emphasized the wide applicability of projection
methods, there is an aspect that has been overshadowed. Recall that the idea is to replace
34
See Judd (1992) for more details.
4 2
the quantity that needs to be approximated by parameterized functions (basis functions
?
i
(
x
) ) with arbitrary coefficients ( a
i
). In projection methods, the coefficients are chosen
to be the best possible choices relative to the basis ?
i
(
x
) and relative to some criterion.
However, the bases are usually chosen to satisfy some general criteria, such as
smoothness and orthogonality conditions. Such bases may be good but very rarely are
they the best possible for the problem under consideration.
An important advantage of parameterized expectations approach is that, for
specific models, it may implicitly deal with the presence of inequality constraints
eliminating the need to constantly check whether the Kuhn-Tucker conditions are
satisfied
35
.
A key component of the conventional parameterized expectations approach
presented in this section is a cumbersome nonlinear regression step. The regression step
implies simulations involving a huge amount of synthetic data points. The problem with
this approach is that it inefficiently concentrates on a residual amount that is obtained
from visiting only high probability points of the invariant distribution of the model. As
Pointed out by Judd (1992) and Christiano and Fisher (2000), it is important to consider
the tail areas of the distribution as well. Christiano and Fisher (2000) offer a modified
version of the parameterized expectations approach that they call the Chebyshev
parameterized expectations approach, specifically designed to eliminate the shortcoming
discussed above. In fact, Christiano and Fisher (2000) explicitly transform the
parameterized expectations approach into a projection method that they refer to as the
weighted residual parameterized expectations approach. As mentioned above, expressing
35
See Christiano and Fisher (2000) for details.
43
the parameterized expectations approach as a projection method opens the door to a
variety of possible implementations.
36
.
I.7. Comparing Numerical Methods: Accuracy and Computational Burden
It is difficult to define the global criteria of success for numerical methods.
Accuracy is in general at the top of the checklist in defining a good numerical method.
However, it may not always be the most important criterion when choosing a numerical
method. For example, even though a method may not provide the best approximation for
the policy function, it may still be preferred to other methods as long as the loss in
accuracy relative to the policy function does not affect too much the value of the
objective function. In such cases, speed or ease of implementation may take precedence.
There does not seem to be a general agreement in the literature on how to evaluate
the accuracy of numerical methods. Consequently, a number of criteria have been
proposed in order to asses the performance of numerical algorithms.
One widely used strategy for determining accuracy is to test the outcome of a
computational algorithm in a particular case where the model displays an analytical
solution. For example, Collard and Juillard (2001) use an average relative error and a
maximal relative error criterion in order to asses the accuracy of several numerical
methods. While this approach may be useful for certain specifications, the problem is that
for alternative parameterizations of the model the approximation error of the computed
decision and value functions may change substantially. Changes in the curvature of the
objective function and in the discount factor are the usual culprits in influencing
36
In fact, Christiano and Fisher (2000) provide two other modified versions of the
parameterized expectations approach (PEA): PEA Galerkin and PEA collocation.
44
considerably the accuracy of the algorithm. Collard and Juillard (2001) determine that for
an asset pricing model the Galerkin method using fourth order Chebyshev polynomials
clearly outperforms linearization methods as well as lower order perturbation methods.
However, higher order (order four and higher) perturbation methods prove to be quite
accurate.
Another strategy used for analyzing the accuracy of numerical methods is to look
at the residuals of the Euler equation. This seems like a natural choice especially for
approaches that are based on approximating certain terms entering, or the whole, Euler
equation
37
.
A procedure for checking accuracy of numerical solutions based on the Euler
equation residuals was proposed by den Haan and Marcet (1990, 1994). It consists of a
test for the orthogonality of the Euler equation residuals over current and past
information. The idea behind this test is to compute simulated time series for all the
choice and state variables as well as Euler equation residuals, based on a candidate
approximation. Then, using estimated values of the coefficients resulting from regressing
the Euler equation residuals on lagged simulated time series, one can construct measures
of accuracy. As pointed out by Santos (2000), the problem with this approach is that
orthogonal Euler equation residuals may be compatible with large deviations from the
optimal policy. In addition, as referenced by Judd (1992), Klenow (1991) found that the
procedure failed to reject candidate solutions that resulted in relatively high errors for the
choice variable while rejecting solutions resulting in occasional high large errors but
without any discernible pattern.
37
For a detailed discussion on criteria involving Euler equation residuals, please see
Reiter (2000) and Santos (2000).
45
Judd (1992, 1998) suggested an alternative test that consists of computing a one
period optimization error relative to the decision rule. The error is obtained by dividing
the current residual of the Euler equation to the value of next period's decision function.
Subsequently, two different norms are applied to the error term: one gives the average
and the other supplies the maximum.
In a study aimed at comparing various approximation methods, Taylor and Uhlig
(1990) found that performance varies greatly depending on the criterion used for
assessing accuracy. For example, the decision rules indicated that some of the easier to
implement methods such as the linear-quadratic method and the extended-path method
were fairly close to the "exact" decision rule
38
as given by the quadrature-value-function-
grid method of Tauchen (1990) or the Euler-equation grid method of Coleman (1990).
However, neither the linear-quadratic nor the extended-path method performed well
when using the martingale-difference tests for the Euler-equation residual. Not
surprisingly, the parameterized expectations approach performed well when using the den
Haan and Marcet criterion but not as well when measured against the exact decision rule.
While accuracy is very important, computational time may also play an important
role in the eyes of some researchers. While the extended-path method has relatively low
cost when compared to grid methods, it is fair to state that both grid methods and the
extended-path method are computationally quite involved, whereas linear-quadratic
methods are typically quite fast. Most projection methods also fare well in terms of
38
Solutions obtained through discretization methods are sometimes referred to as
"exact". The reason behind this labeling is that models obtained as a result of
discretization may be solved exactly by finite-state dynamic programming methods.
However, one has to keep in mind that reducing a continuous-state problem to a finite-
state problem still involves an approximation error.
46
computational burden when compared to discretization methods or even parameterized
expectations methods. As the state space increases, discretization methods suffer heavily
from the curse of dimensionality.
The fact that none of the methods outperforms the others does not mean that every
method could be applied to any model out there with a good degree of success
39
. One has
to use good judgment when deciding on using a certain numerical method.
I.8. Concluding Remarks
As it has become clear over the course of this chapter, there are quite a few
methodologies available for solving non-linear rational expectations models. However, if
one looks closer, it becomes obvious that all methods share some common elements. For
example, certainty equivalence is at the core of the extended path method but it can also
be used in perturbation methods to find the equilibrium of a (deterministic) system
similar to the one under investigation. The discrete state space approach can be viewed as
a projection method with step functions as a basis. Similarly, the first order perturbation
method is nothing more than a simple linearization around steady state. In addition, the
parameterized expectations approach can be easily transformed into a projection method.
Moreover, since all the functional equations for rational expectations models imply the
existence of some integrals, the quadrature approximation may make an appearance in
almost every methodology.
39
Judd (1998) contains an example of a partially revealing rational expectations problem
which cannot be solved by discretizing the state space, but which can be approximated by
more general projection methods.
47
Several studies have tried to asses the performance of these numerical methods.
However, even for relatively simple models their performance may vary greatly
40
.
Despite all of their sophistication, none of these methods can consistently outperform the
others.
Even comparing the methods is not a walk in the park. Several authors including
Judd (1992), Den Haan and Marcet (1994), Collard and Juillard (2001), Santos (2000)
and Reiter (2000) proposed different criteria for evaluating the performance of numerical
solutions. Unfortunately, each criterion has its caveats and it has to be applied selectively,
based on the specificity of the model under investigation. Therefore, one has to choose
carefully the proper methodology when in need of numerical solutions.
40
See the studies by Taylor and Uhlig (1990), Judd (1992), Rust (1997), Christiano and
Fischer (2000), Santos (2000), Collard and Juillard (2001), Fair (2003), Schmitt-Grohé
and Uribe (2004).
48
Chapter II. Using Scenario Aggregation Method to Solve a Finite
Horizon Life Cycle Model of Consumption
II.1. Introduction
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 horizon
41
. 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
41
In the next chapter I will focus more on the length of the span over which the decision
making process takes place.
49
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 life-
cycle 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.
II.2. A Simple Life-Cycle Model with Precautionary Saving
I consider the following version of a life-cycle model. Suppose an individual
agent is faced with the following intertemporal optimization problem:
max E
9 |
t
F
t
(c
t
) | I
0
? T
?
(2.2.1)
{c
t
}
T
=0
t
_t=0
e
?
where F
t
is a utility function which has the typical properties assumed in the literature,
i.e. it is twice differentiable, it is increasing with consumption and exhibits negative
second derivative. The information set I
0
contains the level of consumption, assets, labor
income and interest rate for period zero and all previous periods.
Maximization is subject to the following transition equation:
A
t =
(1+ r
t
) A
t
÷
1
+ y
t
÷ c
t
, t = 0,1,...,T ÷1, (2.2.2)
A
t
> ÷b, with A
-1
, A
T
given (2.2.3)
50
where A
t
represents the level of assets at the beginning of period t , y
t
the labor income
at time t , and c
t
represents consumption in period t . The initial and terminal conditions,
A
÷
1
and A
T
, are given. Uncertainty is introduced in the model through the labor income.
The realizations of the labor income are described by the following process:
y
t
= y
t
÷
1
+ ç
t
, t = 1,...,T , with y
o
given (2.2.4)
and ç
t
being drawn from a normal distribution, ç
t
~
N
(0,o2
y
) . For now, I will not make
any particular assumption about the process generating the interest rate, r
t
. Therefore, to
summarize the model, a representative consumer derives utility in period t from
consuming c
t
, discounts future utility at a rate | and wants, in period zero, to maximize
his present discounted value of future utilities for a horizon of T +1 periods. At the
beginning of each period t the consumer receives a stochastic labor income y
t
, and
based on the return on his assets A
t
÷
1
, from the beginning of period t ÷1 to the beginning
of period t , he chooses the consumption level c
t
, and thus determines the level of assets
A
t
according to equation (2.2.2).
Of particular importance in this problem is the random variable ç
t
. In the
standard formulation of the problem, ç
t
is assumed to be distributed normally with mean
zero and some variance o2
y
. Instead of making the standard assumption, if I assume that
ç
t
's sample space has only a few elements, then the optimization problem (2.2.1) -
(2.2.4) is a perfect candidate for being solved using the scenario aggregation method. Let
me assume for the moment that the sample space is given
by
{e
1
,e
2
,...,e
n
} with the
associated probabilities { p
1
, p
2
,..., p
n
} . If S is the set of all scenarios then its cardinal is
51
given by n
T
. It is obvious that as the sample space for the forcing variable increases, the
number of scenarios increases proportional with power T . Therefore, applying the
scenario aggregation method to find an approximate solution for this problem may only
be feasible when T and n are relatively small. In the next chapter, I will present a
solution for T relatively large.
II.3. The Concept of Scenarios
II.3.1. The Problem
In this section, I formally introduce a multistage optimization problem and then,
in the following sections, I will present the idea of scenario aggregation and how it can be
applied to such a problem.
The multistage stochastic optimization problem consists of minimizing an
objective function, F : R
m
C R subject to some constraints, which usually describe the
dynamic links between stages.
The objective function F is time separable and is given by a sum of functions,
T
F
=
9 F
t
with each function F
t
, F
t
: R
m
C R corresponding to stage t of the
t =0
optimization problem. These functions depend on a set of variables u
t
, which in turn
represent the decisions that need to be made at each stage t . For simplicity I assume that
u
t
is a m
u
·1 vector, with m
u
independent of t , that is, the same number of decisions is
to be made at each stage.
52
If U (t) represents the set of all feasible actions at stage t , then u
t
has to be part
of the set U (t) , that is, u
t
¸ U (t), t = 0,...,T , U (t) [ R
m
u . The temporal dimension of
the problem is characterized by stages t and state variables X (t) .
The link between stages is given by:
x
t
+
1
= G
t
(x
t
,u
t
,u
t
+
1
) .
Hence, the problem can be formulated as:
min
E
?
¿ F
t
( x
t
,u
t
) | I
0
? T
subject to:
??
t =0
x
t
+
1
= G
t
(x
t
,u
t
,u
t
+
1
,ç
t
)
?
?
(2.3.1)
(2.3.2)
where I
0
is the information set at time t = 0 and ç
t
is the forcing variable.
In the next few sections, I will present the concept of scenarios as well as possible
decomposition methods along with the idea of scenario aggregation.
II.3.2. Scenarios and the Event Tree
In this section, I present an intuitive description for the concept of scenarios. A
formal description is presented in Appendix, section A1. Suppose the world can be
described at each point in time by the vector of state variables x
t
. In the case of a
multistage optimization problem, let u
t
denote the control variable and let ç
t
be the
forcing variable. I assume that an agent makes decisions reflected in the control variable
u
t
. For simplicity let ç
t
be a random variable witch can take two values ç
a
and ç
b
with
probabilities p
a
and 1÷ p
a
.
53
If the horizon has T +1 time periods and {ç
a
,ç
b
} is the set of possible
realizations for ç
t
then the sequence
ç
s
=
(ç
0
s
,ç
1
s
,K,ç
T
s
)
is called a scenario
42
. From now on, for notation simplification, I will refer to a scenario
s simply by ç
s
or by the index s . Given that the set of all realizations for ç
t
is finite,
one can define an event
tree
{N,
A
} characterized by the set of nodes N and the set of
arcs A . In this representation, the nodes of the tree are decision points and the arcs are
realizations of the forcing variables. The arcs join nodes from consecutive levels such
that a node n
t
j
at level t is linked to N
t
+
1
nodes, n
k
+
1
, k = 1,..., N
t
+
1
at level t +1. In t
Figure 1 I represent such a tree for a span of T = 3 periods. As mentioned above, the
forcing variable takes only two
values,
{ç
a
,ç
b
} and hence the tree has 15 nodes. The arcs
that join nodes from consecutive levels represent realizations of the forcing variable and
are labeled accordingly.
The set of nodes N can be divided into subsets corresponding to each level
(period). Suppose that at time t there are N
t
nodes. For example, for t = 1, there are two
nodes, node2 and node3. The arcs reaching these two nodes belong each to several
scenarios s . The bundle of scenarios that go through one node plays a very important
role in the decomposition as well as in the aggregation process. The term equivalence
class has been used in the literature to describe the set of scenarios going through a
particular node.
42
Other definitions of scenarios can be found in Helgason and Wallace (1991a, 1991b )
and Rosa and Ruszczynski (1994).
54
node1, t=0
çt=ça çt=çb
node2, t=1 node3, t=1
çt=ça çt=çb çt=ça çt=çb
node4, t=2 node5, t=2 node6, t=2 node7, t=2
çt=ça çt=çb çt=ça çt=çb çt=ça çt=çb çt=ça çt=çb
node8, t=3 node9, t=3 node10, t=3 node11, t=3 node12, t=3 node13, t=3 node14, t=3 node15, t=3
Figure 1 Event tree
By definition, an equivalence class at time t is the set of all scenarios having the
first t +1 realizations common. As mentioned in the above description of the event tree,
at time t there are N
t
nodes. Every node is associated with an equivalence class. Then,
the number of distinct equivalence classes at time t is also N
t
.
In Figure 2 one can see that for t = 1 there are two nodes and consequently two
equivalence classes, {s
1
, s
2
, s
3
, s
4
} and {s
5
, s
6
, s
7
, s
8
} . The number of elements of an
equivalence class is given by the number of leaves stemming from the node associated
with it. In this example, the number of leaves stemming from both nodes is four, which is
also the number of scenarios belonging to each class.
55
node1, t=0
(s1,s2,s3,s4,s5,s6,s7,s8)
çt=ça çt=çb
node2, t=1 node3, t=1
(s1,s2,s3,s4) (s5,s6,s7,s8)
çt=ça çt=çb çt=ça çt=çb
node4, t=2 node5, t=2 node6, t=2 node7, t=2
(s1,s2) (s3,s4) (s5,s6) (s7,s8)
çt=ça çt=çb çt=ça çt=çb çt=ça çt=çb çt=ça çt=çb
node8, t=3 node9, t=3 node10, t=3 node11, t=3 node12, t=3 node13, t=3 node14, t=3 node15, t=3
(s1) (s2) (s3) (s4) (s5) (s6) (s7) (s8)
Figure 2 Equivalence classes
The transition from a state at time t to one at time t + 1 is governed by the control
variable u
t
but is also dependent on the realization of the forcing variable, that is, on a
particular scenario s . Since scenarios will be viewed in terms of a stochastic vector ç
with stochastic components ç
0
s
,ç
1
s
,K,ç
T
s
, it is natural to attach probabilities to each
scenario. I denote the probability of a particular realization of a scenario, s , with
p(s) = prob(ç
s
) .
Let us consider the case of the event trees represented in Figure 1 and Figure 2 and
assume the probability of realization ç
a
is prob(ç
t
= ç
a
) = p
a
while the probability of
realization ç
b
, is prob(ç
t
= ç
b
) = p
b
, with p
a
+ p
b
= 1. Then, due to independence
56
across time, one can compute the probability of realization for scenario s
1
,
prob(ç
s
= s
1
) = p
3
a . Similarly, the probability of realization for scenario s
2
is
prob(ç
s
= s
2
) = p
2
p
b
, or prob(ç
s
= s
2
) = p
2
(1÷ p
a
) .
a a
Further on, I define
43
the probabilities associated with a scenario conditional upon
belonging to a certain equivalence class at time t . For example, the probability associated
with scenario s
1
, conditional on s
1
belonging to equivalence
class
{s
1
, s
2
, s
3
, s
4
} is given
by prob ( s
1
| s
1 e
{s
1
, s
2
, s
3
, s
4
}) = p
2
a
II.4. Scenario Aggregation
44
In this section, I will show how a solution can be obtained by using special
decomposition methods, which exploit the structure of the problem by splitting it into
manageable pieces, and then aggregate their solutions. In the multistage stochastic
optimization literature, there are two groups of methods that have been discussed: primal
decomposition methods that work with subproblems that are assigned to time stages
45
and dual methods, in which subproblems correspond to scenarios
46
. Most of the methods,
regardless of which group belong to, use the general theory of augmented Lagrangian
decomposition. In this chapter I will concentrate on a methodology that belongs to the
second group and has been derived from the work of Rockafellar and Wets (1991).
43
For a more formal definition, see the Appendix, section A1.
44
Section A2 in the Appendix offers a more formal description of scenario aggregation.
45
46
See the work of Birge (1985), Ruszczynski (1986, 1993), Van Slyke and Wets (1969).
See the work of Mulvey and Ruszczynski (1992), Rockafellar and Wets (1991),
Ruszczynski (1989), Wets (1988).
57
Let us assume for a moment that the original problem can be decomposed into
subproblems, each corresponding to a scenario. Then the subproblems can be described
as:
T
u ŒU [R
mu
min 9 F
t
x
s
t ,u
s
t ,
t t
t=1
( )
s ŒS
(2.4.1)
where u
st
and x
st
are the control and the state variable respectively, conditional on the
realization of scenario s while S is a finite, relatively small set of scenarios. Moreover,
suppose that each individual subproblem can be solved relatively easy. The question then
becomes how to blend the individual solutions into a global optimal solution. Let the
term policy
47
describe a set of chosen control variables for each scenario and indexed by
the time dimension.
The policy function has to satisfy certain constraints if two different scenarios s
and s ' are indistinguishable at time t on information available about them at the time.
Then u
s
t = u
s
t
'
, that is, a policy can not require different actions at time t relative to
scenarios s and s ' if there is no way to tell at time t which of the two scenarios will be
followed. In the literature, this constraint is sometimes referred to as the non-
anticipativity constraint. Going back to Figure 2, for t = 1, if the realization of ç
t
is ç
a
,
the decision maker will find himself at the decision point node2. There are four scenarios
that pass through node2 and the non-anticipativity constraint requires that only one
decision be made at that point since the four scenarios are indistinguishable. A policy is
47
A formal description of the policy function is presented in Appendix.
58
defined as implementable if it satisfies the non-anticipativity constraint, that is, u
t
must
be the same for all scenarios that have common past and present
48
.
In addition, a policy has to be admissible. A policy is admissible if it always
satisfies the constraints imposed by the definition of the problem. It is clear that not all
admissible policies are also implementable.
By definition, a contingent policy
49
is the solution, u
s
, to a scenario subproblem.
It is obvious that a contingent policy is always admissible but not necessarily
implementable. Therefore, the goal is to find a policy that is both admissible and
implementable. Such a policy is referred to as a feasible policy. One way to create a
feasible policy from a set on contingent policies is to assign weights (or probabilities) to
each scenario and then aggregate the contingent policies according to these weights.
The question that the scenario aggregation methodology answers is how to obtain
the optimal solution U from a collection of implementable policies U . In this chapter, I ˆ
will present a version of the progressive hedging algorithm originally developed by
Rockafellar and Wets (1991).
48
For certain problems the non-anticipativity constraint can also be defined in terms of
the state variable, that is, x
t
(e ) must be the same for all scenarios that have common past
and present.
49
I borrow this term from Rockafeller and Wets (1991).
59
II.5. The Progressive Hedging Algorithm
The algorithm is based on the principle of progressive hedging
50
which consists of
starting with an implementable policy and creating sequences of improved policies in an
attempt to reach the optimal policy.
Let us go back to the definition of an implementable policy. By computing
u
s
t =
ˆ
¿
}
p s
?
{s
t
}
i
u
s
t
'
= E u
s
t
'
| {s
t
}
i
s?e
{s
t
i
( )
( )
for all s
e
{s
t
}
i
(2.5.1)
for all scenarios s e S and all periods t = 1,...,T , one creates a starting collection of
implementable policies, denoted by U
0
. In equation (2.5.1) E represents the expectation ˆ
operator. Therefore, in order to obtain an initial collection of implementable policies one
should first compute some contingent policies for each scenario and then apply the
expectation operator for each period t and each scenario s conditional on it belonging to
the corresponding equivalence class, {s
t
}
i
.
The progressive hedging algorithm finds a path from U
0
, the set of ˆ
implementable policies, to U , the set of optimal policies, by solving a sequence of
problems in which the scenarios subproblems are not the original ones, but a modified
version of those by including some penalty terms. The algorithm is an iterative process
starting from U
0
and computing at each iteration k a collection of contingent policies ˆ
U
k
which are then aggregated into a collection of implementable policies U
k
that are ˆ
supposed to converge to the optimal solution U . The contingent policies U
k
are found as
optimal solutions to the modified scenario subproblems:
50
This term was coined by Rockafellar and Wets (1991). The idea is based on the theory
of the proximal point algorithm in nonlinear programming.
60
min F
s
( x
s
,u
s
) + w
s
u
s
+
1
µ u
s
÷ u
s
ˆ
2
2
(2.5.2)
where ? is the ordinary Euclidian norm, µ is a penalty parameter and w
s
is an
information price
51
. The use of µ is justified by the fact that the new contingent policy
should not depart too much from the implementable policy found in the previous
iteration. The modified scenario subproblems (2.5.2) have the form of an augmented
Lagrangian.
In the next subsection, I present a detailed description of the progressive hedging
algorithm, which uses subproblems in the form of an augmented Lagrangian as shown
above.
II.5.1. Description of the Progressive Hedging Algorithm
The optimal solution of the problem described by equations (2.3.1) - (2.3.2), U ,
represents the best response an optimizing agent can come up with in the presence of
uncertainty. An advantage of this algorithm is that one does not necessarily need to solve
subproblems (2.5.2) exactly. A good approximation
52
of the solution is enough in
allowing one to solve for the solution of the global problem.
Let U
k
denote a collection of admissible policies and W
k
a collection of
information prices corresponding to iteration k . The progressive hedging algorithm, as
designed by Rockafellar and Wets (1991), consists of the following steps:
51
52
I borrow this term from Rockafellar and Wets (1991).
One can envision transforming the scenario subproblems into quadratic problems by
using second order Taylor approximations.
61
Step 0. Choose a value for µ , W
0
and for U
0
. The value of µ may remain
constant throughout the algorithm but it can also be adjusted from iteration to iteration
53
.
Changing the value of µ may improve the speed of convergence. Throughout this
chapter, I will consider µ as being constant. U
0
can be composed of the contingent
policies u
s
(0
)
= u
1
s(0
)
,u
1
s(0
)
,...,u
T
s(0
)
obtained from solving all the scenarios subproblems,
( )
whether modified or not. W
0
can be initialized to zero, W
0
= 0 . Calculate the collection
of implementable policies, U
0
= JU
0
, where J is the aggregation operator
54
. ˆ
Step 1. For every scenario s e S , solve the subproblem:
min ¿
?
F
t
s
( x
s
t ,u
s
t ) + w
s
t u
s
t +
1
µ u
s
t ÷ u
s
t ? T
?
t=1 ?
2
ˆ 2?
?
(2.5.3)
For iteration k +1 , let u
s
(k+1
)
= u
1
s(k+1
)
,u
s
2(k+1
)
,...,u
T
s(k+1
)
denote the solution to the
( )
subproblem corresponding to scenario s . This contingent policy is admissible but not
necessarily implementable. Let U
k
+
1
be the collection of all contingent policies u
s
(k+1
)
.
Step2. Calculate the collection of implementable policies, U
k
+
1
= JU
k
+
1
. While ˆ
these policies are implementable, they are not necessarily admissible in some cases
55
. If
the policies obtained are deemed a good approximation, the algorithm can stop. A
stopping criterion should be employed in this step.
53
See Rockafeller and Wets (1991) and Helgason and Wallace (1991a, 1991b) for a
discussion on the values of µ . Rosa and Ruszczynski (1994) also provide some
algorithm for updating similar penalty parameters.
54
55
See the appendix for more details on the aggregation operator.
Contingent policies are always admissible. If the domain of admissible policies is
convex then any linear combination of the contingent policies will also belong to that
domain. As noted above, by definition, the aggregation operator is linear. Therefore, for a
convex problem the implementable policies computed in step 1 are also admissible.
62
Step3. Update the collection of information prices W
k
+
1
by the following rule:
W
k
+
1
= W
k
+ µ U
k
÷U
k
(
For each scenario s e S rule (2.5.4) translates into:
ˆ
)
(2.5.4)
w
s
t (k+1
)
= w
s
t (k
)
+ µ u
s
t (k
)
÷ u
s
t(k)
(
ˆ
)
for t = 1,...,T
(2.5.5)
This updating rule is derived from the augmented Lagrangian theory. In principle, the
rule can be changed with something else as long as the decomposition properties are not
altered.
Step 4. Reassign k := k +1 and go back to step one.
Next, I investigate how this methodology can be applied to a type of dynamic
programming problem closed to what is often employed by economists for their models.
II.6. Using Scenario Aggregation to Solve a Finite Horizon Life Cycle Model
In this section, I will take a closer look at the viability of scenario aggregation in
approximating a rational expectations model. I choose a standard finite horizon life cycle
model that has an analytical solution, which will be used as a benchmark for the
performance of the scenario aggregation method.
I start by presenting an algorithm for solving the problem given by (2.2.1) -
(2.2.4) under the assumption that the length of the horizon, T , and the number of
realizations of the forcing variable, n , are relatively small. The algorithm used is similar
to that developed by Rockefeller and Wets (1991). As mentioned above, the idea is to
split the problem into many smaller problems based on scenario decomposition and solve
those problems iteratively imposing the non-anticipativity constraint. For computational
convenience, I will reformulate the problem (2.2.1) - (2.2.4) as a minimization rather than
63
maximization. Hence, for each scenario s e S , represented by the sequence of
realizations y
s
= y
s
0 , y
1
s
,K, y
T
s
, the problem becomes:
( )
min
?
¿ |
t
?÷F
t
(c
s
t ) + w
s
t c
s
t +
1
µ (c
s
t ÷ c
s
t )
?
?
?
T
?
2
??
(2.6.1)
subject to
c
t
? t =0 ?
2
??
A
ts
= 1+ r
t
s
A
t
s÷
1
+ y
s
t ÷ c
s
t , t = 0,1,...,T
( ) (2.6.2)
Expressing c
st
and c
st
as a function of A
t
s
and A
t
s
, the augmented Lagrangian function,
for a fixed scenario s , becomes:
L
=
¿ |
t
÷F
t
?
(1+ r
t
s
) A
t
s÷
1
+ y
s
t ÷ A
ts
? + w
s
t
?
(1+ r
t
s
) A
t
s÷
1
+ y
s
t ÷ A
ts
? +
T
t =0
{
? ? ? ? (2.6.3)
+
1
µ ?(1+ r
t
s
) A
t
s÷
1
+ y
s
t ÷ A
ts
÷
(1+ r
t
s
) A
t
s÷
1
+ y
s
t ÷ A
ts
? ?
2?
( )
2
??
?
All the underlined variables in the above equations represent implementable policies or
states derived from applying implementable policies.
Before going through the steps of the algorithm, I will make a few assumptions
about the functional form of the utility function as well as about the interest rate. First, it
is assumed that preferences are described by a negative exponential utility function.
Hence:
F
t
(c
t
) = ÷ 1 exp (÷u c
t
)
u
(2.6.4)
where u is the risk aversion coefficient. Secondly, the interest rate, r
t
, is taken to be
constant. Finally, the distribution of the forcing variable is approximated by a discrete
counterpart. The realizations as well as the associated probabilities are obtained using a
Gauss-Hermite quadrature and matching the moments up to order two. The number of
points used to approximate the original distribution determines the number of scenarios.
64
By decomposing the original problem into scenarios, the subproblems become
deterministic versions of the original model.
II.6.1. The Algorithm
Given the assumptions made in the previous section, problem (2.6.1) becomes:
min
?
¿
?
? T |
t
?
1 exp ÷u c
s
+ w
s
c
s
+
1
µ c
s
÷ c
s
2 ??
c
t
? t =0
?u
(
t
) t
t
2 ( t
t
)
?
???
(2.6.5)
Consequently the Lagrangian for scenario s is:
L
=
¿ |
t
? 1 exp ?÷
u
((1+
r
) A
t
s÷
1
+ y
s
t ÷ A
t
s
)? + w
s
t
?
(1+
r
) A
t
s÷
1
+ y
s
t ÷ A
ts
? + T
t =0
?
?
u
? ? ? ? (2.6.6)
+
1
µ ?(1+
r
) A
t
s÷
1
+ y
s
t ÷ A
ts
÷
((1+
r
) A
t
÷
1
+ y
s
t ÷ A
t
)? ? 2
2? ??
?
Since the consumption variable was replaced by a function of the asset level, the
algorithm will be presented in terms of solving for the level of assets.
Step 0. Initialization: Set w
st
= 0 for all stages t and scenarios s . Choose a value
for µ that remains constant throughout the algorithm, let it be µ = 5 . Later on, in this
chapter, I will discuss the impact the value of µ has on the convergence process. At this
point, one needs a first set of policies. The convergence process, and implicitly the speed
of the algorithm, is impacted by the choice of the first set of policies.
One suggestion made in the literature by Helgason and Wallace (1991a, 1991b) is
to use the solution to the deterministic version of the model. This would amount to using
the certainty equivalence solution in this case. I will first implement the algorithm using
as starting point the certainty equivalence solution and then I will take advantage of the
fact that for certain specifications of the model each scenario subproblem has an exact
solution. I will then compare the convergence properties of the algorithm in these two
cases.
65
Let
{c
ceq
}
t
=
0
denote the solution to the deterministic problem. Then, using the T
t
transition equation (2.6.2) one can compute the level of assets for each scenario s ,
A
s
(0
)
= A
0
s(0
)
, A
1
s(0
)
,..., A
T
s(÷01
)
. Next, it becomes possible to compute the implementable
{ }
states A
(
0
)
= A
(
00
)
, A
1
(0
)
,...,
A
T
(0÷)1{
}
as a weighted average of A
0
s(0
)
corresponding to all
scenarios s , using as weights the probabilities of realization for each scenario.
Alternatively, one can compute the first set of contingent policies by solving a
deterministic life cycle consumption model for each scenario s :
mi
sn
¿ |
t
1 exp ÷u
?
(1+
r
) A
t
s÷
1
+ y
s
t ÷ A
ts
?
A
t
T
t =0
u
{
?
?
}
(2.6.7)
with A
(
÷s1
)
and A
T
(s
)
given. As before, let A
s
(0
)
= A
0
s(0
)
, A
1
s(0
)
,..., A
T
s(÷01
)
denote the solution
{ }
to this problem. This solution is admissible but not implementable. The implementable
solution for each period t , A
t
0
, is computed as the weighted average of all the contingent
solutions for period t , A
t
s(0
)
, with the weights being given by the probability of
realization for each particular scenario s .
Step 1. For every scenario s e S , solve the subproblem:
mi
sn
¿ |
t
? 1 exp ÷u
?
(1+
r
) A
t
s÷
1
+ y
s
t ÷ A
ts
? +
A
t
T
t =0
?
u
?
{
?
?}
+W
ts ?
(1+
r
) A
t
s÷
1
+ y
s
t ÷ A
ts
? +
? ?
(2.6.8)
+
1
µ (1+
r
) A
t
s÷
1
+ y
s
t ÷ A
ts
÷
?
(1+
r
) A
t
÷
1
+ y
s
t ÷ A
t
? ?
2
{
?
2
??
}
?
A detailed description of how the solution is computed can be found in the Appendix.
The advantage of the scenario aggregation method is that the solution to problem (2.6.8)
does not have to be computed exactly.
66
Let A
s
(k
)
= A
0
s(k
)
, A
1
s(k
)
,...,
A
T
s(÷k1) {
} denote the contingent solution to this problem,
where k denotes the iteration. Based on this solution I also compute the consumption
path for each scenario, c
s
(k
)
. This solution is admissible but not implementable and
therefore the next step is to compute the implementable solution based on the contingent
solutions A
s
(k
)
.
Step 2. First, compute the implementable states A
k
. As it was mentioned in step
0, A
t
k
is computed as the weighted average of all the contingent solutions for period t ,
A
t
s(k
)
, with the weights being given by the probability of realization for each particular
scenario s . Since the space of the solutions for the problem being solved is convex, the
implementable solution is also admissible. At this point, if solution A
k
is considered
good enough, the algorithm can stop and A
k
becomes officially the solution of the
problem described by (2.2.1) - (2.2.4). In order to make a decision on the viability of A
k
as the optimal solution, one needs to define a stopping criterion. Based on the value of
A
k
I compute the implementable consumption path c
k
and then use the following error
sequence
56
:
c(k
)
=
¿ |t
?
( c
(
tk
)
÷ c
(
tk÷1) )2 + A
(
tk
)
÷ A
(
tk÷1) ?
T
t =0
?
?
(
)
?
2
?
(2.6.9)
where k is the iteration number. The termination criterion is c (k
)
< d where d is
arbitrarily chosen. In the next section, I will discuss the importance of the stopping
criterion in determining the accuracy of the method.
56
This is similar to what Helgason and Wallace (1990a) proposed. Later on in this
chapter we will discuss the impact the choice of the value for d has on the results.
67
Step 3. For t = 0,1,...,T and all scenarios s update the information prices:
w
s
t (k+1
)
= w
s
t (k
)
+ µ ?(1+
r
) A
t
s÷(k
)
÷ A
t
(÷k1
)
÷ A
t
s(k
)
÷ A
t
(k
)
? for t = 1,...,T
?
(
1
)( )
?
Step 4. Reassign k := k +1 and go back to step one.
II.6.2. Simulation Results
In this section, I present a brief picture of the results obtained by the
implementation of the scenario aggregation method compared to the analytical solution.
These results show that the numerical approximation obtained through scenario
aggregation is close to the analytical solution for certain parameterizations of the model.
In order to asses the accuracy of the scenario aggregation method I will use several
criteria put forward in the literature. First, I compare the decision rule, i.e. the
consumption path obtained through scenario aggregation with the values obtained from
the analytical solution. In this context, I use two relative criteria similar to what Collard
and Juillard (2001) use. One, E
a
, gives the average departure from the analytical solution R
and is defined as:
a
T +
1
¿
c
*t1
T
c
*
t ÷ c
t
E= R
t =0
(2.6.10)
The other, E
m
, represents the maximal relative error and is defined as: R
E = max ? c
t
÷*c
t
m
?*
?
?
T
(2.6.11)
R
?c
?
? ?
t
?
t
=0 ?
where c
*
t is the analytical solution and c
t
is the value obtained through scenario
aggregation. Alternatively, since the problem is ultimately solved in terms of the level of
assets, the two criteria could also be expressed using the level of assets:
68
E
=
¿t*a
?
t
??T
÷1
R
1
T
÷
1
A
*
÷ A
t
, E
m
= max ? A
*
÷ A
t
T
t
=
0
A
t
R
? A
*
?
?
?
t
?
t
=0 ?
where A
*
is given by the analytical solution and A
t
by the scenario aggregation. Even t
though the scenario aggregation methodology does not use the Euler equation in
obtaining the solution, I will use the Euler equation based criteria proposed by Judd
(1998) as an alternative for determining the accuracy of the approximation. The criterion
is defined as a one period optimization error relative to the decision rule. The measure is
obtained by dividing the current residual of the Euler equation to the value of next
period's decision function. Subsequently, two different norms are applied to the error
term: one, E
a
, gives the average and the other, E
m
, supplies the maximum. Judd (1998)
E E
labeled these criteria as measures of bounded rationality.
The simulations were done using the following common set of parameter values:
the discount factor | = 0.96 ; the initial and terminal values for the level of assets
A
÷
1
= 500 and A
T
= 1000 ; the income generating process has a starting value of
y
0
= 200 . In addition, the interest rate is assumed deterministic. I used two values for the
interest rate, r = 0.04 and r = 0.06 . The distribution of the forcing variable was
approximated by a 3 point discrete distribution. As I mentioned in the description of the
progressive hedging algorithm, a few factors can influence the performance of the
scenario aggregation method. Let us first look at how the starting values and stopping
criterion influence the results.
69
II.6.2.1. Starting Values and Stopping Criterion
As I mentioned above, the starting values and the stopping criterion are very
important elements in the implementation of the algorithm. I consider for the moment
that the starting values are given by the certainty equivalence solution of the life cycle
consumption model. I analyze the case where the value for the coefficient of risk aversion
is u = 0.01, the variance for the income process is o2
y
= 100 and the interest rate is
r = 0.06 . The stopping criterion is given by the sequence c
(
k
)
as defined in (2.6.9) and I
arbitrarily choose d = 0.004 . Therefore when c
(
k
)
becomes smaller than d = 0.004 I stop
and declare the solution obtained in iteration k as the solution to the problem described
by (2.2.1) - (2.2.4). In Table 1 I provide the values for the accuracy measures discussed
above, using the level of assets, as opposed to the level of consumption. One can see that
the approximation to the analytical solution obtained by stopping when c
(
k
)
is smaller
than the arbitrarily chosen d is very good.
Table 1. Accuracy measures for d=.004
u
0.01
o
2
y
E
a
R
E
m
R
E
a
E
E
m
E
100 0.001445515 0.002392885 0.000005019 0.000008735
70
The results presented in Table 1 are obtained after 159 iterations. Next, I will look
at the behavior of the sequence c
(
k ) for the case presented above.
3.5
3
2.5
2
1.5
1
0.5
0
Evolution for the c(k)sequence
5
4
3
2
1
0
x 10
-3
Evolution for the c(k)sequence (zoom)
0 50 100 150 200 250 300 350 150 200 250 300
Iteration k Iteration k
Evolution of the value for Evolution of the value for
the objective function the objective function (zoom)
-111.5
-112
-112.5
-113
-113.5
-114
-114.5
-111.876
-111.878
-111.88
-111.882
-111.884
-111.886
-111.888
-115 -111.89
0 50 100 150 200 250 300 350 150 200 250 300 350
Iteration k Iteration k
Figure 3. Evolution of the c
(
k ) sequence and the value of the objective for u = .01 and o2
y
= 100
One can see in Figure 3 that the value for sequence c
(
k
)
continues to decrease
until iteration 250 when it attains the minimum value. At the same time, the value of the
objective continues to increase until iteration 266 when it attains its maximum. It is worth
noting that the value of the objective is computed as in equation (2.6.12). Based on these
observations one may elect to choose as stopping criterion the point where c
(
k
)
attains its
minimum or when the objective function attains its maximum as opposed to an arbitrary
value d . Next, I look at how close is the approximation to the analytical solution when
71
v
a
l
u
e
o
f
c
( k
) s
e
q
u
e
n
c
e
v
a
l
u
e
o
f
c
( k
) s
e
q
u
e
n
c
e
V
a
l
u
e
o
f
t
h
e
o
b
j
e
c
t
i
v
e
f
u
n
c
t
i
o
n
V
a
l
u
e
o
f
t
h
e
o
b
j
e
c
t
i
v
e
f
u
n
c
t
i
o
n
using these criteria. In Table 2 one can see that there is not much difference between the
last two criteria when compared to the analytical solution. The only difference is that the
value of the expected utility is marginally higher in the second case.
Table 2. Accuracy measures for various stopping criteria
u
0.01
o
2
y
E
a
R
E
m
R
E
a
E
E
m
E
Stopping criterion
100 0.001445515 0.002392885 0.000005019 0.000008735 Arbitrary d = 0.004
100 0.002137894 0.002691210 0.000007190 0.000013733 Minimum of c
(
k)
100 0.002137894 0.002691210 0.000007190 0.000013733 Maximum objective
A somewhat interesting result is that the ad-hoc stopping criterion d = 0.004
leads to a better approximation of the analytical solution. This is explained by the fact
that the progressive hedging algorithm leads to the solution that would be obtained
through the aggregation of the exact solutions for every scenario. Here the starting point
is the certainty equivalent solution and the path to convergence, at some point, is very
close to the analytical solution.
II.6.3. The Role of the Penalty Parameter
In the implementation of the progressive hedging algorithm, I chose the penalty
parameter to be constant. Its role is to keep the contingent solution for each iteration close
to the previous implementable policy. However, its value also has an impact on the speed
of convergence. I will now consider the previous parameterization of the model and I am
72
going to change the value of the penalty parameter to see how it changes the speed of
convergence. In Figure 4 one can see that as µ increases so does the number of iterations
needed to achieve convergence. While a higher value of the penalty parameter helps the
convergence of contingent policies to the implementable policy, it also slows the global
convergence process, requiring more iterations.
Evolution for the c(k)sequence Evolution for the c(k)sequence
-3 for µ = 0.1 -5x 10 x 10 for µ = 0.5
5 3
2.54
2
3
1.5
2
1
10.5
0 0
150 200 250 300 1000 1200 1400 1600 1800 2000 2200
Iteration k Iteration k
Evolution for the c(k)sequence Evolution for the c(k)sequence
x 10
-9
for µ = 2 for µ = 5 x 10
-11
4 8
3.5 7
3 6
2.5 5
2 4
1.5 3
1 2
0.5 1
0 0
7000 7500 8000 8500 9000 9500 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8
Iteration k Iteration k x 10
4
Figure 4. Convergence for different values of the penalty parameter.
For µ = 0.1 , 250 iterations are needed to achieve convergence, while for µ = 0.5 ,
1780 iterations are needed. For higher values, such as µ = 5 , the number of iterations
needed to achieve convergence increases to over 25000 iterations.
73
v
a
l
u
e
o
f
c
( k
) s
e
q
u
e
n
c
e
v
a
l
u
e
o
f
c
( k
) s
e
q
u
e
n
c
e
V
a
l
u
e
o
f
t
h
e
o
b
j
e
c
t
i
v
e
f
u
n
c
t
i
o
n
V
a
l
u
e
o
f
t
h
e
o
b
j
e
c
t
i
v
e
f
u
n
c
t
i
o
n
II.6.4. More simulations
In this section I investigate how close the scenario aggregation solution is to the
analytical solution for various parameters. Table 3 shows the values for the four criteria
enumerated above for different values of the coefficient of risk aversion and of the
variance of the random variable entering the income process. All the simulations whose
results are presented in Table 3 were done using a three point approximation of the
distribution of the random variable entering the income process. The relative measures
are computed using the level of assets.
Table 3. Accuracy measures for various parameters when interest rate r=0.04
u
o2y 0.01 0.05 0.1
E
a
R
E
m
R
E
a
E
E
m
E
E
a
R
E
m
R
E
a
E
E
m
E
E
a
R
E
m
R
E
a
E
E
m
E
1 .0000 .0000 .0000 .0000 .0001 .0001 .0000 .0000 .0002 .0003 .0000 .0000
4 .0000 .0001 .0000 .0000 .0004 .0005 .0000 .0000 .0009 .0011 .0000 .0000
25 .0005 .0007 .0000 .0000 .0029 .0037 .0000 .0000 .0058 .0074 .0000 .0000
100 .0023 .0029 .0000 .0000 .0116 .0147 .0000 .0000 .0230 .0290 .0000 .0000
For lower values of the coefficient of risk aversion the approximation is relatively good.
As the coefficient of risk aversion increases in tandem with the variance of the income
process, the accuracy suffers when looking at relative measures. The Euler equation
measure still indicates a very good approximation.
Let us now look at how this approximation affects the value of the original
objective, i.e. the expected discounted utility over the lifetime horizon. Table 4 shows the
74
ratio of the expected utilities for the whole horizon with the scenario aggregation as the
as the denominator and the analytical solution as the numerator.
Table 4. The ratio of lifetime expected utilities
F
as
F
sc
u
o2y 0.01 0.03 0.05 0.1
1 1.00000 1.00000 1.00000 1.00003
4 1.00000 1.00001 1.00003 1.00058
25 1.00000 1.00002 1.00141 1.02027
100 1.00003 1.00051 1.02273 1.39364
The discounted utilities are computed as in the original formulation of the problem:
F
sc
= 1
N
¿1 ?
?
¿
0
??
u
(
t
)
?
??? ?÷
T |
t
? 1 exp ÷u c
i
??
(2.6.12)
and
N
i= t=
F
as
= 1
N
?÷T |
t
? 1 exp ÷u c
i
* ?? (2.6.13)
N
¿1 ?
?
¿
0
??
u
(
t
)
?
???
i= t=
where N is the number of simulations, F
sc
is the discounted utility obtained with
scenario aggregation and
F
as
is the discounted utility obtained with the analytical
solution. In this formulation, both quantities are negative so their ratio is positive. Note
however that the initial formulation of the problem using the objective function specified
in (2.6.12) and (2.6.13) was a maximization. Therefore, higher ratio in Table 4 means that
the solution obtained through scenario aggregation leads to higher discounted lifetime
utility than the analytical solution. I simulate 2000 realizations of the income process and
then I average the discounted utilities over this sample. The result shows that the solution
75
obtained through scenario aggregation leads to higher overall expected utility as the
coefficient of risk aversion increases. This is explained by the fact that the level of
consumption in the first few periods is higher in the case of scenario aggregation. In the
context of a short horizon, this leads to higher levels of discounted utility.
II.7. Final Remarks
The results show that scenario aggregation can be used to provide a good
approximation to the solution of a life-cycle model for certain values of the parameters.
There are a few remarks to be made regarding the convergence. As pointed out earlier in
this chapter the value of µ has an impact on the speed of convergence. Higher values of
µ lead to faster convergence of the contingent policies towards an implementable policy
but that also means that the overall convergence is slower and hence it impacts the
accuracy if an ad-hoc stopping criterion is used. Therefore, one needs to choose carefully
the values of the ad-hoc parameters. On the other hand, if the scenario problems have an
exact solution then the final implementable policy can be obtained through a simple
weighted average with the weights being the probabilities of realization for each scenario.
76
Chapter III. Impact of Bounded Rationality
57
on the Magnitude of
Precautionary Saving
III.1. Introduction
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 supporters
58
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 rationality.
57
The concept of bounded rationality in this chapter should be understood as a set of
assumptions that departs from the usual rational expectation paradigm. Its meaning will
become clear later in the chapter when the underlying assumptions are spelled out.
58
Sargent (1993) for example, identifies several areas in which bounded rationality can
potentially help, such as equilibrium selection in the case of multiple possible equilibria
and behavior under "regime changes".
77
One 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, simulation-
based models, artificial intelligence based methodologies such as neural networks and
genetic algorithms, evolutionary models drawing their roots from biology, behavioral
78
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 thumb
59
. 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.
59
Some of the examples are Gali et al. (2004), Allen and Carroll (2001), Lettau and
Uhlig (1999) and Ingram (1990).
79
III.2. Empirical Results on Precautionary Saving
There seems to be little agreement in the empirical literature on precautionary
saving, especially when it comes to its relationship to uncertainty. Skinner (1988) found
that saving was lower than average for certain groups
60
of households that are perceived
to have higher than average income uncertainty. In the same camp, Guiso, Jappelli and
Terlizzese (1992), using data from the 1989 Italian Survey of Household Income and
Wealth, found little correlation between the level of future income uncertainty and the
level of consumption
61
. In addition, Dynan (1993), using data from the Consumer
Expenditure Survey, estimated the coefficient of relative prudence and found it to be "too
small to be consistent with widely accepted beliefs about risk aversion".
On the other hand, Dardanoni (1991) basing his analysis on the 1984 cross-
section of the UK FES (Family Expenditure Survey) suggested that the majority of
saving in the sample arises for precautionary motives. He found that average
consumption across occupation and industry groups was negatively related to the within
group variance of income. Carroll (1994) found that income uncertainty was statistically
important in regressions of current consumption on current income, future income and
uncertainty. Using UK FES data, Merrigan and Normandin (1996) estimated a model
where expected consumption growth is a function of expected squared consumption
growth and demographic variables and their results, based on the period 1968-1986,
60
61
Specifically, the groups identified were farmers and self-employed.
In fact the study on Italian consumers did find that consumption was marginally lower
while wealth was marginally higher for those who were facing higher income uncertainty in
the near future.
80
indicate that precautionary saving is an important part of household behavior. Miles
(1997), using several years of cross-sections of the UK micro data and regressing
consumption on several proxies for permanent income and uncertainty, found that, for
each cross-section, the latter variable played a statistically significant role in determining
consumption. In a study trying to measure the impact of income uncertainty on household
wealth, Carroll and Samwick (1997), using the Panel Study of Income Dynamics, found
that about a third of the wealth is attributable to greater uncertainty. Later on, Banks et al.
(2001), exploiting not only the cross-sectional, but also the time-series dimension of their
data set, find that section specific income uncertainty as opposed to aggregate income
uncertainty plays a role in precautionary saving. Finally, Guariglia (2001) finds that
various measures of income uncertainty have a statistically significant effect on savings
decisions.
In this chapter, I am going to show that, by introducing bounded rationality in a
standard life cycle model, one can increase the richness of the possible results. Even if
the setup of the model would imply the existence of precautionary savings, under certain
parameter values and rules followed by consumers, the precautionary saving is apparently
almost inexistent. As opposed to most of the literature
62
studying precautionary savings, I
introduce uncertainty in the interest rate, beside income uncertainty. In this context, the
size of precautionary saving no longer depends exclusively on income uncertainty.
62
A notable exception is Binder et al. (2000).
81
III.3. The Model
I start this section by presenting the formulation of a standard finite horizon life-
cycle consumption model. Then I will introduce a form of bounded rationality
63
and
investigate the path for consumption and savings.
Consider the finite-horizon life-cycle model under negative exponential utility.
Suppose an individual agent is faced with the following intertemporal optimization
problem:
ma
T
x E
?
¿ ÷|
t
1 exp (÷u c
t
) | I
0
? T
subject to
{c
t
}
t
=0
??
t =0
u ?
?
(3.3.1)
A
t =
(1+ r
t
) A
t
÷
1
+ y
t
÷ c
t
, t = 0,1,...,T ÷1, (3.3.2)
A
t
> ÷b, with A
-1
, A
T
given (3.3.3)
where u is the coefficient of risk aversion, A
t
represents the level of assets at the
beginning of period t , y
t
the labor income at time t , and c
t
represents consumption in
period t . The initial and terminal conditions, A
÷
1
and A
T
are given. The information set
I
0
contains the level of consumption, assets, labor income and interest rate for period
zero and all previous periods. The labor income is assumed to follow an arithmetic
random walk:
63
As it was already mentioned above, the approach in defining bounded rationality in this
chapter has some similarities to the approach followed by Lettau and Uhlig (1999) in the
sense that several rules are used to account for the inability of the boundedly rational
agent to optimize over long horizons.
82
y
t
= y
t
÷
1
+ ç
t
, t = 1,...,T , with y
o
given (3.3.4)
and ç
t
being drawn from a normal distribution, ç
t
~
N
(0,o2
y
) . When the interest rate is
deterministic, this problem has an analytical solution
64
. However, if the interest rate is
stochastic, the solution of this finite horizon life cycle model becomes more complicated
and it can not be computed analytically. For now, I will not make any particular
assumption about the process generating the interest rate. Therefore, to summarize the
model, a representative consumer derives utility in period t from consuming c
t
,
discounts future utility at a rate | and wants, in period zero, to maximize his present
discounted value of future utilities for a horizon of T +1 periods. At the beginning of
each period t the consumer receives a stochastic labor income y
t
, finds out the return r
t
on his assets A
t
÷
1
, from the beginning of period t ÷1 to the beginning of period t , and, by
choosing c
t
, determines the level of assets A
t
according to equation (3.3.2).
Now, I introduce a rationally bounded agent in the following way. First, I assume
that the agent does not have either the resources or the sophistication to be able to
optimize over a long horizon. For example, if the agent enters the labor force at time zero
and faces the problem described by (3.3.1) - (3.3.4) over a time span extending until his
retirement, let it be period T , the assumption is that the agent does not have the ability to
optimally choose, at time zero, a consumption plan over that span. Instead, he focuses on
choosing a consumption plan over a shorter horizon, let it be T
h
+1 periods.
Secondly, because of his limited ability to process large amounts of information
he repeats this process every period in order to take advantage of any new available
64
See the appendix for a detailed description of the analytical solution.
83
information. This idea of a shorter and shifting optimization horizon is similar to the
approach taken by Prucha and Nadiri
65
(1984, 1986, and 1991). Now, the question is how
an individual who lacks sophistication, can optimally
66
choose a consumption plan even
for a short time span. In order to model the decision process I make use of the scenario
aggregation method. Under this assumption, the agent evaluates several possible paths
based on the realization of the forcing variables specified in the model. By assigning
probabilities to each of the possible paths, the agent is in the position to aggregate the
scenarios (paths), i.e., to compute the expected value for his decision.
In order to be able to use the scenario aggregation method, the forcing variables
need to have a discrete distribution but in the model presented above, they are described
as being drawn from a normal distribution. This leads to the third element that can be
brought under the umbrella of bounded rationality. Since the agent has limited
computational ability, the distribution of the forcing variable is approximated by a
discrete distribution with the same mean and variance as the original distribution. This
approximation does not necessarily have to be viewed as a bounded rationality element
since similar approaches have been employed repeatedly in numerical solutions using
state space discretization
67
.
Given the assumptions made about the abilities of the rationally bounded
representative agent, I will now go through the details of solving the problem described
65
In their work, a finite and shifting optimization horizon is used to approximate an
infinite horizon model.
66
67
Optimality here means the best possible solution given the level of ability.
Tauchen, among others, used this kind of approximation on various occasions, such as
Tauchen (1990), Tauchen and Hussey (1991).
84
by equations (3.3.1) - (3.3.4). Hence, at every point in time, t , the agent solves the
problem:
ma}xh
E
?
¿ ÷|
t
1 exp (÷u c
t
+
t
) | I
t
?
for
t = 0,1,...,T ÷ T
h
T
h
or
{c
t
+tT=0t
?
?
t
=0
u ?
?
(3.3.5)
ma}x
÷
t
E
?
¿ ÷|
t
1 exp (÷u c
t
+
t
) | I
t
? for t = T ÷ T
h
+1,...,T ÷1 T ÷t
subject to
{c
t
+t
T
=0t
??
t
=
0
u ?
?
(3.3.6)
A
t
+
t
=
(1+ r
t
+
t
) A
t
+t÷
1
+ y
t
+
t
÷ c
t
+
t
,
(3.3.7)
t = 0,1,...,T ÷1, t = 0,..., min (T
h
,T ÷
t
)
with A
-1
, A
t
-1, A
t
+T
h
and A
T
given (3.3.8)
where A
t
+
t
represents the level of assets at the beginning of period t +t , y
t
+
t
the labor
income at time t +t , and c
t
+
t
represents consumption in period t +t . The initial and
terminal conditions, A
-1
, A
t
÷
1
, A
t
+T
h
and A
T
are given. The information set I
t
contains the
level of consumption, assets, labor income and interest rate for period t and all previous
periods. The labor income is assumed to follow an arithmetic random walk:
y
t
+
t
= y
t
+t ÷
1
+ ç
t
b+
t
,
t = 1,...,T , t = 0,..., min (T
h
,T ÷
t
)
with y
0
given
(3.3.9)
ç
t
b+
t
being drawn from a discrete distribution,
D
(0,o2
y
) with a small number of
realizations.
In making the above assumptions, the belief is that they would better describe the
way individuals make decisions in real life. It is often the case that plans are made for
shorter horizons, but not entirely forgetting about the big picture.
85
Recalling the results of Skinner (1988) who found that saving was lower than
average for farmers and self employed, groups that are otherwise perceived to have
higher than average income uncertainty, one can assume that planning for those groups
does not follow the recipe given by the standard life cycle model. Given the high level of
uncertainty, I believe it would be more appropriate to model these consumers as if they
plan their consumption path only for a short period of time and then reevaluate. This
would be consistent with the fact that farmers change their crop on a cycle of several
years and may be influenced by the fluctuations in the commodities markets and other
government regulations. Similarly, some among the self employed are likely to have
short term contracts and are more prone to reevaluate their strategy on a high frequency
basis. Therefore, the model above seems like a good description on how the decision
making process works. The only detail that remains to be decided is how the consumer
chooses the short horizon terminal condition, that is, the level of assets, or the wealth. For
this purpose, I propose three different rules and I investigate their effect on the saving
behavior.
So far, no assumption has been made about the process governing the realizations
of the interest rate. From now on, I assume that the interest rate is also described by an
arithmetic random walk:
r
t
= r
t
÷
1
+u
t
, t = 1,...,T , with r
o
given (3.3.10)
Since in this formulation the problem does not have an analytical solution, the classical
approach would be to employ numerical methods in order to describe the path of
consumption, even for a very short horizon. In order to find the solution corresponding to
the model incorporating the bounded rationality assumption I will use the scenario
86
aggregation
68
methodology. Then I will compare this solution with the numerical
solution
69
that would result from the rational expectation version of the model when
optimizing over the whole T period horizon.
III.3.1. Rule 1
Under rule 1, the consumer considers several possible scenarios for a short
horizon and assumes that for later periods certainty equivalence holds. In this context, he
makes a decision for the current period and moves on to the next period when he
observes the realization of the forcing variables. Then he repeats the process by making a
decision based on considering all the relevant scenarios for the near future and assuming
certainty equivalence for the distant future. Hence, the decision making process takes
place every period. More precisely, when optimizing in period t , the consumer considers
all the scenarios in the tree event determined by the realizations of the forcing variable
for the first T
h
periods. From period t + T
h
he considers that certainty equivalence holds
for the remaining T ÷ t ÷ T
h
periods. This translates specifically to considering that
income and interest rate are frozen for each existing scenario for the remaining T ÷ t ÷ T
h
periods. To be more specific, for time t = 0 , the consumer considers all the scenarios
available in the event tree for the first T
h
periods and assumes certainty equivalence for
68
Since an analytical solution can be obtained when income follows an arithmetic
random walk and interest rate is deterministic, it is not necessary to discretize both
forcing variables, but only the interest rate. This approach reduces considerably the
computational burden. A short description on the methodology used along with the
solution for one scenario with deterministic, interest rate is presented in the appendix.
More details on the scenario aggregation methodology can be found in the second
chapter.
69
The numerical solution is obtained using projection methods and is due to Binder et al.
(2000).
87
the remaining T ÷ T
h
periods. When it advances to period t = 1, he optimizes again
considering all the scenarios available in the tree event for periods 1, 2,...,T
h
+1 and
assumes certainty equivalence for the remaining T ÷ T
h
÷1 periods.
In fact, this rule can be considered as an extension to the scenario aggregation
method in order to avoid the dimensionality curse. One may recall that due to its
structure, the number of scenarios in the scenario aggregation method increases
exponentially with the number of periods. In effect, this rule is limiting the number of
scenarios considered and it is consistent with a rationally bounded decision maker who
can only consider a limited and, most likely, low number of possible scenarios.
Following are some graphical representations of the simulations for rule 1. Each
graph contains the values for the coefficient of risk aversion, u . The graphs also contain
the numerical solution and, for comparison purposes, the evolution of assets if the
solution were computed in the case of certainty equivalence. I first consider a group of 12
cases varying certain parameters of the model. For all simulations in this group, the total
number of periods considered is T = 40 and the optimizing horizon is T
h
= 6 . The
starting level of income is y
0
= 200 , the initial level of assets is A
÷
1
= 500 while the
terminal value is A
T
= 1000 . The discount factor is | = 0.96 , the starting value for the
interest rate, r
0
= 0.06 while the standard deviation for the interest process is given by
o
r
= 0.0025. I use a discrete distribution with three possible realizations to approximate
the original distribution of the forcing variable and that implies that in each period t , for
t s T ÷ T
h
= 34 , the optimization process goes over 3
T
h = 729 scenarios. For periods
34 = T ÷ T
h
< t s T ÷1 = 39 the number of scenarios considered decreases to 3
T
÷
t
. The
88
parameters that are changing in the simulations are the variance for the income process
and the coefficient of risk aversion. I consider all cases obtained combining three values
for the standard deviation of income, o
y e
{1, 5, 10} and four values for the coefficient
of risk aversion, u
e
{0.005, 0.01, 0.05,
0.1
} . The results presented in this section as well
as for the rest of the chapter are based on 1000 simulations. This means that for both the
income generating process and the interest rate generating process, I consider 1000
realizations for each period. The decision to use only 1000 realizations was based on the
observation that the sample drawn provided a good representation of the arithmetic
random walk process assumed in the model. Specifically, both the mean and the standard
deviation of the sample were close to their theoretical values.
Some general results have emerged from all these simulations. First, the path for
the level of assets for the solution obtained in the bounded rationality case always lies
below the path for the level of assets for the numeric al solution obtained in the rational
expectation case. Consequently, the consumption path in the bounded rationality case
starts with values of consumption higher than in the rational expectations case.
Eventually the paths cross and the consumption level in the rational expectations case
ends up being higher toward the end of the horizon.
89
360
340
320
300
280
260
240
220
200
180
160
ConsumptionPathfor u =0.005
numerical solution
boundedrationality
360
340
320
300
280
260
240
220
200
180
160
ConsumptionPathfor u =0.01
numerical solutionbounded
rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
360
340
320
300
280
260
240
220
200
180
160
Time t (periods)
ConsumptionPathfor u =0.05
numerical solution
boundedrationality
360
340
320
300
280
260
240
220
200
180
160
Time t (periods)
ConsumptionPathfor u =0.1
numerical solutionbounded
rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 5. Consumption paths for o
y
= 1, r
0
= 0.06 and o
r
= 0.0025.
One can see in Figure 5 that consumption is increasing over time for both
solutions, with the steepest path corresponding to the lowest value of the coefficient of
risk aversion.
When looking at the asset path for the same value of the standard deviation of the
income process, one notices in Figure 6 that the level of saving in the certainty
equivalence case is mostly higher than the level of saving obtained in the bounded
rationality case as well as under the rational expectations assumption.
90
C
o
n
s
u
m
p
t
i
o
n
l
e
v
e
l
C
o
n
s
u
m
p
t
i
o
n
l
e
v
e
l
C
o
n
s
u
m
p
t
i
o
n
l
e
v
e
l
C
o
n
s
u
m
p
t
i
o
n
l
e
v
e
l
1600
1400
1200
1000
800
600
Level of Assets for u =0.005
1600
1400
1200
1000
800
600
Level of Assets for u =0.01
400
numerical solution
boundedrationality
certaintyequivalence
400
numerical solution
boundedrationality
certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
1600
1400
1200
1000
800
600
Time t (periods)
Level of Assets for u =0.05
1600
1400
1200
1000
800
600
Time t (periods)
Level of Assets for u =0.1
400
numerical solution
boundedrationality
certaintyequivalence
400
numerical solution
boundedrationality
certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 6. Asset paths for o
y
= 1, r
0
= 0.06 and o
r
= 0.0025.
While for lower levels of the coefficient of risk aversion u
e
{0.005,
0.01
} , the
asset path obtained assuming certainty equivalence crosses under the other two paths in
the later part of the horizon, the same is not true for higher values of the coefficient of
risk aversion, u
e
{0.05,
0.1
} .
It is not only the relative position of the three paths that changes in the context of
an increasing coefficient of risk aversion, but also the absolute size of the level of
savings. Moreover, the shape of the paths for both the rational expectation and bounded
rationality case changes from concave to convex.
I present now a new set of simulations with the standard deviation of income
being increased to o
y
= 5 . One can see in Figure 7 that the consumption paths for
91
A
s
s
e
t
l
e
v
e
l
A
s
s
e
t
l
e
v
e
l
A
s
s
e
t
l
e
v
e
l
A
s
s
e
t
l
e
v
e
l
u
e
{0.005,
0.01
} are not much different from those presented in Figure 5 while for
higher values of the risk aversion coefficient, u
e
{0.05,
0.1
} , the consumption paths are
steeper than in the previous case.
Looking now at the level of savings, one notices in Figure 8 a similar change to
that observed in the case of consumption. While not much has changed for the lower
values of the coefficient for risk aversion, the asset paths for higher values of the risk
aversion coefficient, u
e
{0.05,
0.1
} , have changed, effectively becoming concave, as
opposed to convex in the previous case. Besides the concavity change, one can observe
that for u = 0.1 the level of assets resulting from the numerical approximation of the
rational expectations model is higher than in the case of certainty equivalence for the
bigger part of the lifetime horizon.
360
340
320
300
280
260
240
220
200
180
160
Consumption Path for u=0.005
numerical solution
bounded rationality
360
340
320
300
280
260
240
220
200
180
160
Consumption Path for u=0.01
numerical solution
bounded rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
360
340
320
300
280
260
240
220
200
180
160
Time t (periods)
Consumption Path for u=0.05
numerical solution
bounded rationality
360
340
320
300
280
260
240
220
200
180
160
Time t (periods)
Consumption Path for u=0.1
numerical solution
bounded rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 7. Consumption paths for o
y
= 5 , r
0
= 0.06 and o
r
= 0.0025.
92
C
o
n
s
u
m
p
t
i
o
n
l
e
v
e
l
C
o
n
s
u
m
p
t
i
o
n
l
e
v
e
l
C
o
n
s
u
m
p
t
i
o
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l
e
v
e
l
C
o
n
s
u
m
p
t
i
o
n
l
e
v
e
l
1600
1400
1200
1000
800
600
Level of Assets for u=0.005
1600
1400
1200
1000
800
600
Level of Assets for u=0.01
numerical solution numerical solution
400 bounded rationality 400 bounded rationality
certainty equivalence certainty equivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
1600
1400
1200
1000
800
600
Time t (periods)
Level of Assets for u=0.05
1600
1400
1200
1000
800
600
Time t (periods)
Level of Assets for u=0.1
numerical solution numerical solution
400 bounded rationality 400 bounded rationality
certainty equivalence certainty equivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 8. Asset paths for o
y
= 5 , r
0
= 0.06 and o
r
= 0.0025.
By raising the variance of the income again, one can see in Figure 9 that the path
for consumption becomes a lot steeper for u
e
{0.05,
0.1
} . On the other hand, there
seems to be little change in the consumption pattern for u = 0.005 .
On the savings front, the level of precautionary saving increases tremendously for
the highest coefficient of risk aversion, u = 0.1, and quite substantially for u = 0.05 .
Consequently, in these two cases, the level of savings for the rational expectation model,
as well as the bounded rationality version, becomes noticeably higher than what certainty
equivalence produces. Yet, the level of savings continues to be higher for the much lower
coefficient of risk aversion, u = 0.005 , when compared with the savings pattern for
u = 0.01 and u = 0.05 .
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Consumption Path for u=0.005
numerical solution
boundedrationality
400
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Consumption Path for u=0.01
numerical solution bounded
rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
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200
150
Time t (periods)
Consumption Path for u=0.05
numerical solution
boundedrationality
400
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250
200
150
Time t (periods)
Consumption Path for u=0.1
numerical solution bounded
rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 9. Consumption paths for o
y
= 10 , r
0
= 0.06 and o
r
= 0.0025.
Another interesting observation is that if one compares the level of savings from
the panel corresponding to u = 0.05 and o
y
= 10 in Figure 10, to the level of savings
from the panel corresponding to u = 0.005 and o
y
= 1 in Figure 6, the two are almost the
same, if not the later higher. This is to say that for values of coefficient of risk aversion
and of standard deviation for income ten times as high as the ones in Figure 6, the level
of precautionary saving is almost unchanged.
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solution
boundedrationality boundedrationality400 400
certaintyequivalence certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Level of Assets for u =0.05 Level of Assets for u =0.1
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1200 1200
1000 1000
800 800
600 600
numerical solution numerical solution
400 boundedrationality 400 boundedrationality
certaintyequivalence certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 10. Asset paths for o
y
= 10 , r
0
= 0.06 and o
r
= 0.0025.
As a general observation, it seems that the level of precautionary saving derived
from the rational expectation model is consistently higher, even if not by high margins,
than the level of savings obtained in the case of bounded rationality. For consumption,
the paths can be steeper or flatter but the general allure remains the same. The rationally
bounded consumer tends to start with a higher consumption while after a few periods the
unboundedly rational consumer tends to take over and continue to consume more until
the end of the horizon.
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III.3.2. Rule 2
Under rule 2, the consumer considers all the relevant scenarios for the immediate
short horizon and then, for the later periods, he only takes in account what I call the
extreme cases. Rule 2 is similar to rule 1 in the way the decision maker emphasizes the
importance of scenarios only for the short term horizon. The difference is that under rule
2, rather than assuming certainty equivalence for the later periods, the consumer
considers the extreme case scenarios as a way of hedging against uncertainty in the
distant future. More precisely, when optimizing in period t , the consumer considers all
the scenarios in the event tree determined by the realizations of the forcing variable for
the first T
h
periods but then he becomes selective and only considers the extreme cases
70
for the remaining T ÷ t ÷ T
h
periods. To be more specific, for time t = 0 , the consumer
considers all the scenarios available in the event tree for the first T
h
periods and only the
extreme cases for the remaining T ÷ T
h
periods. When it advances to period t = 1, he
optimizes again considering all the scenarios available in the tree event for periods
1, 2,...,T
h
+1 and only the extreme cases for the remaining T ÷ T
h
÷1 periods.
In fact, this rule can also be considered as an extension to the scenario
aggregation method in an attempt order to avoid the dimensionality curse. One may recall
that due to its structure, the number of scenarios in the scenario aggregation method
increases exponentially with the number of periods. This rule is in fact limiting the
number of scenarios considered by trying to keep intact the possible variation in the
forcing variable. As opposed to rule 1 where from time t + T
h
the assumption is that the
70
The notion of extreme cases covers scenarios for which the realization of the forcing
variable remains the same. For more details see section 0 in the appendix.
96
forcing variable keeps its unconditional mean value, that is, zero, until the end of the
horizon, this rule expands the number of scenarios by adding all the extreme case
scenarios stemming from the nodes existent at time t + T
h
. This expansion can also be
seen as the equivalent of placing more weight on the tails of the original distribution of
the forcing variable. This rule is consistent with a rationally bounded decision maker who
can only consider a limited and, most likely, low number of possible scenarios but wants
to account for the variance in the forcing variable in the later periods of the optimization
horizon.
Following are some graphical representations of the simulations for rule 2. The
graphs depicting the consumption paths contain the bounded rationality solution as well
as the numerical solution. For comparison purposes, the graph panels containing the
evolution of assets display the savings pattern resulting from the solution obtained in the
case of certainty equivalence on top of the solutions for the rational expectations and the
bounded rationality models.
As in the case of rule 1, one can see in Figure 11 that consumption is increasing
over time for both solutions, with the steepest path corresponding to the lowest value of
the coefficient of risk aversion.
As opposed to the previous rule, the rationally bounded consumer does not always
start with a higher level of consumption. In fact, in this panel, for u = 0.05 and u = 0.1,
the solution of the rational expectations model has higher starting values for
consumption.
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ConsumptionPathfor u =0.005
numerical solution
boundedrationality
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ConsumptionPathfor u =0.01
numerical solution
boundedrationality
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180
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Time t (periods)
ConsumptionPathfor u =0.05
numerical solution
boundedrationality
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240
220
200
180
160
Time t (periods)
ConsumptionPathfor u =0.1
numerical solution
boundedrationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 11. Consumption paths for o
y
= 1, r
0
= 0.06 and o
r
= 0.0025.
Looking at the asset paths for the same value of the standard deviation of the
income process, one can notice in Figure 12 that the level of saving in the certainty
equivalence case is mostly higher than the level of saving obtained in the bounded
rationality case as well as under the rational expectations assumption. While for lower
levels of the coefficient of risk aversion u
e
{0.005,
0.01
} , the asset path obtained
assuming certainty equivalence crosses under the other two paths in the later part of the
horizon, the same is not true for higher values of the coefficient of risk aversion. For
u
e
{0.05,
0.1
} there is only one period, the one next to last, when the level of savings
under certainty equivalence is lower than in the other two cases.
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Level of Assets for u =0.01
numerical solution numerical solution
400 boundedrationality 400 boundedrationality
certaintyequivalence certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
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Time t (periods)
Level of Assets for u =0.05
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Time t (periods)
Level of Assets for u =0.1
numerical solution numerical solution
400 boundedrationality 400 boundedrationality
certaintyequivalence certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 12. Asset paths for o
y
= 1, r
0
= 0.06 and o
r
= 0.0025.
As it was the case with rule 1, an increase in the coefficient of risk aversion
results in a decrease of the absolute size of the level of savings. Moreover, the shape of
the paths for both the rational expectation and bounded rationality cases changes from
concave to convex. As opposed to rule 1, for u
e
{0.05,
0.1
} the level of savings under
bounded rationality is higher than under rational expectations.
The next set of simulations has the standard deviation of income increased to
o
y
= 5 . The consumption paths for u
e
{0.005,
0.01
} in Figure 13 are not much different
from those presented in Figure 11 while for higher values of the risk aversion coefficient,
u
e
{0.05,
0.1
} , the consumption paths are steeper than in the previous case.
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ConsumptionPathfor u =0.005
numerical solution
boundedrationality
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ConsumptionPathfor u =0.01
numerical solutionbounded
rationality
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Time t (periods)
ConsumptionPathfor u =0.05
numerical solution
boundedrationality
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Time t (periods)
ConsumptionPathfor u =0.1
numerical solutionbounded
rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 13. Consumption paths for o
y
= 5 , r
0
= 0.06 and o
r
= 0.0025.
For the level of savings, the change is similar to that observed in the case of
consumption. In Figure 14 one can see that, while not much has changed for the lower
values of the coefficient for risk aversion, the asset paths for higher values of the risk
aversion coefficient, u
e
{0.05,
0.1
} , have changed, effectively becoming concave, as
opposed to convex in the previous case. Besides the concavity change, one can observe
that for u = 0.1 the level of assets resulting from the numerical approximation of the
rational expectations model is higher than in the case of certainty equivalence for the
bigger part of the lifetime horizon.
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Level of Assets for u =0.01
numerical solution numerical solution
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certaintyequivalence certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
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Level of Assets for u =0.05
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Time t (periods)
Level of Assets for u =0.1
numerical solution numerical solution
400 boundedrationality 400 boundedrationality
certaintyequivalence certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 14. Asset paths for o
y
= 5 , r
0
= 0.06 and o
r
= 0.0025.
For a yet higher variance of income, one can notice in Figure 15 that the path for
consumption becomes a lot steeper for u
e
{0.05,
0.1
} . On the other hand, there seems to
be little change in the consumption pattern for u = 0.005 . On the savings front, the level
of precautionary saving increases tremendously for the highest value of the coefficient of
risk aversion considered here, u = 0.1, and quite substantially for u = 0.05 . As it can be
easily seen in Figure 16, in these two cases, the level of savings for the rational
expectation model, as well as the bounded rationality version, becomes noticeably higher
than what certainty equivalence produces.
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Consumption Path for u=0.005
numerical solution
bounded rationality
400
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250
200
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Consumption Path for u=0.01
numerical solution
bounded rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
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250
200
150
Time t (periods)
Consumption Path for u=0.05
numerical solution
bounded rationality
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250
200
150
Time t (periods)
Consumption Path for u=0.1
numerical solution
bounded rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 15. Consumption paths for o
y
= 10 , r
0
= 0.06 and o
r
= 0.0025.
Yet, the level of savings continues to be higher for the much lower coefficient of
risk aversion, u = 0.005 , when compared with the savings pattern for u = 0.01 and
u = 0.05 .
As in the case of rule 1, comparing the level of savings from the panel
corresponding to u = 0.05 and o
y
= 10 in Figure 16, to the level of savings from the
panel corresponding to u = 0.005 and o
y
= 1 in Figure 12, leads to the observation that
the two are almost the same. This is to say that for values of the coefficient of risk
aversion and of standard deviation for income ten times as high as the ones in Figure 12,
the level of precautionary saving is almost unchanged.
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certaintyequivalence certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Level of Assets for u =0.05 Level of Assets for u =0.1
1800 1800
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1000 1000
800 800
600 600
numerical solution numerical solution
400 boundedrationality 400 boundedrationality
certaintyequivalence certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 16. Asset paths for o
y
= 10 , r
0
= 0.06 and o
r
= 0.0025.
As in the case of rule 1 the level of savings under bounded rationality is fairly
close to the level of precautionary saving derived from the rational expectation model.
However, in contrast to rule 1, the relative size depends on the parameters of the model
and hence the level of precautionary saving derived from the rational expectation model
is no longer consistently higher when compared to the level of savings obtained in the
case of bounded rationality. Consequently, the rationally bounded consumer no longer
starts consistently with a higher consumption level.
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III.3.3. Rule 3
In this section, I will consider a simpler rule than the previous two, meaning that
the level of wealth A
t
+T
h
is chosen such that, given the number of periods left until time
T , a constant growth rate would insure that the final level of wealth is A
T
.
Following are some graphical representations of the simulations for rule 3. All the
graphs contain a representation of the numerical solution and, for comparison purposes,
the graphs detailing the evolution for the level of assets also contain the certainty
equivalent solution.
The simulations for rule 3 use the same values of the parameters as in the
previous two sections. Consequently, the numerical solution for the rational expectations
model exhibits the same characteristics as discussed before. Therefore, when presenting
the results in this section I will concentrate on the solution derived from assuming
bounded rationality.
As one can see in Figure 17, the consumption paths have kept their upward slope
but for lower values of the coefficient of risk aversion, the difference between the rational
expectation and bounded rationality solutions is considerably higher than for the previous
two rules. The difference can be clearly seen in the picture, with the rationally bounded
consumer consuming more in the beginning while the unboundedly rational consumers
consumes more from the 12
th
period until the end of the horizon. On the other hand, for
higher values of the coefficient of risk aversion, consumption paths are almost
indistinguishable.
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Consumption Path for u=0.005
numerical solution
boundedrationality
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Consumption Path for u=0.01
numerical solution bounded
rationality
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Time t (periods)
Consumption Path for u=0.05
numerical solution
boundedrationality
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Time t (periods)
Consumption Path for u=0.1
numerical solution bounded
rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 17. Consumption paths for o
y
= 1, r
0
= 0.06 and o
r
= 0.0025.
Looking at the asset paths in Figure 18 one will notice that, for low values of the
coefficient of risk aversion, the bounded rationality assumption leads to much lower
levels of precautionary saving than in the case of rational expectations or certainty
equivalence. However, the surprising result is that for higher values of the coefficient of
risk aversion, there is almost no difference between the level of savings under rational
expectations and bounded rationality.
By increasing the standard deviation of income to o
y
= 5 , one can see in Figure
19 a clear difference between the consumption paths for bounded rationality and rational
expectations for all levels of risk aversion. As before, the two consumption paths have an
upward slope with the rational expectation solution being the steeper one.
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Level of Assets for u =0.01
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certaintyequivalence certaintyequivalence
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Time t (periods)
Level of Assets for u =0.1
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numerical solution
boundedrationality
certaintyequivalence
400
numerical solution
boundedrationality
certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 18. Asset paths for o
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= 1, r
0
= 0.06 and o
r
= 0.0025.
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Consumption Path for u=0.005
numerical solution
bounded rationality
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Consumption Path for u=0.01
numerical solution
bounded rationality
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Time t (periods)
Consumption Path for u=0.05
numerical solution
bounded rationality
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Time t (periods)
Consumption Path for u=0.1
numerical solution
bounded rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 19. Consumption paths for o
y
= 5 , r
0
= 0.06 and o
r
= 0.0025.
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The asset paths represented in Figure 20 show clearly a higher level of
precautionary saving in the case of rational expectations. The path corresponding to
certainty equivalence produces higher levels of saving than the bounded rationality path.
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Level of Assets for u =0.01
numerical solution numerical solution
400 bounded rationality 400 bounded rationality
certainty equivalence certainty equivalence
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Time t (periods)
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numerical solution numerical solution
400 bounded rationality 400 bounded rationality
certainty equivalence certainty equivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 20. Asset paths for o
y
= 5 , r
0
= 0.06 and o
r
= 0.0025.
Increasing again the standard deviation for income to o
y
= 10 , one will notice in
Figure 21 that there is not much change in the paths for consumption at low levels of risk
aversion. However, the slope of consumption for u = 0.1 increases quite a lot.
107
A
s
s
e
t
l
e
v
e
l
A
s
s
e
t
l
e
v
e
l
A
s
s
e
t
l
e
v
e
l
A
s
s
e
t
l
e
v
e
l
400
350
300
250
200
150
Consumption Path for u=0.005
numerical solution
bounded rationality
400
350
300
250
200
150
Consumption Path for u=0.01
numerical solution bounded
rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
400
350
300
250
200
150
Time t (periods)
Consumption Path for u=0.05
numerical solution
bounded rationality
400
350
300
250
200
150
Time t (periods)
Consumption Path for u=0.1
numerical solution bounded
rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 21. Consumption paths for o
y
= 10 , r
0
= 0.06 and o
r
= 0.0025.
On the saving side, one can see in Figure 22 that for the highest coefficient of risk
aversion, the rational expectations solution provides a much higher level of savings,
while the rationally bounded consumer still saves less than in the case of certainty
equivalence for u = 0.01.
While the level of precautionary saving depends heavily on the parameter values
of the model for the unboundedly rational consumer, the same can not be said for the
rationally bounded consumer in the case of rule 3. The asset path for the rationally
bounded consumer is barely concave and increasing the variance of income does not
seem to create the same type of changes as the ones observed for the fully rational
consumer. This behavior is the result of optimizing for only short periods of time coupled
with the fact that the intermediary asset level targets are chosen assuming a constant
growth rate.
108
C
o
n
s
u
m
p
t
i
o
n
l
e
v
e
l
C
o
n
s
u
m
p
t
i
o
n
l
e
v
e
l
C
o
n
s
u
m
p
t
i
o
n
l
e
v
e
l
C
o
n
s
u
m
p
t
i
o
n
l
e
v
e
l
Level of Assets for u=0.005 Level of Assets for u=0.01
1800 1800
1600 1600
1400 1400
1200 1200
1000 1000
800 800
600 600
numerical solution numerical solution
400 boundedrationality 400 boundedrationality
certaintyequivalence certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
1800
1600
1400
1200
1000
800
600
Timet (periods)
Level of Assets for u=0.05
1800
1600
1400
1200
1000
800
600
Timet (periods)
Level of Assets for u=0.1
numerical solution numerical solution
400 boundedrationality 400 boundedrationality
certaintyequivalence certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Timet (periods) Timet (periods)
Figure 22. Asset paths for o
y
= 10 , r
0
= 0.06 and o
r
= 0.0025.
In conclusion, in the case of rule 3, the rule employed by the rationally bounded
consumer for the accumulation of assets is overshadowing the precautionary motives
embedded in the functional specification of the model.
109
A
s
s
e
t
l
e
v
e
l
A
s
s
e
t
l
e
v
e
l
A
s
s
e
t
l
e
v
e
l
A
s
s
e
t
l
e
v
e
l
III.4. Final Remarks
The level of precautionary saving under bounded rationality depends quite heavily
on the behavior assumptions. While in many of the simulations presented in this chapter
the level of precautionary saving chosen on average by the rationally bounded consumer
is below that resulting from a rational expectations model, there a few parameterizations
of the model, under rule 2, for which the rationally bounded consumer saves more.
The simulations also show that for low coefficients of risk aversion, variation in
income uncertainty does not affect much the level of saving. If one adds to this
observation the possibility that self selection exists (individuals with high risk aversion
choose occupations with low income uncertainty), it is easy to see why some empirical
studies would find relatively low levels of precautionary saving.
Another interesting result is that under rule 3, where the rationally bounded
consumer follows some form of financial planning, there is not much difference for asset
paths across various levels of risk aversion and income uncertainty. This result is
consistent with the observation made by Lusardi (1997) that the saving rates do not
change much across occupations.
Most of the studies looking to asses the importance of precautionary saving, or the
impact of income uncertainty on precautionary saving, have assumed that interest rate
uncertainty does not play an important role in the decision making process. For the model
discussed in this chapter, the assumption of a constant interest rate would result in an
asset path that is constant regardless of the realizations for the income process. By
introducing uncertainty in the interest rate process, that is no longer the case. The
dynamic of the asset path is especially influenced by the realization of the interest rate
110
process for lower levels of risk aversion. Therefore, the empirical literature should also
consider the impact of interest rate uncertainty when studying the importance of
precautionary motives on the level of saving.
While the results presented in this chapter point to an important role for the
bounded rationality in the decision making process, it would be difficult to test the
model's validity in a standard empirical setting. The problem is that the results depend
heavily on the rules adopted as well as on the parameterization of the model and it would
be difficult to distinguish between the effects of the general assumptions corresponding to
bounded rationality and those specific to a particular rule. Therefore, a more appropriate
framework for testing the validity of the model would be an experimental setting. In such
a framework, one can potentially "calibrate" the model by identifying the level of risk
aversion and the level of patience for each subject. Once these parameters are determined
it becomes easier to test hypotheses regarding the decision making process. There have
been several studies in the field of experimental economics investigating consumption
behavior under uncertainty (Hey and Dardanoni (1988), Ballinger et al. (2003) and
Carbone and Hey (2004)) that concluded that actual behavior differs significantly from
what is considered optimal. While these studies provide some insights in the decision
making process, they do not test for any particular alternative to the optimal behavior
corresponding to an unboundedly rational individual. Therefore a future area of research
is the design of an experimental framework that could test the hypotheses regarding the
decision making process advanced in this chapter.
111
Appendices
Appendix A. Technical notes to chapter 2
Appendix A1. Definitions for Scenarios, Equivalence Classes and Associated Probabilities
Suppose the world that can be described at each point in time by the vector of
state variables x
t
, and let u
t
denote the control variable while ç
t
is the forcing variable.
Suppose ç
t
is a random variable, with the underlying probability space
71
(O, E,
P
) . ç
t
is
defined as ç
t
: O C R where O is countable and finite. If the horizon has T +1 time
periods and ç
t
(
e
) is a realization of ç
t
for the event e ŒO in time period t , then the
sequence
ç
s
(
e
)
=
(ç
0
s
(
e
),ç
1
s
(
e
),K,ç
T
s
(
e
))
is called a scenario
72
. From now on, for notation simplification, I will refer to a scenario
s simply by ç
s
or by the index s and, in vector form, by ç
s
= ç
0
s
,ç
1
s
,K,ç
T
s
.
( )
Let S (e ) denote the set of all scenarios. Given that O is finite, the set S (e ) is
also finite. Therefore, one can define an event tree {N,
A
} characterized by the set of
nodes N and the set of arcs A . In this representation, the nodes of the tree are decision
points and the arcs are realizations of the forcing variables. The arcs join nodes from
71
72
O is the sample space, E is the sigma field and P is the probability measure.
Other definitions of scenarios can be found in Helgason and Wallace (1991a, 1991b)
and Rosa and Ruszczynski (1994).
112
consecutive levels such that a node n
it
at level t is linked to N
t
+
1
nodes n
k
+
1
, k = 1,..., N
t
+1 t
at level t +1.
The set of nodes N can be divided into subsets corresponding to each level
(period). Suppose that at time t there are N
t
nodes. The arcs reaching the nodes
n
it
, i = 1,K, N
t
belong each to several scenarios ç
q
(e ), q = 1,..., L
t
where L
t
represents
the number of leaves stemming from a node at level t . The bundle of scenarios that go
through one node plays a very important role in the decomposition as well as in the
aggregation process. The term equivalence class has been used in the literature to
describe the set of scenarios going through a particular node.
By definition, the equivalence class {s
t
}
i
, i = 1,K, N
t
is the set of all scenarios
having the first t +1 coordinates, ç
0
,K,ç
t
common. This means that for two scenarios
ç
j
=
(ç
0
j
,ç
1
j
,K,ç
t
j÷
1
,ç
t
j
,...,ç
T
j
) and ç
k
=
(ç
0
k
,ç
1
k
,K,ç
t
k÷
1
,ç
t
k
,...ç
T
k
) that belong to the
equivalence class
{
s
} , i =1,K, N t
i
t
the first t +1 elements are common, that is,
ç
lj
= ç
l
k for l = 0,...,t . Formally,
{
s
}
=
{ç
t
i
k
|ç
lk
= ç
l
i
for l = 0,...,t }
As mentioned in the above description of the event tree, at time t there are N
t
nodes. Then, the number of distinct equivalence classes
{
s
}
t
i
is also N
t
, that is,
i = 1,K, N
t
. Every node n
it
, i = 1,K, N
t
is associated with an equivalence
class
{s
t
}
i
.
The number of elements of the
set
{s
t
}
i
is given by the number of leaves stemming from
node i , level (stage) t .
113
Since scenarios are viewed in terms of a stochastic vector ç with stochastic
components ç
0
s
,ç
1
s
,K,ç
T
s
, it is natural to attach probabilities to each scenario. I denote
the probability of a particular realization of a scenario, s , with
p(s) = prob(ç
s
) .
These probabilities are non-negative numbers and sum to one. Formally,
p(s) > 0 and
9 p(s) =1. I assume that for each scenario ç
sŒS
s
the stochastic components
ç
0
s
,ç
1
s
,K,ç
T
s
are independent. Then
(
)
T
p(s) = prob ç
s
(
e
)
=
÷ prob ç
t
s
(
e
) ( ) (A.1.1)
t =0
Further on, I define the probabilities associated with a scenario conditional upon
belonging to a certain equivalence
class
{s
t
}
i
at time t :
p s s Πs
t
( {
}
) =
prob
(ç i
s
ç
s
Œ
{s
t
}
i
= )
p
(p{(ss)
}
), t
i
where p s
t
({
}
) i
is the probability mass of all scenarios belonging to the class
t
{
s
} .
t
i
Under the assumptions outlined above, p s
t s
({
}
)
=
÷
prob
(ç t
(
e
)) .
Therefore, the
conditional probability is easily computed as
i
t=0
p
s
{s
t
} = prob ç
s
ç
s
e
{s
t
}
i
=
(
i
)
( )
T
[
prob
(ç
t
(
e
)) s
t =t +1
The transition from the state at time t to that at time t + 1 is governed by the control
variable u
t
but is also dependent on the realization of the forcing variable, that is, on a
particular scenario s .
114
Appendix A2. Description of the Scenario Aggregation Theory
The idea is to show how a solution can be obtained by using special
decomposition methods that exploit the structure of the problem by splitting it into
manageable pieces and coordinate their solution.
Let us assume for a moment that the original problem can be decomposed into
subproblems, each corresponding to a scenario. Then the subproblems can be described
as:
T
min 9 F
t
x
s
t ,u
s
t ,
u ŒU [R
mu
( )
s ŒS
(A.2.1)
t t
t=1
where u
st
and x
st
are the control and the state variable respectively, conditional on the
realization of scenario s while S is a finite, relatively small set of scenarios.
Formally, by definition, a policy is a function or a mapping U : S ÷ R
m
assigning
to each scenario s e S a sequence of controls U (s)
=
(u
s
0 ,u
1
s
,K,u
s
t ,K,u
T
s
) , where u
st
denotes the decision to be made at time t if the scenario happens to be s . Similarly, the
state variable at each stage is associated with a particular scenario s . I use the notation
x
st
to show the link between the state variable and scenario s at time t . One can think of
T
the mappings U : S ÷ R
m
as a set of time linked mappings U
t
: S ÷ R
m
t with m
=
¿ m
t
.
t =1
The policy function has to satisfy certain constraints if two different scenarios s
and s ' are indistinguishable at time t on information available about them at time t .
Then u
s
t = u
s
t
'
, that is, a policy can not require different actions at time t relative to
scenarios s and s ' if there is no way to tell at time t which of the two scenarios will be
115
followed. This constraint is referred to as the non-anticipativity constraint. One way to
model this constraint is to introduce an information structure by bundling scenarios into
equivalence classes
73
as defined above. In this way, the scenario set S is partitioned at
each time t into a finite number of disjoint
sets,
{s
t
}
i
. Let the collection of all scenario
equivalence classes at time t be denoted by B
t
, where B
t
=
U{s
t
}
i
. In most cases i
partition B
t
+
1
is a refinement of partition B
t
, that is, every equivalence
class
{s
t
}
i
e B
t
is
a union of some equivalence classes
{
s
} t
+1
j
e B
t
+
1
. Formally,
{
s
}
t
i
=
U
{
s
} t
+1
j =1...m
i
j
.
Looking back to the event tree representation discussed in the previous section, m
i
represents the number of nodes n
t
j+
1
at level t +1 that are linked to the same node n
it
.
A policy is defined as implementable if it satisfies the non-anticipativity
constraint, that is, u
t
(e ) must be the same for all scenarios that have common past and
present
74
. In other words, a policy is implementable if for all t = 0,K,T the t
th
element
is common to all scenarios in the same class
{
s
} , i.e. if t
i
u
t
(ç
i
) = u
t
(ç
k
) whenever
{
s
}
=
{
s
}
t i t k
.
Let E be the space of all mappings U : S ÷ R
n
with components U
t
: S ÷ R
n
t .
Then the subspace
73
74
Some authors, such as Rockaffeler and Wets (1991), use the term scenario bundle.
For certain problems the non-anticipativity constraint can also be defined in terms of
the state variable, that is, x
t
(e ) must be the same for all scenarios that have common past
and present.
116
H = U e E |U
t
is constant on each class {s
t
}
i
e B
t
, for t = 1,...,T {
identifies the policies that meet the non-anticipativity constraint.
}
A policy is admissible if it always satisfies the constraints imposed by the
definition of the problem. It is clear that not all admissible policies are also
implementable. By definition, a contingent policy is the solution, u
s
, to a scenario
subproblem. It is obvious that a contingent policy is always admissible but not
necessarily implementable. Therefore, the goal is to find a policy that is both admissible
and implementable. Such a policy is referred to as a feasible policy.
One way to create a feasible policy from a set on contingent policies is to assign
weights (or probabilities) to each scenario and then blend the contingent policies
according to these weights. Specifically, if the probabilities associated with each scenario
are defined as in (A.2.1), one calculates for every period t and for every equivalence
class
{s
t
}
i
e B
t
the new policy u
t
by computing the expected value:
u
t
({
s
}
)
=
¿
}
p
(s
?
{
s
}
)u (s?)
t i
s?e
{
s
t
i
t i t
(A.2.2)
Then one defines the new policy for all scenarios s that belong to the equivalence class
{
s
} e B
t
i
t
as:
u
s
t = u
t
ˆ
({
s
}
) for all s
e
{
s
}
t i t i
(A.2.3)
Based on its definition, u
st
is implementable. The operator J :U ÷ U defined by (A.2.2)
ˆ
and (A.2.3) is called the aggregation operator.
Let us
rewrite equation
(2.4.1) as:
ˆ
117
min F
s
( x
s
,u
s
), seS (A.2.4)
u
t
eU
t
_R
mu
T
by defining the functional F
s
( x
s
,u
s
)
=
¿ F
t
( x
t
(s),u
t
(s
)
) .
t =1
Then the overall problem can be reformulated as:
min ¿ p
s
F
s
( x
s
,u
s
) over all U e E I H
seS
(A.2.5)
Let us assume for a moment that u
s
is an implementable policy obtained as in (A.2.3) ˆ
from contingent policies u
s
and u
s
is the optimal policy for the particular scenario s of
the problem described by (A.2.5). Let U and U be the collections of policies u
s
and
ˆ ˆ
u
s
respectively. One can easily see that U represents the optimal policy for the problem
described by (A.2.5). The question that the scenario aggregation methodology answers is
how to obtain the optimal solution U from a collection of implementable policies U . ˆ
Appendix A3. Solution to a Scenario Subproblem
In order to take advantage of the fact that scenario aggregation does not require
the computation of an exact solution for each scenario, I transform the Lagrangian (2.6.8)
by replacing the utility function with a first order Taylor series expansion around the
solution obtained in the previous iteration. Hence:
e
÷
uc
t
= e
÷
uc
t
s
s(k÷1)
?1÷u c
st
÷ c
s
t (k÷1
)
?
?
( )
?
From the transition equation, consumption can be expressed as:
c
s
t
=
(1+
r
) A
t
s÷
1
+ y
s
t ÷ A
t
s
118
Then
e
÷
uc
t
= e
÷
uc
t
s
s(k÷1)
{1÷u ??(1+
r
)
(A
s
and
t ÷1
÷ A
t
s÷(k÷1
)
÷ A
ts
÷ A
t
s(k÷1
)
? .For iteration
scenario s the Lagrangian becomes:
1
)(
)
}
?
(
k
)
min ¿ |
?
T
t
?e
÷
uC
s
t(k÷1) ?
?
u ?1÷
u
(1+
r
) A
t
s÷
1
÷ A
t
s÷(k÷1
)
+u A
ts
÷ A
t
s(k÷1
)
? +W
ts ?
(1+
r
) A
t
s÷
1
+ y
s
t ÷ A
ts
? +
t =0
?
?
(
1
)( )
? ? ?
+
1
µ ?(1+
r
) A
t
s÷
1
÷ A
(
t÷k1÷1
)
÷ A
ts
÷ A
(
tk÷1
)
?
2?
( )( )
?
2
}
Then, the first order condition with respect to A
t
s
is given by:
|
t
e
÷
uc
(
{
s k÷1)
t
÷W
t
s(k
)
÷ µ ?(1+
r
) A
t
s÷
1
÷ A
(
t÷k1÷1
)
÷ A
ts
÷ A
(
tk÷1
)
? +
?
( )(
)
} ?
|t+
1 ÷
(1+
r
) e
÷
uc{
s(k÷1)
t+1
+
(1+
r
)W
t
s+(1k
)
+
µ
(1+
r
)
?
(1+
r
) A
ts
÷ A
(
tk÷1
)
÷ A
t
s+
1
÷ A
(
t+k1
)
? = 0
Rearranging the terms leads to:
?
( )(
1
?
)
}
1 ?e
÷
uc
s
t(k÷1)
÷
(1+
r
) | e
÷
uc
s
t(+k1÷1) ÷W
s
(k
)
+ (1+
r
) |W
s
(k
)
? +
µ
?
?
t
t +
1
? ?
+
(1+
r
) A
(
t÷k1÷1
)
÷ A
(
tk÷1
)
÷ (1+
r
)2 | A
(
tk÷1
)
+ (1+
r
) | A
(
t+k1÷1
)
= (A.3.1)
=
(1+
r
) A
t
s÷
1
÷ A
ts ÷
(1+
r
)2 | A
ts +
(1+
r
) | A
t
s+1
Let
I
s
t (k
)
= 1 ?e
÷
uc
t
÷
(1+
r
) | e
÷
uc
t
+
1
÷W
t
s(k
)
+ (1+
r
) |W
t
s+(1k
)
?
s(k÷1) s(k÷1)
µ
?
?
Then the first order condition with respect to A
t
s
can be written as:
?
?
(A.3.2)
I
s
t (k
)
+ (1+
r
) A
(
t÷k1÷1
)
÷ ?1
+
(1+
r
)2 | ? A
(
tk÷1
)
+ (1+
r
) | A
(
t+k1÷1
)
=
? ?
=
(1+
r
) A
t
s÷
1
÷ ?1
+
(1+
r
)2 | ? A
ts +
(1+
r
) | A
t
s+1
? ?
For t = T ÷1 the first order condition becomes:
I
s
T(÷k1
)
+ (1+
r
) A
T
(k÷÷21
)
÷ ?1
+
(1+
r
)2 | ? A
T
(k÷÷11
)
+ (1+
r
) | A
T
(k÷1
)
=
? ?
(A.3.3)
=
(1+
r
) A
T
s÷
2
÷ ?1
+
(1+
r
)2 | ? A
T
s÷
1
+
(1+
r
) | A
T
? ?
119
Noting that A
T
(k÷1
)
= A
T
= A
T
equation (A.3.3) can be written as:
I
s
T(÷k1
)
+ (1+
r
) A
T
(k÷÷21
)
÷ ?1
+
(1+
r
)2 | ? A
T
(k÷÷11
)
= (1+
r
) A
T
s÷
2
÷ ?1
+
(1+
r
)2 | ? A
T
s÷1
?
Similarly, for t = 0 one obtains:
? ? ?
I
s
0(k
)
+ (1+
r
) A
(
÷k1÷1
)
÷ ?1
+
(1+
r
)2 | ? A
(
0k÷1
)
+ (1+
r
) | A
1
(k÷1
)
=
? ?
(A.3.4)
(1+
r
) A
÷
s
1
÷ ??1
+
(1+
r
)2 | ?? A
0s +
(1+
r
) | A
1
s
Again, noting that A
÷
1
is given, A
(
÷k1÷1
)
= A
÷
s
1
so equation (A.3.4) becomes:
I
s
0(k
)
÷ ?1
+
(1+
r
)2 | ? A
(
0k÷1
)
+ (1+
r
) | A
1
(k÷1
)
= ÷ ?1
+
(1+
r
)2 | ? A
0s +
(1+
r
) | A
1
s
? ? ? ?
Rewriting the system of equations in matrix form, leads to:
?÷?1
+
(1+
r
)2 |?
??
?
?
(1+
r
)
?
(1+
r
)|
÷?1
+
(1+
r
)2 |?
0 0
0
K
0
0
0
0
?
?
s
?? A
0
?
?
?
0
?
(1+
r
)
?
(1+
r
)|
÷?1
+
(1+
r
)2 |? (1+
r
) |
K
0
?? A
s
1 ?
?? s ?
? ? ? K
0
?? A
2
? =
?
M M M M M M M
?? M ?
?? s ?
?
?
??A
T
÷
1
? ?
?
?
?
0 0 0 0
K
(1+
r
)
÷?1
+
(1+
r
) |
?
?
2
??
?
I
s
0(k
)
÷?1
+
(1+
r
)2 |?
A
(
0k÷1
)
+(1+
r
) |A
(
1k÷1)
?
? ?
? ? ? ?
?I
s
(k
)
+(1+
r
) A
(
k÷1
)
÷?1
+
(1+
r
)2 |? A
(
k÷1
)
+(1+
r
) |A
(
k÷1
)
? ?1
=?
?
?
0
?
M
?1
2
?
? ? ?
?
I
T
÷
1
+
(1+
r
) A
T
÷
2
?
( )
?
T÷1
?
s(k)
(k÷1
)
÷?1+ 1+r 2 |? A
(
k÷1)
?
?
120
Appendix B. Technical notes to chapter 3
Appendix B1. Analytical Solution for a Scenario with Deterministic Interest Rate
Consider the problem described by (3.3.1) - (3.3.4). Solving the period-by-period
budget constraint (3.3.2) for c
t
, t = T ÷ 1 and t = T , and substituting back into the utility
function, the period T ÷ 1 optimization problem is given by:
e exp ÷u
(1+ r
T
÷
1
) A
T
÷
2
+ y
T
÷
1
÷ A
T
÷
1
?
max ¹÷
c
{
e
}
?÷
A
T
÷1
©
¹
u
_ exp ÷u
(1+ r
T
) A
T
÷
1
+ y
T
÷
A
T
?
(B.1.1)
{
e
}
¹
E·| u
?
I
ˆ÷ T÷
1
˜ ?
subject to
·
.
A
T
÷
1
> ÷b
+?
˜¹
(B.1.2)
Taking derivatives with respect to A
T
÷
1
, the Euler equation for (B.1.1) is given by:
exp ÷
u
(1+ r
T
÷
1
) A
T
÷
2
÷u y
T
÷
1
+u A
T
÷
1
?
e
e
¹
?
exp ÷
u
(1+ r
T
÷
1
) A
T
÷
2
÷u y
T
÷
1
÷u b? ,
= max c
e ? ÷
¹
?
(B.1.3)
¹
|
(1+ r
T
) E exp ÷
u
(1+ r
T
) A
T
÷
1
÷u y
T
+u A
T
? I
T
÷1 ©
{
e ?
}
? u
2
o
2
y
?
¹
?
Note that y
T
= y
T
÷
1
+ ç
T
while E ?
exp
(÷uç
T
) | I
T
÷
1
? = exp ?
? ?
? 2 ? and hence solving ?
? ?
(B.1.3) for the optimal wealth level at the beginning of period T ÷ 1 yields:
e*
A
= max c
¹÷b
,
(1+ r
T
÷
1
) A
T
÷
2
+ I*
T
+ A
T
?
÷
.
T ÷1
e
?¹
?
(B.1.4)
©
¹
(2 + r
T
)
¹?
where I*
T
= I + log
|
(1+ r
T
)? /u , and I = uo 2
y
/ 2 .
{
e
?}
Going now to period T ÷ 2 , the optimization problem is given by
121
? exp ÷u
?
(1+ r
T
÷
2
) A
T
÷
3
+ y
T
÷
2
÷ A
T
÷
2
?
{
?
}
{
*
?÷
? ÷ | E ? exp ÷u
?
(1+ r
T
÷
1
) A
T
÷
2
+ y
T
÷
1
÷ A
T
÷
1
? +
max ?
? ? ?
}
A
T
÷2
?
?
u
exp ?÷
u
((1+ r
T
) A
*
÷
1
+ y
T
÷ A
T
)? T
?
?
u
+| ?
?
I ?? ???
subject to
u
T ÷2
?
?
?
?
(B.1.5)
A
T
÷
2
> ÷b (B.1.6)
Taking derivatives with respect to A
T
÷
2
, and noting that
E exp ÷u A
*
÷
1
I
T
÷
2
? = exp ÷u A
*
÷
1
,
e
(
T
)
?
(
T
)
the Euler equation for (B.1.5) is given by:
exp ?÷
u
(1+ r
T
÷
2
) A
T
÷
3
÷u y
T
÷
2
+u A
T
÷
2
?
?
?
?
exp ?÷
u
(1+ r
T
÷
2
) A
T
÷
3
÷u y
T
÷
2
÷ub? ,
=
max ? ?
?
*
?
?
?
?
(
B.
1.
7)
?
?|
(1+ r
T
÷
1
) exp ?÷
u
(1+ r
T
÷
1
) A
T
÷
2
+u A
T
÷
1
? E ?exp (÷u y
T
÷
1
) I
T
÷
2
?? ?
? ? ??
Since y
T
÷
1
= y
T
÷
2
+ ç
T
÷
1
, (B.1.7) can be rewritten as:
exp ?÷
u
(1+ r
T
÷
2
) A
T
÷
3
÷u y
T
÷
2
+u A
T
÷
2
?
?
?
?
exp ?÷
u
(1+ r
T
÷
2
) A
T
÷
3
÷u y
T
÷
2
÷ub? ,
= max ?
? ?
?
?
?
*
? ?
?|
(1+ r
T
÷
1
) exp ?÷
u
(1+ r
T
÷
1
) A
T
÷
2
÷u y
T
÷
2
+u A
T
÷
1
? E ?exp (÷uç
T
÷
1
) I
T
÷
2
??
? ? ? ??
Assuming that liquidity constraint is not binding, solving (B.1.7) for A
T
÷
2
yields:
ln
|
(1+ r
T
÷
1
)? uo 2y
(1+ r
T
÷
2
) A
T
÷
3
÷ A
T
÷
2
= ÷
e u ?
+
(1+ r
T
÷
1
) A
T
÷
2
÷
A
*
÷
1
÷ T
2
(B.1.8)
Using the notation from above, equation (B.1.8) can be written as:
I*T ÷
1
= ÷ A
*
÷
1
+
(2 + r
T
÷
1
) A
T
÷
2
÷
(1+ r
T
÷
2
) A
T
÷3 T
Similarly, for period t , the equivalent of equation (B.1.9) is given by:
122
(B.1.9)
I*t+
1
= ÷ A
*
+
1
+
(2 + r
t
+
1
) A
t
÷
(1+ r
t
) A
t
÷1 t
(B.1.10)
It is clear that the optimal wealth level at the beginning of period t does not depend on
labor income received at the beginning of the period. This result is not general, but is
rather specific to the life-cycle model with a negative exponential utility function and
labor income following an arithmetic random walk process.
Solving for the beginning-of-period wealth levels from t = 0 to t = T ÷1 means
solving the system of linear equations:
_ A
0
ˆ
_
(1+ r
0
) A
÷
1
+ I*
1
ˆ
· A
1
˜ ·
I*
2
˜
· ˜ · ˜
· A
2
˜ · I*3 ˜
D·
·
M
˜ =
·
˜·
· A
T
÷
3
˜ ·
M
I*T ÷2
˜
˜
˜
(B.1.11)
·A ˜ ·
˜
· T ÷
2
˜ · I
T
÷1
*
˜
· A
T
÷1 ˜ ·
. +. A
T
+ I*
T
+˜
where D is a tridiagonal coefficient matrix,
_ 2 + r
1
÷1 0 L 0 0 0ˆ
·
÷
(1+ r
1
) 2 + r
2
÷1 L 0 0 0
˜
· ˜
D= · M M M M M M˜ (B.1.12)
·
0 0 0
· L
÷
(1+ r
T
÷
2
) 2 + r
T
÷1 ÷1
˜
˜
·0 .
0 0
L
0
÷
(1+ r
T
÷
1
)
2 + r
T
˜ +
Once the values for wealth levels are computed, the consumption levels follow.
The solution presented in this section is in fact the solution for a scenario obtained by
discretizing the distribution of the forcing variable for the interest rate. Since an
analytical solution can be obtained when income follows an arithmetic random walk and
interest rate is deterministic, it is no longer necessary to discretize both forcing variables,
but only the interest rate. This approach reduces considerably the computational burden.
123
For different labor income processes, a dual discretization is necessary, that is, for both
forcing variables.
Appendix B2. Details on the Assumptions in Rule 1
In period t the consumer wants to solve the optimization problem given by:
ma
T
x
E
_9
T ÷| t ÷
t
_ 1 ˆ
exp
÷u c |
I
?
subject to
{c
t
}
t
=t
et=t
·
u
˜
.+
(
t
) t ??
(B.2.1)
A
t =
(1+ r
t
) A
t
÷
1
+ y
t
÷ c
t
, t = t,t +1,...,T ,
(B.2.2)
with A
t
÷
1
, A
T
given t = 0,1,...,T ÷1,
y
t
= y
t
÷
1
+ ç
t
, t = t +1,...,T ,
(B.2.3)
with y
t
given t = 0,1,...,T ÷1,
r
t
= r
t
÷
1
+u
t
,
with r
t
given
t = t +1,...,T ,
t = 0,1,...,T ÷1,
(B.2.4)
The assumption is that the forcing variable u
t
has three possible realizations,
{u
a
,u
b
,u
c
} . The set of its realizations determines the event tree and consequently the set
of scenarios. For T
h
periods the number of all scenarios is 3
T
h . The consumer considers
all the possible scenarios from period t to period t + T
h
. From there on it assumes that
for every leaf the scenario will be determined by u
t
taking its unconditional mean, that
is, zero. For example, if the short optimizing horizon is given by T
h
= 4 and the sequence
of realizations for u
t
up to period t + 4 , for a particular scenario,
is
{u
a
,u
c
,u
b
,u
c
} , the
assumption made by consumer is that for this particular scenario the realizations of u
t
for
the rest of the periods will be 0 , that is, the whole scenario
is
{u
a
,u
c
,u
b
,u
c
, 0, 0,...,
0
} .
124
This process is repeated as the consumer advances to period t +1 and goes again
through the optimization procedure. The number of scenarios considered remains the
same unless T ÷ t < T
h
, which is to say that there are fewer than T
h
periods left until the
terminal period.
Appendix B3. Details on the Assumptions in Rule 2
In period t the consumer wants to solve the optimization problem given by:
ma
T
x
E
9 ÷|t÷
t
_ 1 ˆ
exp
(÷u c
t
) | I
t
? T
?
subject to
{c
t
}
t
=t
e_
t=t
·
u
˜
.+
?
(B.3.1)
A
t =
(1+ r
t
) A
t
÷
1
+ y
t
÷ c
t
, t = t,t +1,...,T ,
(B.3.2)
with A
t
÷
1
, A
T
given t = 0,1,...,T ÷1,
y
t
= y
t
÷
1
+ ç
t
, t = t +1,...,T ,
(B.3.3)
with y
t
given t = 0,1,...,T ÷1,
r
t
= r
t
÷
1
+u
t
,
with r
t
given
t = t +1,...,T ,
t = 0,1,...,T ÷1,
(B.3.4)
The assumption is that the forcing variable u
t
has three possible realizations,
{u
a
,u
b
,u
c
} . The set of its realizations determines the event tree and consequently the set
of scenarios. For T
h
periods the number of all scenarios is 3
T
h . The consumer considers
all the possible scenarios from period t to period t + T
h
. From there on it assumes that
for every leaf only three more scenarios emerge, with u
t
taking only one of the three
values
{u
a
,u
b
,u
c
} every period until the end of the horizon. For example, if the short
optimizing horizon is given by T
h
= 4 and the sequence of realizations for u
t
up to
125
period t + 4 , for a particular scenario, is {u
a
,u
c
,u
b
,u
c
} , the assumption made by
consumer is that only three more scenarios will stem from the leaf corresponding to
scenario
{u
a
,u
c
,u
b
,u
c
} . These three scenarios are given
by
{u
a
,u
c
,u
b
,u
c
,u
a
,u
a
,...,u
a
} ,
{u
a
,u
c
,u
b
,u
c
,u
b
,u
b
,...,u
b
}
and
{u
a
,u
c
,u
b
,u
c
,u
c
,u
c
,...,u
c
} . Effectively, the total number
of scenarios considered is 3
T
h +
1
as opposed to 3
T
÷
t
which would represent the total
number of scenarios for the horizon from period t to period T .
This whole process is repeated as the consumer advances to period t +1 and goes
again through the optimization procedure. The number of scenarios considered remains
the same unless T ÷ t < T
h
, which is to say that there are fewer than T
h
periods left until
the terminal period.
126
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doc_731940249.docx
Bounded rationality is the idea that in decision-making, rationality of individuals is limited by the information they have, the cognitive limitations of their minds, and the finite amount of time they have to make a decision.
CASE STUDY ON DECISION MAKING UNDER
UNCERTAINTY AND BOUNDED RATIONALITY
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 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.
ii
Table of Contents
Dedication ..................................................................................................................... ii
Table of Contents......................................................................................................... iii
Chapter I. Review of Methods Used for Solving Non-Linear Rational Expectations
Models........................................................................................................................... 1
I.1. Introduction ........................................................................................................ 1
I.2. Generic Model .................................................................................................... 3
I.3. Using Certainty Equivalence; The Extended Path Method ................................ 6
I.3.1. Example ....................................................................................................... 7
I.3.2. Notes on Certainty Equivalence Methods ................................................. 10
I.4. Local Approximation and Perturbation Methods ............................................. 11
I.4.1. Regular and General Perturbation Methods .............................................. 11
I.4.2. Example ..................................................................................................... 13
I.4.3. Flavors of Perturbation Methods ............................................................... 15
I.4.4. Alternative Local Approximation Methods............................................... 16
I.4.5. Notes on Local Approximation Methods .................................................. 18
I.5. Discrete State-Space Methods .......................................................................... 19
I.5.1. Example. Discrete State-Space Approximation Using Value-Function
Iteration ............................................................................................................... 20
I.5.2. Fredholm Equations and Numerical Quadratures ..................................... 21
I.5.3. Example. Using Quadrature Approximations ........................................... 24
I.5.4. Notes on Discrete State-Space Methods.................................................... 26
I.6. Projection Methods........................................................................................... 27
I.6.1. The Concept of Projection Methods.......................................................... 28
I.6.2. Parameterized Expectations....................................................................... 39
I.6.3. Notes on Projection Methods .................................................................... 42
I.7. Comparing Numerical Methods: Accuracy and Computational Burden.......... 44
I.8. Concluding Remarks ........................................................................................ 47
Chapter II. Using Scenario Aggregation Method to Solve a Finite Horizon Life Cycle
Model of Consumption ............................................................................................... 49
II.1. Introduction ..................................................................................................... 49
II.2. A Simple Life-Cycle Model with Precautionary Saving ................................ 50
II.3. The Concept of Scenarios ............................................................................... 52
II.3.1. The Problem ............................................................................................. 52
II.3.2. Scenarios and the Event Tree ................................................................... 53
II.4. Scenario Aggregation...................................................................................... 57
II.5. The Progressive Hedging Algorithm............................................................... 60
II.5.1. Description of the Progressive Hedging Algorithm................................. 61
II.6. Using Scenario Aggregation to Solve a Finite Horizon Life Cycle Model .... 63
II.6.1. The Algorithm .......................................................................................... 65
II.6.2. Simulation Results.................................................................................... 68
II.6.3. The Role of the Penalty Parameter........................................................... 72
II.6.4. More simulations...................................................................................... 74
II.7. Final Remarks ................................................................................................. 76
iii
Chapter III. Impact of Bounded Rationality on the Magnitude of Precautionary
Saving ......................................................................................................................... 77
III.1. Introduction.................................................................................................... 77
III.2. Empirical Results on Precautionary Saving................................................... 80
III.3. The Model...................................................................................................... 82
III.3.1. Rule 1 ...................................................................................................... 87
III.3.2. Rule 2 ...................................................................................................... 96
III.3.3. Rule 3 .................................................................................................... 104
III.4. Final Remarks .............................................................................................. 110
Appendices................................................................................................................ 112
Appendix A. Technical notes to chapter 2............................................................ 112
Appendix A1. Definitions for Scenarios, Equivalence Classes and Associated
Probabilities ...................................................................................................... 112
Appendix A2. Description of the Scenario Aggregation Theory ..................... 115
Appendix A3. Solution to a Scenario Subproblem........................................... 118
Appendix B. Technical notes to chapter 3 ............................................................ 121
Appendix B1. Analytical Solution for a Scenario with Deterministic Interest
Rate ................................................................................................................... 121
Appendix B2. Details on the Assumptions in Rule 1 ....................................... 124
Appendix B3. Details on the Assumptions in Rule 2 ....................................... 125
iv
Chapter I. Review of Methods Used for Solving Non-Linear
Rational Expectations Models
I.1. Introduction
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 categorization
1
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
solution.
1
This classification draws on Binder et al. (2000), Burnside (1999.), Marcet et al. (1999),
McGrattan (1999), Novales et al. (1999), Uhlig (1999) and Judd (1992, 1998).
1
The 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
2
.
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
3
.
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.
2
Examples include Baxter et al. (1990), Christiano (1990a, 1990b), Coleman (1990),
Tauchen (1990) and Taylor and Uhlig (1990), Tauchen and Hussey (1991), Deaton and
Laroque (1992), and Rust (1996)
3
This approach is used, for example, by Binder et al. (2000), Christiano and Fisher
(2000), Judd (1992) and Miranda and Rui (1997).
2
I.2. Generic Model
I start by presenting a generic model in discrete time that will be used along the
way to exemplify the application of some of the methods discussed in this chapter. I
assume that the problem consists of maximizing the expected present discounted value of
an objective function:
max E
?
¿ |
tt
(u
t
) O
0
? ·
subject to
u
t
?
?t =0
x
t
=
h
(x
t
÷
1
,u
t
, y
t
)
f (x
t
, x
t
÷
1
) > 0
?
?
(1.2.1)
(1.2.2)
(1.2.3)
where u
t
and x
t
denote the values of the control and state variables u and x
respectively, at the beginning of period t . y
t
is a vector of forcing variables, | e (0,1) a
constant discount factor while t represents the objective function. I further assume that
t
(
?
) is twice continuously differentiable, strictly increasing, and strictly concave with
respect to u
t
.
E
(? O
0
) denotes the mathematical expectations operator, conditional on
the information set at the beginning of period 0, O
0
. At any point in time, t , the
information set is given by O
t =
{u
t
,u
t
÷
1
,...; x
t
, x
t
÷
1
,...; y
t
, y
t
÷
1
,...}
4
. Finally, y
t
is assumed
to be generated by a first-order process
y
t
=
q
( y
t
÷
1
, z
t
) , (1.2.4)
4
The elements of the information set point to the fact that variables become known at the
beginning of the period. During the chapter this assumption may change to allow for an
easier setup of the problem.
3
where the elements of z
t
are distributed independently and identically across t and are
drawn from a distribution with a finite number of parameters.
The preceding generic optimization problem covers various examples of models
in economics, including the life-cycle model of consumption under uncertainty with or
without liquidity constraints, stochastic growth model with or without irreversible
investment and certain versions of asset pricing models. The present specification does
not cover models that have more than one control variable. However, some of the
techniques presented in this chapter could be used to solve such models.
If the underlying assumptions are such that the Bellman principle holds, one can
use the Bellman equation method to solve the dynamic programming problem. The
Bellman equation for the problem described by (1.2.1) - (1.2.2) is given by
V
( x
t
, y
t
) = muax
t
(u
t
) + | E ?
V
(
h
( x
t
,u
t
+
1
, y
t
+
1
), y
t
+
1
) | O
t
?
t
{
?
?
}
(1.2.5)
where
V
(
?
) is the value function. An alternative way to solve the model is to use the
Euler equation method. If u can be expressed as a function of x , i.e. u
t
=
g
(x
t
, x
t
÷
1
, y
t
) ,
the Euler equation for period t for the same problem is:
t'
u
?
g
( x
t
, x
t
÷
1
, y
t
)? g
'
x ( x
t
, x
t
÷
1
, y
t
) +
? ?
t
+ | E t'
u
?
g
( x
t
, x
t
+
1
, y
t
+
1
)? g
'
x
t
( x
t
, x
t
+
1
, y
t
+
1
) | O
t
= 0
{
? ?
}
(1.2.6)
So far, it has been assumed that the inequality constraint was not binding. If one
considers the possibility of constraint (1.2.3) being binding, then one must employ either
the Kuhn-Tucker method or the penalty function method. In the case of the former, the
Euler equation for period t becomes:
t'
u
?
g
( x
t
, x
t
÷
1
, y
t
)? g
'
x ( x
t
, x
t
÷
1
, y
t
) + ·
t
f
x
'
( x
t
, x
t
÷
1
) + ·
t
+
1
f
x
'
( x
t
, x
t
+
1
) +
? ?
t t t
(1.2.7)
+ | E t ?
g
( x
t
, x
t
+
1
, y
t
+
1
)? g
{
'
u
?
4
?
'
x
t
( x
t
, x
t
+
1
, y
t
+
1
) | O
t
} = 0
where ·
t
and ·
t
+
1
are Lagrange multipliers. The additional Kuhn-Tucker conditions are
given by:
·
t
> 0,
f
( x
t
, x
t
÷
1
) > 0, ·
t f
( x
t
, x
t
÷
1
) = 0 (1.2.8)
Alternatively, one can use penalty methods to account for the inequality constraint. One
approach is to modify the objective function by introducing a penalty term
5
. Then the
new objective function becomes:
E
e
9 |
t
t
t
(u
t
) + µ min
f
(x
t
, x
t
÷
1
), 0
?
O
0
÷
c
•
3
©
t =
0
e _
( )
?
?
?
?
where · is the penalty parameter. Consequently, the Bellman equation is given by:
V
( x
t
, y
t
) = muax
t
(u
t
) + · min (
f
( x
t
, x
t
÷
1
),
0
)3 + | E ?
V
(
h
( x
t
,u
t
+
1
, y
t
+
1
), y
t
+
1
) | O
t
?
t
{
?
?
}
(1.2.9)
Let u
*
t =
d
( x
t
, y
t
) denote the solution of the problem. When an analytical solution
for
d
(
?
) can not be computed, numerical techniques need to be used. Three main
approaches have been used in the literature to solve the problem (1.2.1) - (1.2.4) and to
obtain an approximation of the solution. First approach consists of modifying the
specification of the problem (1.2.1) - (1.2.2) so that it becomes easier to solve, as is the
case with the linear quadratic approximation
6
. Second approach is to employ methods
that seek to approximate the value and policy functions by using the Bellman equation
7
.
5
6
This approach is used by McGrattan (1990).
This approach has been used, among others, by Christiano (1990b) and McGrattan
(1990).
7
Examples of this approach are: Christiano (1990a), Rust (1997), Santos and Vigo
(1998), Tauchen (1990).
5
Finally, the third approach focuses on approximating certain terms appearing in the Euler
equation such as decision functions or expectations
8
.
These approaches have shaped the design of numerical algorithms used in solving
dynamic non-linear rational expectation models. In the next few sections, I will present
several of the numerical methods employed by researchers in their attempt to solve
functional equations such as the Euler and Bellman equations (1.2.5) - (1.2.9) presented
above.
I.3. Using Certainty Equivalence; The Extended Path Method
Certainty equivalence has been used especially for its convenience since it may
allow researchers to compute an analytical solution for their models. It has also been used
to compute the steady state of a model as a prerequisite for applying some linearization or
log-linearization around its equilibrium state
9
or to provide a starting point for more
complex algorithms
10
. One methodology that received a lot of attention in the literature is
the extended path method developed by Fair and Taylor (1983). Solving a model such as
(1.2.1) - (1.2.3) usually leads to a functional equation such as a Bellman or an Euler
equation.
8
Examples of this approach are Binder et al. (2000), Christiano and Fisher (2000), Judd
(1992), Marcet (1994), Mc-Grattan (1996).
9
This is the case in the linear quadratic approach where the law of motion is linearized
and the objective function is replaced by a quadratic approximation around the
deterministic steady state.
10
Certainty equivalence has also been used to provide starting values or temporary values
in algorithms used to solve models leading to nonlinear stochastic equations as in early
work by Chow (1973, 1976), Bitros and Kelejian (1976) and Prucha and Nadiri (1984).
6
Let
F x
t
, x
t
÷
1
,u
t
,u
t
÷
1
, y
t
, y
t
÷
1
, E
t
t
'
?
h
( x
t
, x
t
+
1
, y
t
+
1
)? h '
x
t
, E
t
t ?
h
( x
t
, x
t
+
1
, y
t
+
1
)? = 0
(
{
? ?
}
? ?
)
(1.3.1)
denote such a functional equation for period t . As before, x
t
is the state variable, u
t
is
the control variable, y
t
is a vector of forcing variables,
t
(
?
) is the objective function, t'
is the derivative of t with respect to the control variable, and E
t
is the conditional
expectations operator based on information available through period t . F is a function
that may be nonlinear in variables and expectations. For numerous models if the
expectations terms appearing in F were known, (1.3.1) could be easily solved. Since that
is not the case, the approach of the extended path method is to first set current and future
values of the forcing variables to their expected values. This is equivalent to assuming
that all future values of z
t
in equation (1.2.4) are zero. Then equation (1.3.1) becomes:
F x
t
, x
t
÷
1
,u
t
,u
t
÷
1
, y
t
, y
t
÷
1
,t
'
?
h
( x
t
, x
t
+
1
, E
t
y
t
+
1
)? h '
x
t
,t ?
h
( x
t
, x
t
+
1
, E
t
y
t
+
1
)? ,... = 0
(
? ? ? ?
) (1.3.2)
Then, the idea is to expand the horizon and iterate over solution paths. Let us consider an
example to see how this method can be applied.
I.3.1. Example
11
Consider the following problem where the social planner or a representative agent
maximizes an objective function
max E
e
9 |
tt
(u
t
) O
0
÷ •
u
t
c
©t=0
?
?
(1.3.3)
11
The application of the extended path method in this example draws to some extent on
the model presented in Gagnon (1990).
7
subject to
x
t
=
h
(x
t
÷
1
,u
t
, y
t
) (1.3.4)
where y
t
is a Gaussian AR (
1
) process with the law of motion y
t
= µ y
t
÷
1
+ z
t
where z
t
is i.i.d. N 0,o
2
. It is further assumed that u can be expressed as a function of x , i.e.
( )
u
t
=
g
(x
t
, x
t
÷
1
, y
t
) . Then the Euler equation for period t is:
0 = t ' ?
g
( x
t
, x
t
÷
1
, y
t
)? ? g '
x
t
( x
t
, x
t
÷
1
, y
t
)
? ?
+ | E t ' ?
g
( x
t
+
1
, x
t
, y
t
+
1
)? ? g '
x
t
( x
t
+
1
, x
t
, y
t
+
1
) O
t
{
? ?
}
(1.3.5)
If the expectation term were known in equation (1.3.5), it would be easy to find a
solution. The idea of the extended path method is to expand the horizon and then iterate
over solution paths. As in Fair and Taylor (1983), I consider the horizon t,...,t + k +1 and
assume that x
t
÷
1
and y
t
÷
1
are given and that z
t
+
s
= 0 for s = 1,..., k +1 . Following is an
algorithm that would implement the extended path methodology. The first step is to
choose initial values for x
t
+
s
and y
t
+
s
for s = 1,..., k +1 and denote them by ˆ
t
+
s
and x
yˆ
t
+
s
. Then, for period t , the Euler equation becomes:
0 = t ' ?
g
( x
t
, x
t
÷
1
, y
t
)? ? g '
x
t
( x
t
, x
t
÷
1
, y
t
)
? ?
(1.3.6)
+ |t ' ?
g
(ˆ
t
+
1
, x
t
, ˆ
t
+
1
)? ? g '
x
t
( x
t
+
1
, x
t
, ˆ
t
+
1
)
?x y?
ˆ y
Similarly, for period t + s , the Euler equation is given by:
0 = t ' ?
g
( x
t
+
s
, x
t
+s÷
1
, y
t
+
s
)? ? g '
x
t+
s
( x
t
+
s
, x
t
+s÷
1
, y
t
+
s
)
? ?
(1.3.7)
+ |t ' ?
g
(ˆ
t
+s+
1
, x
t
+
s
, ˆ
t
+s+
1
)? ? g '
x
t+
s
(ˆ
t
+
1
, x
t
+
s
, ˆ
t
+s+
1
)
In addition,
?x
y
?
x y
y
t
+
s
= µ y
t
+s÷
1
+ z
t
+s (1.3.8)
u
t
+
s
=
g
(x
t
+
s
, x
t
+s÷
1
, y
t
+
s
) (1.3.9)
8
Therefore, for period t + s , equations (1.3.7) - (1.3.9) define a system where x
t
+s÷
1
, y
t
+s÷
1
,
ˆ
t
+s+
1
, ˆ
t
+s+
1
are known so one can determine the unknowns x
t
+
s
, y
t
+
s
and u
t
+
s
. Let x
t
j+
s
,
x y
y
t
j+
s
and u
t
j+
s
denote the solutions of the system for s = 0,..., k +1 , where j represents the
iteration for a fixed horizon, in this case t,...,t + k +1. If the solutions x
t
j+s
{}
k +1
k +1
s =0
, y
t
j+s
{}
k +1
s =0
and u
t
j+s
{}
s =0
obtained in iteration j are not satisfactory then proceed with the next
iteration
where
{ˆ
t
j++s
1
}
s
=
1
=
{x
t
j+
s
}
s
=
1
,
{ˆ
t
j++s
1
}
s
=
1
=
{y
t
j+
s
}
s
=
1
. Notice that the horizon remains
k +1 k +1 k +1 k +1
x y
the same for iteration j +1. The iterations will continue until a satisfactory solution is
obtained. At this point, the methodology calls for the extension of the horizon without
modifying the starting period. Fair and Taylor extend the horizon by a number of periods
that is limited to the number of endogenous variables. This is in essence an ad-hoc rule.
In the present example, the horizon is extended by 2 periods, that is, t,...,t + k + 3. The
same steps are followed for the new horizon with the exception of the end criterion,
which should consist of a comparison between the last obtained solution, using the
t,...,t + k + 3 horizon, and the solution provided using the previous horizon, t,...,t + k +1.
The expansion of the horizon continues until a satisfactory solution is obtained. At that
point, the procedure will start over with a new starting period and a new horizon. In our
example the next starting period should be t +1 and the initial horizon t +1,...,t + k + 2 .
One of the less mentioned caveats of this method is that no general convergence
proofs for the algorithm are available. In addition, the method relies on the certainty
equivalence assumption even though the model is nonlinear. Since expectations of
functions are treated as functions of the expectations in future periods in equation (1.3.2),
9
the solution is only approximate unless function F is linear. This assumption is similar
to the one used in the case of linear-quadratic approximation to rational expectations
models that has been proposed, for example, by Kydland and Prescott (1982).
In the spirit of Fair and Taylor, Fuhrer and Bleakley (1996), following an
algorithm from an unpublished paper by Anderson and Moore (1986), sketch a
methodology for finding the solution for nonlinear dynamic rational expectations models.
I.3.2. Notes on Certainty Equivalence Methods
All the methods that use certainty equivalence either as a main step or as a
preliminary step in finding a solution, incur an approximation error due to the assumption
of perfect foresight. The magnitude of this error depends on the degree of nonlinearity of
the model being solved. Fair (2003), while acknowledging its limitations, argues that the
use of certainty equivalence may provide good approximations for many
macroeconometric models.
In the case of the extended path algorithm, the error propagates through each level
of iteration and therefore it forces the use of strong convergence criteria. Due to this fact,
the extended path algorithm tends to be computationally intensive. Other methodologies
that only use certainty equivalence as a preliminary step as in the case of linearization
methods or linear quadratic approaches are not subject to the same computational burden.
In conclusion, while there are cases where certainty equivalence may be used to
obtain good approximations, one needs to be careful when using this methodology since
there are no guarantees when it comes to accuracy.
10
I.4.Local Approximation and Perturbation Methods
Economic modeling problems have used a variety of approximation methods in
the absence of a closed form solution. One of the most used approximation methods,
coming in different flavors, is the local approximation. In particular, the first order
approximation has been extensively used in economic modeling. Formally, a function
a(x) is a first order approximation of b(x) around x
0
if a(x
0
) = b(x
0
) and the
derivatives at x
0
are the same, a '(x
0
) = b '(x
0
) . In certain instances, first order
approximations may not be enough so one would have to compute higher order
approximations. Perturbation methods often use high order local approximation and
therefore rely heavily on two very well own theorems, Taylor's theorem and implicit
function theorem.
I.4.1. Regular and General Perturbation Methods
Perturbation methods are formally addressed by Judd (1998). In this section,
following Judd's framework, I try to highlight the basic idea of regular perturbation
methods. I start by assuming that the Euler equation of the model under consideration is
given by:
F
( u,
c
) = 0 (1.4.1)
where
u
(
c
) is the policy I want to solve for and c is a parameter. Further on, I assume
that a solution to (1.4.1) exists, that F is differentiable,
u
(
c
) is a smooth function and
u
(
0
) can be easily determined or is known. Differentiating equation (1.4.1) leads to:
F
u
(
u
(
c
),
c
)u
'
(
c
) + F
c
(
u
(
c
),
c
) = 0 (1.4.2)
11
Making c = 0 in equation (1.4.2) allows one to compute u
'
(
0
) :
F
c
(
u
(
0
),
0
)
u
'
(
0
) = ÷
F
u
(
u
(
0
),
0
)
(1.4.3)
The necessary condition for the computation of u
'
(
0
) is that F
u
(
u
(
0
),
0
) = 0 . Assuming
that indeed F
u
(
u
(
0
),
0
) = 0 , it means that now u
'
(
0
) is known and one can compute the
first order Taylor expansion, of
u
(
c
) around c = 0 :
F
c
(
u
(
0
),
0
)
u
(
c
) ?
u
(
0
) ÷
F
u
(
u
(
0
),
0
)
c
(1.4.4)
This is a linear approximation of
u
(
c
) around c = 0 . In order to be able to compute
higher order approximations of
u
(
c
) one needs to know at least the value of u
''
(
0
) . That
can be found by differentiating (1.4.2):
u
''
(
0
) = ÷
F
uu
(
u
(
0
),
0
)(u
'
(
0
))2 + 2F
u
c
(
u
(
0
),
0
)u
'
(
0
) + F
cc
(
u
(
0
),
0
)
F
u
(
u
(
0
),
0
)
(1.4.5)
The necessary condition for the computation of u
''
(
0
) is, once again, that
F
u
(
u
(
0
),
0
) = 0 . In addition, second order derivatives shall exist. Then the second order
approximation of
u
(
c
) around c = 0 is given by:
u
(
c
) ?
u
(
0
) ÷
F
c
(
u
(
0
),
0
)
c÷
1
c
2
F
uu
(
u
(
0
),
0
)(u
'
(
0
)) + 2F
u
c
(
u
(
0
),
0
)u
'
(
0
) + F
cc
(
u
(
0
),
0
) 2
F
u
(
u
(
0
),
0
)
2
F
u
(
u
(
0
),
0
)
In general, higher order approximations of
u
(
c
) can be computed if higher
derivatives of
F
(u,
c
) with respect with u exist and if F
u
(
u
(
0
),
0
) = 0 . The advantage
of regular perturbation methods based on an implicit function formulation is that one
12
directly computes the Taylor expansions in terms of whatever variables one wants to use,
and that expansion is the best possible asymptotically.
I.4.2. Example
Consider the following optimization problem
max E
e
9 |
tt
(u
t
) | O
0
÷ •
subject to
u
t
c
©t=0
x
t
=
h
(x
t
÷
1
,u
t
÷
1
, y
t
)
?
?
(1.4.6)
(1.4.7)
with y
t
= y
t
÷
1
+ c z
t
, where u
t
is the control variable, x
t
is the state variable, c is a scalar
parameter and z
t
is a stochastic variable drawn from a distribution with zero mean and
unit variance. x
t
, u
t
, c and z
t
are all scalars. The Bellman equation is given by:
V (x
t
) = muax
t
(u
t
) + | E ?V (
h
( x
t
,u
t
+
1
,c z
t
+
1
)) | O
t
?
t
{
?
?
}
(1.4.8)
Then the first order condition is:
0 = t
u
(u
t
) + | E ?V
'
(
h
( x
t
,u
t
,c z
t
+
1
)) h
u
( x
t
,u
t
,c z
t
+
1
)?
?
Differentiating the Bellman equation with respect to x
t
, one obtains:
?
(1.4.9)
V
'
(x
t
) = | E ?V
'
(
h
( x
t
,u
t
,c z
t
+
1
)) h
x
( x
t
,u
t
,c z
t
+
1
)?
? ?
(1.4.10)
Let the control law
U
( x,
c
) be the solution of this problem. Then the above equation
becomes:
V
'
(x) = | E
?
V
'
h ( x,
U
( x,
c
),c
z
) h
x
?
?
( )
?
The idea is to first solve for steady state in the deterministic case, which here is
equivalent to c = 0 , and then find a Taylor expansion for
U
( x,
c
) around c = 0 .
13
Assuming that there exists a steady state defined by (x
*
,u
*
) such that x
*
= h x
*
,u
*
, one
( )
can use the following system to obtain steady state solutions:
x
*
= h x
*
, u
*
( )
(1.4.11)
0 = t
u
u
*
+ |V ' h x
*
,u
*
h
u
x
*
,u
*
() (( )) ( ) (1.4.12)
V ' x
*
= |V ' h x
*
,u
*
h
x
x
*
,u
*
() (( )) ( ) (1.4.13)
V x
*
= t u
*
+ | V x
*
() () () (1.4.14)
Further assuming local uniqueness and stability for the steady state, equations (1.4.11)-
(1.4.14) provide the solutions for the four steady state quantities x
*
, u
*
, V x
*
, and ()
V ' x
*
. Given that the time subscript for all variables is the same, I drop it for the ()
moment. Going back to equations (1.4.9) - (1.4.10), in the deterministic case, that is, for
c = 0 , one obtains:
0 = t
u
(
U
(
x
)) + |V ' ?h ( x,
U
(
x
))? h
u
( x,
U
(
x
))
? ?
(1.4.15)
V
'
(x) = |V
'
?
h
( x,
U
(
x
))? h
x
( x,
U
(
x
))
? ?
(1.4.16)
Differentiating (1.4.15) and (1.4.16) with respect to x yields
0 = t
uu
U
'
x + |V
"
(
h
)(h
x
+ h
u
U
'
x ) h
u
+ |V
'
(
h
)(h
ux
+ h
uu
U
'
x ) (1.4.17)
V " = |V
"
(
h
)(h
x
+ h
u
U
'
x ) h
x
+ |V
'
(
h
)(h
xx
+ h
xu
U
'
x ) (1.4.18)
Therefore, the steady state version of the system (1.4.17) - (1.4.18) is given by:
0 = t
uu
x
*
,u
*
U
'
x x
*
+ |V " x
*
h
x
x
*
,u
*
( ) () () ( ) e
(1.4.19)
+h
u
x ,u U
x
x
?
h
u x ,u + |V '
x
e
h
ux x ,u + h
uu
x ,u U
x
x
?
(
* *
) () (
'
*
?
* *
) () (
*
* *
) (
* *
) ()
V " x
*
= |V " x
*
h
x
x
*
,u
*
+h
u
x
*
,u
*
U
'
x x
*
? h
x
x
*
,u
*
'
*
?
() () ( ) ( ) () ( )
e ?
(1.4.20)
+|V ' x
*
h
xx
x
*
,u
*
+ h
xu
x
*
,u
*
U
'
x x
*
?
()
e
( ) ( ) ()
?
These equations define a quadratic system for the unknowns V "(x ) and
U
(
x
) .
* ' *
x
14
Going back to the stochastic case, the first order condition with respect to u is given by:
0 = t
u
(
U
( x,
c
)) + | E V '
h
( x,
U
( x,
c
),c z
t
+
1
) h
u
( x,
U
( x,
c
),c z
t
+
1
) | O
t
(1.4.21)
{( ) }
Taking the derivative of the Bellman equation with respect to x yields:
V
'
(x) = | E V
'
h (x,
U
(x,
c
),c z
t
+
1
) h
x
(x,
U
(x,
c
),c z
t
+
1
) | O
t
{( ) } (1.4.22)
In order to obtain a local approximation of the control law around c = 0 , its derivatives
with respect to c must exist and be known. To find these values one needs to
differentiate equations (1.4.21) - (1.4.22) with respect to c , make c = 0 and solve the
resulting system for the values of the derivatives of U with respect to c when c = 0 ,
i.e., for U
c
'
( x
*
,
0
) . Once that value is found, one can compute a Taylor expansion for
U
( x,
c
)
around
( x
*
,
0
) .
If the model requires the addition of an inequality constraint such as (1.2.3) which
could be the representation of a liquidity constraint or a gross investment constraint, the
Bellman equation (1.4.8) becomes:
V (x
t
) = muax
t
(u
t
) + · min (
f
( x
t
, x
t
÷
1
),
0
)3 + | E ?
V
(
h
( x
t
,u
t
,c z
t
)) | O
t
? (1.4.23)
t
{
?
?}
where · is the penalty parameter.
I.4.3. Flavors of Perturbation Methods
Economic modeling problems have used a variety of approximation methods that
may be characterized as perturbation methods. The most common use of perturbation
methods is the method of linearization around the steady state. Such linearization
provides a description on how a dynamical system evolves near its steady state. It has
often been used to compute the reaction of a system to shocks. While the first-order
15
perturbation method exactly corresponds to the solution obtained by standard
linearization of first-order conditions, one well known drawback of such a solution,
especially in the case of asset pricing models, is that it does not take advantage of any
piece of information contained in the distribution of the shocks. Collard and Juillard
(2001) use higher order perturbation methods and apply a fixed-point algorithm, which
they call "bias reduction procedure", to capture the fact that the policy function depends
on the variance of the underlying shocks. Similarly, Schmitt-Grohé and Uribe (2004)
derive a second-order approximation to the policy function of a general class of dynamic,
discrete-time, rational expectations models using a perturbation method that incorporates
a scale parameter for the standard deviations of the exogenous shocks as an argument of
the policy function.
I.4.4. Alternative Local Approximation Methods
There are also certain local approximations techniques used in the literature that
may look like perturbation methods when in fact they are not. One frequently used
approach is to find the deterministic steady state and then to replace the original nonlinear
problem with a linear-quadratic problem that is similar to the original problem. The
linear-quadratic problem can then be solved using standard methods. This method differs
from the perturbation method in that the idea here is to replace the nonlinear problem
with a linear-quadratic problem, whereas the perturbation approach focuses on computing
derivatives of the nonlinear problem. Let me consider again the problem defined by
equations (1.2.1) - (1.2.2). The idea is to approximate the original problem by a
16
combination of a quadratic objective and a linear constraint, which would take the
following form:
max E ?
¿
( ? · |
t
Q +Wu + Ru
2
| O ?
u
t
?t =0
t
t
) 0
?
?
(1.4.24)
s.t. x
t
= Ax
t
÷
1
+ Bu
t
+ Cy
t
+ D (1.4.25)
where Q, R, W , A, B, C and D are scalars.
In order to obtain the new specification, the first step is to compute the steady
state for the deterministic problem (which means z
t
= 0 in equation (1.2.4)). Therefore,
one has to formulate the Lagrangian:
·
L
=
¿ |t {
t
(u
t
) ÷ ì
t
?x
t
÷
h
( x
t
÷
1
,u
t
, y
0
)
?
}
t =0
? ?
(1.4.26)
The first order conditions for (1.4.26) is a system of 3 equations with unknowns
x,u and ì . The solution of the system represents the steady
state,
( x
*
,u
*
, ì
*
) . The next
step is to take the second order Taylor expansion for
t
(u
t
) and first order Taylor
expansion for
h
(x
t
÷
1
,u
t
, y
t
)
around
( x
*
,u
*
, y
0
) . Thus,
t
(
u
) =
t
(
u
) + t
'
(
u
) (u ÷
u
) + t
"
(
u
) (
*2
t * * t * *
u ÷
u
) t
2
(1.4.27)
h
( x
t
÷
1
,u
t
, y
t
) =
h
( x
*
,u
*
, y
0
) + h
'
x ( x
*
,u
*
, y
0
)( x
t
÷
1
÷ x
*
) +
(1.4.28)
+ h
u
'
( x
*
,u
*
, y
0
)(u
t
÷ u
*
) + h
'
y ( x
*
,u
*
, y
0
)( y
t
÷ y
0
)
These expansions allow one to identify the parameters Q, R, W , A, B, C and D .
Specifically,
*2
Q =
t
( u
*
) ÷ t
'
( u
*
) u
*
+ t
"
(
u
*
)
u
2
t
"
(
u
*
)
(1.4.29)
W = t
'
( u
*
) ÷ t
"
( u
*
)
u
*
17
R=
2
A = h
'
x ( x
*
,u
*
, y
0
) B = h
u
'
( x
*
,u
*
, y
0
) C = h
'
y ( x
*
,u
*
, y
0
)
(1.4.30)
D =
h
( x
*
,u
*
, y
0
) ÷ h
'
x ( x
*
,u
*
, y
0
) x
*
÷ h
u
'
( x
*
,u
*
, y
0
)u
*
÷ h
'
y ( x
*
,u
*
, y
0
) y
0
Once the parameters have been identified, the problem can be written in the form
described by (1.4.24) and (1.4.25) which has a quadratic objective function and linear
constraints
12
.
If the model needs to account for an additional inequality constraint such as
(1.2.3), the Lagrangian (1.4.26) becomes
L
=
¿ |t {
t
(u
t
) ÷ ì
t
?x
t
÷
h
( x
t
÷
1
,u
t
, y
0
)? + ·
t f
( x
t
, x
t
÷
1
)
} ·
t =0
? ?
(1.4.31)
and the additional Kuhn-Tucker conditions have to be taken into account.
I.4.5. Notes on Local Approximation Methods
The perturbation methods provide a good alternative for dealing with the major
drawback of the method of linearization around steady state, that is, its lack of accuracy
in the case of high volatility of shocks or high curvature of the objective function. While
the first order perturbation method coincides with the standard linearization, the higher
order perturbation methods offer a much higher accuracy
13
.
Some of the local approximation implementations such as the linear-quadratic
method
14
do fairly well when it comes to modeling movements of quantities, but not as
12
There are some other variations of this approach used in the literature such as
Christiano (1990b).
13
See Collard and Juillard (2001) for a study on the accuracy of perturbation methods in
the case of an asset-pricing model.
14
Dotsey and Mao (1992), Christiano (1990b) and McGrattan (1990) have documented
the quality of some implementations of the macroeconomic linear-quadratic approach.
18
well with asset prices. The reason behind this result is that approximation of quantity
movements depends only on linear-quadratic terms whereas asset-pricing movements are
more likely to involve higher-order terms.
I.5. Discrete State-Space Methods
15
These methods can be applied in several situations. In the case where the state
space of the model is given by a finite set of discrete points these methods may provide
an "exact" solution
16
. In addition, these methods are frequently applied by discretizing an
otherwise continuous state space. The use of discrete state-space methods in models with
a continuous state space is based on the result
17
that the fixed point of a discretized
dynamic programming problem may converge point wise to its continuous equivalent
18
.
The discrete state-space methods sometimes prove to be a useful alternative to
linearization and log-linear approximations to the first order necessary conditions,
especially for certain model specifications.
15
16
This section draws heavily on Burnside (1999) and on Tauchen and Hussey (1991)
This may be the case in models without endogenous state variables, especially when
there is only one state variable that follows a simple finite state process. Examples are
Mehra and Prescott (1985) and Cecchetti, Lam and Mark (1993).
17
As documented in Burnside (1999), Atkinson (1976) and Baker (1977) present
convergence results related to the use of discrete state spaces to solve integral equations.
Results concerning pointwise and absolute convergence of solutions to asset pricing
models obtained using discrete state spaces are presented in Tauchen and Hussey (1991)
and Burnside (1993).
18
The procedure employed by discrete state-space methods in models with a continuous
state space is sometimes referred to as 'brute force discretization'.
19
I.5.1. Example. Discrete State-Space Approximation Using Value-Function Iteration
As before, I consider the following maximization problem:
max E
?
¿ |
tt
(u
t
) | O
0
? ·
subject to
u
t
?
?t =0
x
t
+
1
=
h
(x
t
,u
t
, y
t
)
?
?
(1.5.1)
(1.5.2)
where y
t
is a realization from an n -state Markov chain, u
t
is the control variable and x
t
is the state variable. Let Y
=
{Y
1
, Y
2
,..., Y
n
} be the set of all possible realizations for y
t
.
In order to be able to apply the above mentioned methodology one has to establish a grid
for the state variable. Let the ordered set X
=
{X
1
, X
2
,..., X
k
} be the grid for x
t
.
Assuming that the control variable u
t
can be explicitly determined from equation (1.5.2)
as a function of x
t
, x
t
+
1
and y
t
, then the dynamic programming problem can be
expressed as:
V (x
t
, y
t
) = max
t
( x
t
, x
t
+
1
, y
t
) + | E ?
V
( x
t
+
1
, y
t
+
1
) | O
t
?
x eXt+1
{
?
?
}
(1.5.3)
Let H(x
t
, y
t
) be the Cartesian product of Y and X , that is, the set of all possible
m = n ? k pairs (x
i
, y
j
) . Formally, H(x
t
, y
t
) = (x
i
, y
j
) | x
i
ŒX ¸ ÷
k
and y
j
ŒY ¸ ÷
n
.
{ }
Hence H(x
t
, y
t
) ¸ ÷
k
· ÷
n
= ÷
m
. If equation (1.5.3) is discretized using the grid given
by H(x
t
, y
t
) one can think of function
V
(
?
) as a point in ÷
m
. Similarly, the expression
t
(x
t
, x
t
+
1
, y
t
) + |
E
(
V
(x
t
+
1
, y
t
+
1
) | O
t
) can be thought of as a mapping M from ÷
m
into
÷
m
. In this context
V
(
?
) is a fixed point for M , that is, V =
M
(
V
) . One of the methods
commonly used to solve for the fixed point in these situations is the value function
iteration.
20
In order to solve the maximization problem one can use various algorithms. The
algorithm I am going to present follows, to some degree, Christiano (1990a). Let
S
j
(X
p
, Y
q
) be the value of x
t
+
1
that maximizes
M
(V
j
) for given values of x
t
and y
t
,
( x
t
, y
t
)
=
(X
p
,Y
q
) c H . Formally,
S
t
j+
1
(X
p
, Y
q
) = arg max
t
(X
p
, x
t
+
1
, Y
q
) + | E ?V
j
( x
t
+
1
, y
t
+
1
) | O
t
?
x
t
+
1
eX
{
?
?
}
(1.5.4)
where j represents the iteration. The idea is to go through all the possible values for x
t
+
1
,
that is, the set X , and find the value that maximizes the right hand side of (1.5.4). That
will become the value assigned to S
j
(X
p
, Y
q
) . Then the procedure will be repeated for a
different value of the pair x
t
and y
t
belonging to set H(x
t
, y
t
) and, finally, a global
maximum will be found. The exposition of the algorithm so far implies an exhaustive
search of the grid. The speed of the algorithm can be improved by choosing a starting
point for the search in every iteration and continue the search only until the first decrease
in the value function is encountered
19
. The decision rule for u
t
can then be derived by
substituting S
t
+
1
for x
t
+
1
in the law of motion.
I.5.2. Fredholm Equations and Numerical Quadratures
Let me consider the model specified by (1.2.1) - (1.2.2). Then the Bellman
equation is given by:
V
( x
t
, y
t
) = muax
t
(u
t
) + | E ?
V
( x
t
+
1
, y
t
+
1
) | O
t
?
t
{
?
?
}
(1.5.5)
19
T his change in the algorithm, as presented by Christiano (1990a), is valid only when
the value function is globally concave.
21
If y
t
follows a process such as (1.2.4), one can rewrite the conditional expectation and
consequently the whole equation (1.5.5) as:
V
( x
t
, y
t
) = muax
t
(u
t
) +
|
}
V
( x
t
+
1
, y
t
+
1
)
q
( y
t
+
1
| y
t
) dy
t
+1{t
}
(1.5.6)
In the above equation, the term needing approximation is the integral
}
V
( x t +1
, y
t
+
1
)
q
( y
t
+
1
| y
t
) dy
t
+1
If V (x
t
+
1
, y
t
+
1
) is continuous in y
t
+
1
for every x , the integral can be replaced by an N-
point quadrature approximation. An N-point quadrature method is based on the notion
that one can find some points y
i
,
N
and some weights w
i
,
N
in order to obtain the following
approximation
N
¿
V
( x t +1
i=1
, y
i
.
N
) w
i
,
N
~
}
Y
V
( x
t
+
1
, y
t
+
1
)
q
( y
t
+
1
| y
t
) dy
t
+1
(1.5.7)
where the points y
i
,
N
eY ,i = 1,K, N , are chosen according to some rule, while the weight
given to each point, w
i
,
N
, relates to the density function
q
(
y
) in the neighborhood of
those points. In general, a quadrature method requires a rule for choosing the points, y
i
,
N
,
and a rule for choosing the weights, w
i
,
N
. The abscissa y
i
,
N
and weights w
i
,
N
depend only
on the density
q
(
y
) , and not directly on the function V .
Quadrature methods differ in their choice of nodes and weights. Possible choices
are Newton-Cotes, Gauss, Gauss-Legendre and Gauss-Hermite approximations. For a
classical N-point Gauss rule along the real line, the abscissa y
i
,
N
and weights w
i
,
N
are
determined by forcing the rule to be exact for all polynomials of degree less than or equal
to 2N ÷1.
22
For most rational expectation models, integral equations are a very common
occurrence both in Bellman equations such as (1.5.6), as well as in Euler equations. One
of the most common forms of integral equations mentioned in the literature is the
Fredholm equation
20
. Therefore, in this section I will present an algorithm similar to the
one used by Tauchen and Hussey (1991) for solving such equation.
Now let me assume for a moment that the Euler equation of the model is given by
a Fredholm equation of the second kind:
v( y
t
)
=
v¢ ( y
t
+
1
, y
t
)v( y
t
+
1
)q( y
t
+
1
y
t
)dy
t
+
1
+¸ ( y
t
) (1.5.8)
where y
t
is an n-dimensional vector of variables, E
t
is the conditional expectations
operator based on information available through period t , and
¢
( y
t
, y
t
+
1
) and
¸
( y
t
) are
functions of y
t
and y
t
+
1
that depend upon the specific structure of the economic model,
and where
v
( y
t
) is the solution function of the model. The
process
{y
t
} is characterized
by a conditional density, q( y
t
+
1
y
t
) .
Following Tauchen and Hussey (1991), let the T[?] operator define the integral
term in equation (1.5.8). Then (1.5.8) can be written as:
v = T [v ] + ¸ (1.5.9)
Under regularity conditions, the operator [I ÷ T ]
÷
1
exists, where I denotes the identity
operator, and the exact solution is:
v = [I ÷ T ]
÷
1¸ (1.5.10)
20
One example where this form of integral equation appears is a version of the asset
pricing model. See Tauchen and Hussey (1991) and Burnside (1999) for more details.
23
An approximate solution is obtained using T
N
in place of T , where T
N
is an
approximation of T using quadrature methods for large N . Then [I ÷ T
N
] can be
inverted.
v
N
= [I ÷ T
N
]
÷
1¸ (1.5.11)
In some cases, the function ¸ is of the form ¸ = T[¸
0
] and then the approximate solution
is taken as [I ÷ T
N
]
÷
1
T
N
[¸
0
] .
I.5.3. Example. Using Quadrature Approximations
This is an example of discrete state-space approximation using quadrature
approximations and value-function iterations. I consider a similar model to the one
described in section I.5.1 with the difference that y
t
is a Gaussian
AR
(
1
) process as
opposed to a Markov chain. Again, the representative agent solves the following
optimization problem
max
E
?
¿ |
tt
(u
t
) O
0
? ·
subject to
u
t
?
?t =0
x
t
+
1
=
h
(x
t
,u
t
, y
t
)
?
?
(1.5.12)
(1.5.13)
where y
t
is a Gaussian
AR
(
1
) process with the law of motion y
t
= µ y
t
÷
1
+ z
t
where z
t
is i.i.d. N 0,o
2
. I assume that u
t
can be expressed as a function of x , i.e.
( )
u
t
=
g
(x
t
, x
t
+
1
, y
t
) . Then the Bellman equation for the dynamic programming is given by
V
( x
t
, y
t
) = mxax
t
(
g
( x
t
, x
t
+
1
, y
t
)) + |
E
{
V
( x
t
+
1
, y
t
+
1
) O
t
+
1
} t+1
(1.5.14)
24
Writing the expectation term explicitly, equation (1.5.14) becomes:
V
( x
t
, y
t
) = mxax
t
(
g
( x
t
, x
t
+
1
, y
t
)) +
|
}
V
( x
t
+
1
, y
+
1
)
f
( y
t
+
1
y
t
) dy
t
+1t+1
(1.5.15)
where
y
t
+
1
= µ y
t
+ z
t
+1 (1.5.16)
To convert the dynamic programming problem in (1.5.15) to one involving
discrete state spaces one needs first to approximate the law of motion of y
t
using a
discrete state-space process. That is, redefine y
t
to be a process which lies in a set
Y = y
i
,N
{}
N
with y
i
,
N
= o a
i
,
N
, where
{
a
}
N
is the set of quadrature points
i=1
i,N
i=1
corresponding to an N-point rule for a standard normal distribution
21
. Let the probability
that y
t
+
1
= y
i
,
N
conditional on y
t
= y
j
,
N
be given by
f y
i
,
N
y
j
,
N
w
i
,N
p
ji
=
(
f y
i
,
N
0 (
) )
s
j
(1.5.17)
where
N
s
j
=
9
f y
i
,
N
y
j
,N (
)
w
(1.5.18)
i=1
f y
i
,
N
0 ( )
i,N
and {w
i
,
N
}N=
1
are the quadrature weights as described in section I.5.2.. With this i
approximation, the Bellman equation can be written as:
V
( x
t
, y
t
) =
mxax ?
?
t g x , x , y + |N
V
( x ,
y
)
p
?
t+
1
?
(
( t t+1
j
) ¿1t+1iji ?? )
i=
(1.5.19)
given y
t
= y
j
, j = 1,..., N .
21
This is in fact the approach used by Tauchen and Hussey (1991) and Burnside (1999),
among others.
25
The next step is to replace the state space by a discrete domain X from which the
solution is chosen. There is no universal recipe for choosing a discrete domain and
therefore it is usually done on a priori knowledge of possible values of the state
variable
22
. The maximization problem can now be solved by value function iteration as
presented in section I.5.1..
I.5.4. Notes on Discrete State-Space Methods
Discrete state-space methods tend to work well for models with a low number of
state variables. As the number of variables increases, this approach becomes numerically
intractable, suffering from what the literature usually refers to as the curse of
dimensionality. In addition, as pointed out in Baxter et al. (1990), when the method is
used to solve continuous models there are two sources of approximation error. One is due
to forcing a discrete grid on continuous state variables and second from using a discrete
approximation of the true distribution of the underlying shocks. There are also instances
where the use of discrete state-space methods is entirely inappropriate since the
discretization process transforms an infinite state space into a finite one and in the
process is changing the information structure. This may not be an issue in most models,
but it definitely has an impact in models with partially revealing rational expectations
equilibria
23
.
22
23
See Tauchen (1990) for an example.
See Judd (1998) pp. 578-581 for an example.
26
I.6. Projection Methods
24
As opposed to the previously presented numerical methods, the techniques that
are going to be presented in this section have a high degree of generality. Projection
methods appear to be applicable to solving a wide variety of economic problems. In fact,
projection methods can be described as general numerical methods that make use of
global approximation techniques
25
to solve equations involving unknown functions.
The idea is to replace the quantity that needs to be approximated by parameterized
functions with arbitrary coefficients that are to be determined later on
26
, or to represent
the approximate solution to the functional equation as a linear combination of known
basis functions whose coefficients need to be determined
27
. In either case, there are
coefficients to be computed in order to obtain the approximate solution. These
coefficients are found by minimizing some form of a residual function.
Further on, a step by step description of the general projection method is
presented, followed by a discussion of the parameterized expectations approach.
24
I borrow this terminology from Judd (1992, 1998). These methods are also called
weighted residual methods by some authors (for example Rust (1996), McGrattan (1999),
Binder et al. (2000)). In fact, one can argue that weighted residual methods are just a
subset of the projection methods with a given norm and inner product.
25
In some cases local approximations are used on subsets of the original domain and then
they are pieced together to give a global approximation. One such case is the finite
element method.
26
See Marcet and Marshall (1994a), Marcet and Lorenzoni (1999), Wright and Williams
(1982a, 1982b, 1984) and Miranda and Helmberger (1988)
27
See McGrattan (1999)
27
I.6.1. The Concept of Projection Methods
Suppose that the functional equation can be described by:
F (d ) = 0 (1.6.1)
where F is a continuous map, F : C
1
÷ C
2
with C
1
and C
2
complete normed function
spaces and d : D c ?
k
÷ ?
m
is the solution to the optimization problem. More generally,
d is a list of functions that enter in the equations that define the equilibrium of a model,
such as decision rules, value functions, and conditional expectations functions, while the
F operator expresses equilibrium conditions such as Euler equations or Bellman
equations.
I.6.1.1. Defining the Problem
The problem is to find d : D c ?
k
÷ ?
m
that satisfies equation (1.6.1). This
translates into finding an approximation
d
ˆ(x;u ) which depends on a finite-dimensional
vector of parameters u
=
|u
1
,u
2
,K,u
n
| such that F
dˆ
( x;
u
) is as close as possible to
( )
zero.
I.6.1.1.1. Example
28
Consider the following finite horizon problem where the social planner or a
representative agent maximizes
E
e
9 |
tt
(u
t
) O
0
÷ T
subject to
c
©t=0
?
?
(1.6.2)
28
x
t
=
h
(x
t
÷
1
,u
t
, y
t
)
The example in section I.6.1 draws heavily on Binder et al. (2000)
28
(1.6.3)
with x
0
and x
T
given. y
t
is an
AR
(
1
) process with the law of motion
y
t
= µ y
t
÷
1
+ z
t
(1.6.4)
and z
t
are i.i.d. with z
t
~ N 0,o 2
y
. I assume that u can be expressed as a function of
( )
x , i.e. u
t
=
g
(x
t
÷
1
, x
t
, y
t
) . Then the Euler equation for period T ÷ 1 is given by
0 = t '
g
(x
T
÷
2
, x
T
÷
1
, y
T
÷
1
) ? g '
x
T÷
1
(x
T
÷
2
, x
T
÷
1
, y
T
÷
1
)
( )
(1.6.5)
+ | E t '
g
(x
T
÷
1
, x
T
, y
T
) ? g '
x
T÷
1
(x
T
÷
1
, x
T
, y
T
) O
T
÷1
{( ) }
Let the optimal decision rule for x
T
÷
1
be given by x
*
÷
1
= d
T
÷
1
(x
T
÷
2
, y
T
÷
1
) where T
d
(
?
) is a smooth function. The projection methodology consists of approximating
d
(
?
)
by d
ˆ
(?,
u
) , where u represents an unknown parameter matrix. The unknown parameters
are computed such that the Euler equation also holds for d
ˆ
(?,
u
) .
Further on in this section I present the necessary steps one needs to take when
applying the projection methods, drawing heavily on the formalization provided by Judd
(1998)
29
. As I mentioned above, the methodology consists of finding an approximation
dˆ(x;u ) such that F d
ˆ
( x;
u
) is as close as possible to zero. It becomes obvious that
( )
there are a few issues that need to be addressed: what form of approximation to choose
for dˆ(x;u ) ; does the operator F need to be approximated; what does one understand by,
or in other words, what is the formal representation of "as close as possible to zero".
29
Judd provides a five step check list for applying the projection methods.
29
I.6.1.2. Finding a Functional Form
The first step comes quite naturally from the need to address the question on how
to represent d (x;u ) . In general d
ˆ
is defined as a finite linear combination of basis
functions, ?
i
(x), i = 0,K, n :
n
dˆ(x;u ) = ?
0
(x)
+
¿u
i
?
i
(x) (1.6.6)
i=1
Therefore, the first step consists of choosing a basis over C
1
.
Functions ?
i
(x), i = 0,K, n are typically simple functions. Standard examples of
basis functions include simple polynomials (such as ?
0
(x) = 1, ?
i
(x) = x
i
), orthogonal
polynomials (for example, Chebyshev polynomials), and piecewise linear functions.
Choosing a basis is not a straightforward task. For example, ordinary polynomials are
sometimes adequate in simple cases where they may provide a good solution with only a
few terms. However, since they are not orthogonal on R
+
and they are all monotonically
increasing and positive for x ΠR
+
, for x big enough, they are almost indistinguishable
and hence they tend to reduce numerical accuracy
30
. Consequently, orthogonal bases are
usually preferred to avoid the shortcomings just mentioned.
One of the more popular orthogonal bases is formed by Chebyshev polynomials.
They constitute a set of orthogonal polynomials with respect to the weight function
30
In order to solve for the unknown coefficients u
i
one needs to solve linear systems of
equations. The accuracy of these solutions depends on the properties of the matrices
involved in the computation, i.e. linear independence of rows and columns. Due to the
properties already mentioned, regular polynomials tend to lead to ill-conditioned
matrices.
30
e(x) = 1 1÷ x
2
, that is,
}
1
p
i
(x) p
j
(x)e(x)dx = 0 for all i = j . Chebyshev polynomials
÷1
are defined on the closed
interval
|÷1,
1
| and can be computed recursively as follows:
p
i
(x) = 2xp
i
÷
1
(x) ÷ p
i
÷
2
(x), i = 2, 3, 4,K (1.6.7)
with p
0
(x) = 1 and p
1
(x) = x or, non-recursively, as:
p
i
(x) = cos (i arccos (
x
)) (1.6.8)
Another set of possible basis functions that can be used to construct a piecewise
linear representation for d
ˆ
is given by:
? x ÷ x
i
÷
1
if x
e
| x ,
x
|
? x
i
÷ x
i
÷1
?
i÷1 i
?
i
(x) = ? x
i
+
1
÷ x if x
e
|x
i
, x
i
+
1
|
? i+1 i
?x ÷ x
(1.6.9)
?
?
?
0 elsewhere
The points x
i
, i = 1,K, n that divide the domain D c ? need not be equally spaced. If,
for example, it is known that the function to be approximated has large gradients or kinks
in certain places then the subdivisions can be smaller and clustered in those regions. On
the other hand, in areas where the function is near-linear the subdivisions can be larger
and hence fewer.
Once the basis is chosen, the next step is to choose how many terms and
consequently how many parameters the functional form will have. In general, if the
choice of the basis is good, the higher the number of terms the better the approximations.
However, due to the fact that the more terms are chosen the more parameters have to be
computed, one should choose the smallest number of terms, n , that yields an acceptable
31
approximation. One possible approach is to begin with a small n and then increase its
value until some approximation threshold is reached.
I.6.1.2.1. Example
Going back to the model defined by equations (1.6.2) and (1.6.3) the next step is
choosing a base. I assume that Chebyshev polynomials are used in constructing the
functional form for dˆ
T
÷
1
(?,
u
) . Then:
dˆ
T
÷
1
(x
T
÷
2
, y
T
÷
1
;u
T
÷
1
) =
n
x
,T ÷
1
n
y
,T ÷1
9 9u
T ÷1,sq
p
s
÷
1
(x
T
÷
1
) p
q
÷
1
( y
T
÷
1
)
s=1 q=1
% %
(1.6.10)
where u
T
÷1,sq is
the
(s,
q
) element of u
T
÷
1
, p
l
(?) is the l -th order Chebyshev polynomial
as defined in (1.6.7) - (1.6.8), while n
x
,T ÷
1
and n
y
,T ÷
1
are the maximum order of the
Chebyshev polynomials assumed for x
T
÷
1
and y
T
÷
1
respectively. In order to restrict the
% %
domain of the polynomials to the unit interval the following transformation is applied:
%T ÷
1
= 2 x
T
ma÷1x ÷ x
T
m÷i1
n
÷ 1
min
x
%
x
T
÷
1
÷ x
T
÷1
min
%
(1.6.11)
y
T
÷
1
= 2 y
T
ma÷1x ÷ y
T
m÷i1
n
÷1 (1.6.12)
y
T
÷
1
÷ y
T
÷1
I.6.1.3. Choosing a Residual Function
In many cases, computing F (dˆ) may require the use of numerical approximations
such as when F (d ) involves integration of d . In those cases, the F operator has to be
approximated. In addition, once the methodology for approximating d and F has been
32
established, one needs to choose a residual function. Therefore, the third step consists of
defining the residual function and an approximation criterion. Let
R(x;u ) } F (dˆ(?,u ))(x) ˆ
(1.6.13)
be the residual function. At this point, a decision has to be made on how an acceptable
approximation is defined. That is accomplished by choosing an approximation criterion.
One choice is to compute the sum of squared residuals, R(?;u ) } R(?;u ), R(?;u ) and
then determine u such that R(?;u ) is minimized. An alternative would be to choose a
collection of n test functions in C
2
, p
i
: D C R
m
, i = 1,..., n , and for each guess of u to
compute the n projections, P
i
(?) } R(?;u ), p
i
(?)
31
.
It is obvious that this step creates the
projections that will be used to determine the value of the unknown coefficients, u .
Another popular choice in the literature is the weighted residual criterion defined as
32
:
v ¢ (x)R(x;u )dx = 0,
iD
i = 1,K, n
(1.6.14)
where ¢
i
(x), i = 1,K, n are weight functions. Alternatively, the set of equations (1.6.14)
can be written as
v
D
e (x)R(x;u )dx = 0
n
(1.6.15)
where D is the domain for function d , e (x)
=
9e ¢
i
(x) and (1.6.15) must hold for i
i=1
any non-zero weights e
i
, i = 1,K, n . Therefore, the method sets a weighted integral of
R(x;u ) to zero as the criterion for determining u .
31
The choice of the criterion gives the method its name. That is why in the literature the
method appears both under the name "projection method" and "weighted residual
method".
32
See McGrattan (1999).
33
I.6.1.3.1. Example
Going back to the example, recall that Chebyshev polynomials were used in
constructing the functional form for dˆ
T
÷
1
(?,
u
) :
dˆ
T
÷
1
(x
T
÷
2
, y
T
÷
1
;u
T
÷
1
) =
n
x
,T ÷
1
n
y
,T ÷1
9 9u
T ÷1,sq
p
s
÷
1
( %T÷
1
) p
q
÷
1
( %T÷
1
)
s=1 q=1
x y
As mentioned above, the Euler equation (1.6.5) needs to hold for d
ˆ
(?,
u
) . Therefore, its
right hand side is a prime candidate for defining the residuals function. Let v
T
÷
1
= _
T
÷
2
ˆ
.
x
·
y
T÷1 ˜
With this notation, the residual function is given by:
. +
R
T
÷
1
?v
T
÷
1
; dˆ
T
÷
1
(v
T
÷
1
;u
T
÷
1
)? =
? ?
g v
T
÷
1
, dˆ
T
÷
1
(v
T
÷
1
;u
T
÷
1
), y
T
÷
1
? g '
x
T÷
1
v
T
÷
1
, dˆ
T
÷
1
(v
T
÷
1
;u
T
÷
1
), y
T
÷1
( ) ( ) (1.6.16)
+
(t
'
)÷1 E
?
|t ' g dˆ
T
÷
1
(v
T
÷
1
;u
T
÷
1
), x
T
, y
T
? g '
x
T÷
1
dˆ
T
÷
1
(v
T
÷
1
;u
T
÷
1
), x
T
, y
T
?
{
?
( ) (
)
}
?
Then the criterion for computing u%
T
÷
1
is given by the weighted residual integral equation:
v
R
T
÷
1
v
T
÷
1
; dˆ
T
÷
1
(v
T
÷
1
;uˆT ÷
1
)
?
W (v
T
÷
1
)dv
T
÷
1
= 0
v
T
÷1
e ?
(1.6.17)
where W is a weighting function. In the next section it will become clear why the choice
of W is important in the computation of u%
T
÷
1
.
I.6.1.4. Methods Used for Estimating the Parameters
Evidently, the next step is to find u ΠR
n
that minimizes the chosen criterion. In
order to determine the coefficients u
1
,K,u
n
several methods can be used, depending on
the criterion chosen.
34
If the projection criterion is chosen, finding the n components of u means
solving the n equations
R
(x,
u
), p
i
= 0 for some specified collection of test functions,
p
i
. The choice of the test functions p
i
defines the implementation of the projection
method. In the least squares implementation the projection directions are given by the
gradients of the residual function. Therefore, the problem is reduced to solving the
c
R
( x ,
u
)
nonlinear set of equations generated by
R
(x,
u
),
cu
i
= 0 i = 1,..., n .
One alternative is to choose the first n elements of the basis u , that is,
?
i
(x) i = 1,..., n , as the weight functions, ¢
i
(x), i = 1,K, n . In other words, n elements of
the basis used to approximate dˆ(x;u ) are also used as test functions to define the
projection direction, ¢
i
(x) = ¢
i
(x), i = 1,K, n . This technique is known as the Galerkin
method. As a result of this choice, the Galerkin method forces the residual to be
orthogonal to each of the basis functions. Therefore u is chosen to solve the following
set of equations:
P
i
(
u
) =
R
(x,
u
),¢
i
(
x
) = 0 i = 1,..., n (1.6.18)
As long as the basis functions are chosen from a complete set of functions, system
(1.6.18) provides the exact solution, given that enough terms are included. If the basis
consists of monomials, the method is also known as the method of moments. Then u is
the solution to the system:
P
i
(
u
) =
R
(x,
u
), x
i
÷
1
= 0 i = 1,..., n (1.6.19)
The collocation method chooses u so that the functional equation holds exactly at
n fixed points, x
i
, called the collocation points. That is, u is the solution to:
35
R(x
i
;u ) = 0, i = 1,..., n (1.6.20)
where
{x
i
}ni=
1
are n fixed points from D . It is easy to see that this is a special case of the
projection approach, since R(x;u ),o (x ÷ x
i
) = R(x
i
;u ) , where o (x ÷ x
i
) is the Dirac
function at x
i
. If the collocation points x
i
are chosen as the n roots of the n
th
orthogonal
polynomial basis element and the basis elements are orthogonal with respect to the inner
product, the method is called orthogonal collocation. The Chebyshev polynomial basis is
a very popular choice for an orthogonal collocation method.
I.6.1.4.1. Example
Going back to the example, it was established that the criterion for computing
u%
T
÷
1
is given by the following integral equation:
v
R
T
÷
1
v
T
÷
1
; dˆ
T
÷
1
(v
T
÷
1
;uˆT ÷
1
)?W (v
T
÷
1
)dv
T
÷
1
= 0
v
T
÷1
e ?
As discussed in this section, given this criterion, the collocation method is a
sensible choice for computing u%
T
÷
1
. Then the choice for the weighting functions, as used
in Binder et al. (2000), is the n
x
,T ÷
1
, n
y
,T ÷
1
Dirac delta functions o x
T
÷
1
÷ x
i
T ÷
1
, y
T
÷
1
÷ y
i
T ÷
1
,
( )
where x
iT
÷
1
and y
iT
÷
1
are chosen such that x
iT
÷
1
and y
iT
÷
1
are the n
x
,T ÷
1
and n
y
,T ÷
1
zeros of
% %
the Chebyshev polynomials forming the basis of the approximation dˆ
T
÷
1
(v
T
÷
1
;u
T
÷
1
) . The
zeros for the Chebyshev polynomials are given by
?
?
co
s
(2
i
÷
1
)
t
?
?
v%T÷
1
=
? i
2n
x
,T ÷1 ? (1.6.21)
? (2i ÷1)t ?
? cos 2n ?
? y,T ÷1 ?
36
Then the integral equation can be reduced to:
R
T
÷
1
v
ij
÷
1
; dˆ
T
ij÷
1
= 0
for all
(
T
)
(1.6.22)
v
ij
÷
1
= x
i
T ÷
1
, y
T
j÷
1
, i = 1, 2,..., n
x
,T ÷
1
,
and
T
( )
dˆ
T
ij÷
1
= dˆ
T
÷
1
v
ij
÷
1
;uˆT ÷1
j = 1, 2,..., n
y
,T ÷1
(1.6.23)
(
T
)
(1.6.24)
The discrete orthogonality of Chebyshev polynomials implies that:
n
x
,T ÷
1
n
y
,T ÷1
9 9 e
p
(x
%
)
p
(y
%
)??e
p
(x
%
)
p
(y
%
)?? =
0
i=1 j =1 w÷1
i
T ÷1
p÷1
j
T ÷1
s÷1
i
T ÷1
q÷1
j
T ÷1
(1.6.25)
for w t s and /or p t q , and
n
x
,T ÷
1
n
y
,T ÷1
9 9 e
p
(x
%
)
p
(y
%
)??e
p
(x
%
)
p
(y
%
)?? =
c
(n
i=1 j =1 w÷1
i
T ÷1
p÷1
j
T ÷1
s÷1
i
T ÷1
q÷1
j
T ÷1
sq
x,T ÷1
, n
y
,T ÷
1
(1.6.26) )
for w = s and p = q , with
?n
x
,T ÷
1
n
y
,T÷
1
, w = s = p = q = 1
?
? ?w = s = 1 and p = q = 1,
c
sq
(n
x
,T ÷
1
, n
y
,T÷
1
) = ?x,T÷1y,T÷
1
) / 2, ?or
?
(n n
?
(1.6.27)
?
?
??w
= s =
1 and
p =
q
=
1,
?
(n
x
,T ÷
1
n
y
,T÷
1
) / 4, w = s = 1 and p = q = 1 ?
Then u is given by:
n
x
,T ÷
1
n
y
,T ÷1
1
uˆT ÷1,sq =
9 9
p
(x
%
)
p
(y
%
)?e
g
(v
( )
s÷1
i
T ÷1
q÷1
j
T ÷1
T ÷1
, dˆ
T
ij÷
1
, y
T
÷
1
? g '
x
T÷
2
v
T
÷
1
, dˆ
T
ij÷
1
,
y
T
÷1
c
sq
n
x
,T ÷
1
, n
y
,T ÷1
i=1 j =1
) ( )
+
(t%
'
)÷1 E |t ' g dˆ
T
ij÷
1
, x
T
, y
T
? g '
x
T÷
1
dˆ
T
ij÷
1
, x
T
, y
T
? v
T
÷1 ?
({
e
( ) ( )
?
}
)
?
?
(1.6.28)
for s = 1, 2,..., n
x
,T ÷
1
, q = 1, 2,..., n
y
,T ÷
1
.
37
The conditional expectation from the above equation needs to be computed
numerically. In order to compute the integral one can use some of the quadrature methods
such as the Gauss quadrature presented in section I.5.2. All that remains is to solve
equation (1.6.28) for u
T
÷1,sq , s = 1, 2,..., n
x
,T ÷
1
, q = 1, 2,..., n
y
,T ÷
1
. Once dˆ
T
÷
1
v
T
÷
1
;uˆT ÷1 (
)
is
computed, one can proceed recursively backwards to period T ÷ 2 . Note that
x
*
÷
1
= dˆ
T
÷
1
v
T
÷
1
;uˆT ÷
1
will be used in the definition of R
T
÷
2
v
ij
÷
2
; dˆ
T
ij÷
2
. The computation
T
( ) (
T
)
of uˆT ÷
2
can now follow the same logic as the computation of uˆT ÷
1
.
So far the flavors of the projection methodology have been categorized either with
respect to the choice of the approximation criterion or with respect to the method
employed for estimating the parameters. The choice of basis functions for the
representation in (1.6.6) can be used to further divide projection methods into two
categories: spectral methods and finite-element methods. Spectral methods use basis
functions that are smooth and non-zero on most of the domain of x such as Chebyshev
polynomials and the same functions are used on all regions of the state space. Finite-
element methods use basis functions that are equal to zero on most of the domain and
non-zero on only a few subdivisions of the domain of x (these are in general piecewise
linear functions such as those defined in (1.6.9)) and they provide different
approximations in different regions of the state space. For problems with many state
variables, there are typically many coefficients to compute and it implies the inversion of
a large, dense matrix. With the finite-element method, however, the same matrix is sparse
and its structure can typically be exploited. For the above-mentioned reasons McGrattan
38
(1996, 1999) argues that a finite-element method is better suited to problems in which the
solution is nonlinear or kinked in certain regions.
I.6.2. Parameterized Expectations
While Marcet (1988) is largely credited in the literature with the introduction of
the parameterized expectations approach, Christiano and Fisher (2000) point out that the
underlying idea of parameterized expectations seems to have surfaced earlier in the work
of Wright and Williams (1982a, 1982b, 1984), and then in the work of Miranda and
Helmberger (1988). Marcet (1988)
33
implemented a variation of that idea and the
approach finally caught on with the publication of Den Haan and Marcet (1990).
In this section, I will concentrate on what Christiano and Fisher (2000) call the
conventional parameterized expectations approach due to Marcet (1988). While one may
argue that this methodology does not belong under the label of projection methods, I
believe that it can be viewed as a special case of projection methods by virtue of its use of
parameterized functions to approximate an unknown quantity, of an implicit choice of a
residual function and an approximation criterion similar to projection methods. In
addition, the techniques used to estimate the parameters are also common to projection
methods. The assumption is that the functional equation has the following form:
g E
t
?
|
(q
t
+
1
,q
t
)? ,q
t
÷
1
,q
t
, z
t
= 0
(
? ?
)
(1.6.29)
where q
t
includes all the endogenous and exogenous variables and z
t
is a vector of
exogenous shocks. As it has been repeatedly asserted in this chapter, the reason why
33
For more information of this variant of the parameterized expectations approach, see
the references cited in Marcet and Marshall (1994b).
39
many dynamic models are difficult to solve is that conditional expectations often appear
in the equilibrium conditions. The assumption under which this methodology operates is
that conditional expectations are a time-invariant function c of some state variables:
c (u
t
) = E
t
||(q
t
+
1
,q
t )
| (1.6.30)
where E
t
||(q
t
+
1
,q
t )
| = E ?|(q
t
+
1
,q
t
) u
t
? is the conditional expectation based on the
? ?
available information at time t , u
t
e R
l
where u
t
is a subset
of
(q
t
÷
1
, z
t
) . As Marcet and
Lorenzoni (1999) point out, a key property of c is that under rational expectations, if
agents use c to form their decisions, the series generated is such that c is precisely the
best predictor of the future variables inside the conditional expectations. So, if c were
known, one could easily simulate the model and check whether this is actually the
conditional expectation.
The basic approach of Marcet and Marshall (1994a) is to substitute the
conditional expectations in equation (1.6.29) by parameterized functions of the state
variables with arbitrary coefficients. Then (1.6.29) is used to generate simulations for u
t
consistent with the parameterized expectations. With these simulations, one can iterate on
the parameterized expectations until they are consistent with the solution they generate.
In this fashion, the process of estimating the parameters is reduced to a fixed-point
problem.
I.6.2.1. Example
Consider again the model specified by (1.6.2) - (1.6.3) with the Euler equation for
period t given by:
40
0 = t '
g
(x
t
, x
t
÷
1
, y
t
) ? g '
x
t
(x
t
, x
t
÷
1
, y
t
)
( )
(1.6.31)
+ | E t '
g
(x
t
+
1
, x
t
, y
t
+
1
) ? g '
x
t
(x
t
+
1
, x
t
, y
t
+
1
) O
t
{( ) }
The idea is to substitute
E
t
t '
g
(x
t
+
1
, x
t
, y
t
+
1
) ? g '
x
t
(x
t
+
1
, x
t
, y
t
+
1
)
{( ) }
by a parameterized function
¢
( x
t
÷
1
, y
t
;
u
) where u is a vector of parameters. For
simplicity, let the function ¢ be given by:
¢
t
( x
t
÷
1
, y
t
;u
1
,u
2
) = u
1
x
t
÷
1
+u
2
y
t
(1.6.32)
The next step is to generate a
series
{z
t
}Tt=
1
as draws from a Gaussian distribution and to
choose starting values for the elements of u , u
i
0
, i = 1, 2 . Then, for uˆi = u
i
0
and assuming
that the initial values for x
t
and y
t
, that is, x
÷
1
and y
0
are given, one can use the
following system
t
'
(
g
(x
t
, x
t
÷
1
, y
t
))? g '
x
(x
t
, x
t
÷
1
, y
t
) +uˆ1x
t
÷
1
+uˆ2 y
t
= 0 for t = 0,...,T ÷1 t
x
t
=
h
(x
t
÷
1
,u
t
, y
t
) for t = 0,...,T , with x
÷
1
given (1.6.33)
y
t
= µ y
t
÷
1
+ z
t
for t = 1,...,T , with y
0
given
to generate
series
{ˆ
tt
j
}
t
=
0
,
{ˆ
t
j
}
t
=
1
and
{u
t
j
}
t
=
0
where j represents the iteration. In order
T T
x y ˆT
to estimate the parameters u , proponents of this methodology run a regression of
?
tj
uˆj = t '
g
(ˆ
t
j
, ˆ
t
j÷
1
, ˆ
t
j
) ? g '
x
t
(ˆ
t
j
, ˆ
t
j÷
1
, ˆ
t
j
)
() ( xx y ) xx y (1.6.34)
on ¢
t
. Formally, the regression can be written as:
?
tj
uˆj = a
1
ˆ
t
j÷
1
+ a
2
ˆ
tj
+ ç
t
() x y
where ç
t
is the error term. The estimates for a
1
and a
2
provide a new set of values for u
for the next iteration. With those values new series will be generated for {ˆ
t
j+
1
}
t
=
0
and T
x
41
{u
ˆ
} j
+1 T
y
T
t t =0
. In this particular case, there is no need to generate new series
for
{ˆ
t
j+
1
}
t
=
1
if the
same vector of shocks {z
t
}Tt=1 is used. In addition, note that a
1
and a
2
are in fact
functions of uˆ . Specifically, for iteration j , the vector of parameters a is a function of
uˆ
j
, a = G uˆ
j
. Hence the final step is to find the fixed point u =
G
(
u
) . One approach ()
suggested by Marcet and Lorenzoni (1999) is to compute the values of uˆ for iteration
j +1 using the following expression uˆj+
1 =
(1÷
b
)uˆj + bG uˆ
j
where b > 0 . The iteration ()
process should stop when uˆ
j
and G uˆ
j
are sufficiently close. ()
I.6.3. Notes on Projection Methods
As Judd (1992) points out, the advantage of the projection method framework is
that one can easily generate several different implementations by choosing among
different basis, residual functions or methods for estimating the parameters. Obviously,
the many choices also imply some trade-offs among speed, accuracy, and reliability. For
example, the orthogonal collocation method tends to be faster than the Galerkin method,
while the Galerkin method tends to offer more accuracy
34
.
The generality of the projection techniques can also be seen from the fact that
even methods that discretize the state space can be thought of as projection methods that
are using step function bases.
While throughout this section I emphasized the wide applicability of projection
methods, there is an aspect that has been overshadowed. Recall that the idea is to replace
34
See Judd (1992) for more details.
4 2
the quantity that needs to be approximated by parameterized functions (basis functions
?
i
(
x
) ) with arbitrary coefficients ( a
i
). In projection methods, the coefficients are chosen
to be the best possible choices relative to the basis ?
i
(
x
) and relative to some criterion.
However, the bases are usually chosen to satisfy some general criteria, such as
smoothness and orthogonality conditions. Such bases may be good but very rarely are
they the best possible for the problem under consideration.
An important advantage of parameterized expectations approach is that, for
specific models, it may implicitly deal with the presence of inequality constraints
eliminating the need to constantly check whether the Kuhn-Tucker conditions are
satisfied
35
.
A key component of the conventional parameterized expectations approach
presented in this section is a cumbersome nonlinear regression step. The regression step
implies simulations involving a huge amount of synthetic data points. The problem with
this approach is that it inefficiently concentrates on a residual amount that is obtained
from visiting only high probability points of the invariant distribution of the model. As
Pointed out by Judd (1992) and Christiano and Fisher (2000), it is important to consider
the tail areas of the distribution as well. Christiano and Fisher (2000) offer a modified
version of the parameterized expectations approach that they call the Chebyshev
parameterized expectations approach, specifically designed to eliminate the shortcoming
discussed above. In fact, Christiano and Fisher (2000) explicitly transform the
parameterized expectations approach into a projection method that they refer to as the
weighted residual parameterized expectations approach. As mentioned above, expressing
35
See Christiano and Fisher (2000) for details.
43
the parameterized expectations approach as a projection method opens the door to a
variety of possible implementations.
36
.
I.7. Comparing Numerical Methods: Accuracy and Computational Burden
It is difficult to define the global criteria of success for numerical methods.
Accuracy is in general at the top of the checklist in defining a good numerical method.
However, it may not always be the most important criterion when choosing a numerical
method. For example, even though a method may not provide the best approximation for
the policy function, it may still be preferred to other methods as long as the loss in
accuracy relative to the policy function does not affect too much the value of the
objective function. In such cases, speed or ease of implementation may take precedence.
There does not seem to be a general agreement in the literature on how to evaluate
the accuracy of numerical methods. Consequently, a number of criteria have been
proposed in order to asses the performance of numerical algorithms.
One widely used strategy for determining accuracy is to test the outcome of a
computational algorithm in a particular case where the model displays an analytical
solution. For example, Collard and Juillard (2001) use an average relative error and a
maximal relative error criterion in order to asses the accuracy of several numerical
methods. While this approach may be useful for certain specifications, the problem is that
for alternative parameterizations of the model the approximation error of the computed
decision and value functions may change substantially. Changes in the curvature of the
objective function and in the discount factor are the usual culprits in influencing
36
In fact, Christiano and Fisher (2000) provide two other modified versions of the
parameterized expectations approach (PEA): PEA Galerkin and PEA collocation.
44
considerably the accuracy of the algorithm. Collard and Juillard (2001) determine that for
an asset pricing model the Galerkin method using fourth order Chebyshev polynomials
clearly outperforms linearization methods as well as lower order perturbation methods.
However, higher order (order four and higher) perturbation methods prove to be quite
accurate.
Another strategy used for analyzing the accuracy of numerical methods is to look
at the residuals of the Euler equation. This seems like a natural choice especially for
approaches that are based on approximating certain terms entering, or the whole, Euler
equation
37
.
A procedure for checking accuracy of numerical solutions based on the Euler
equation residuals was proposed by den Haan and Marcet (1990, 1994). It consists of a
test for the orthogonality of the Euler equation residuals over current and past
information. The idea behind this test is to compute simulated time series for all the
choice and state variables as well as Euler equation residuals, based on a candidate
approximation. Then, using estimated values of the coefficients resulting from regressing
the Euler equation residuals on lagged simulated time series, one can construct measures
of accuracy. As pointed out by Santos (2000), the problem with this approach is that
orthogonal Euler equation residuals may be compatible with large deviations from the
optimal policy. In addition, as referenced by Judd (1992), Klenow (1991) found that the
procedure failed to reject candidate solutions that resulted in relatively high errors for the
choice variable while rejecting solutions resulting in occasional high large errors but
without any discernible pattern.
37
For a detailed discussion on criteria involving Euler equation residuals, please see
Reiter (2000) and Santos (2000).
45
Judd (1992, 1998) suggested an alternative test that consists of computing a one
period optimization error relative to the decision rule. The error is obtained by dividing
the current residual of the Euler equation to the value of next period's decision function.
Subsequently, two different norms are applied to the error term: one gives the average
and the other supplies the maximum.
In a study aimed at comparing various approximation methods, Taylor and Uhlig
(1990) found that performance varies greatly depending on the criterion used for
assessing accuracy. For example, the decision rules indicated that some of the easier to
implement methods such as the linear-quadratic method and the extended-path method
were fairly close to the "exact" decision rule
38
as given by the quadrature-value-function-
grid method of Tauchen (1990) or the Euler-equation grid method of Coleman (1990).
However, neither the linear-quadratic nor the extended-path method performed well
when using the martingale-difference tests for the Euler-equation residual. Not
surprisingly, the parameterized expectations approach performed well when using the den
Haan and Marcet criterion but not as well when measured against the exact decision rule.
While accuracy is very important, computational time may also play an important
role in the eyes of some researchers. While the extended-path method has relatively low
cost when compared to grid methods, it is fair to state that both grid methods and the
extended-path method are computationally quite involved, whereas linear-quadratic
methods are typically quite fast. Most projection methods also fare well in terms of
38
Solutions obtained through discretization methods are sometimes referred to as
"exact". The reason behind this labeling is that models obtained as a result of
discretization may be solved exactly by finite-state dynamic programming methods.
However, one has to keep in mind that reducing a continuous-state problem to a finite-
state problem still involves an approximation error.
46
computational burden when compared to discretization methods or even parameterized
expectations methods. As the state space increases, discretization methods suffer heavily
from the curse of dimensionality.
The fact that none of the methods outperforms the others does not mean that every
method could be applied to any model out there with a good degree of success
39
. One has
to use good judgment when deciding on using a certain numerical method.
I.8. Concluding Remarks
As it has become clear over the course of this chapter, there are quite a few
methodologies available for solving non-linear rational expectations models. However, if
one looks closer, it becomes obvious that all methods share some common elements. For
example, certainty equivalence is at the core of the extended path method but it can also
be used in perturbation methods to find the equilibrium of a (deterministic) system
similar to the one under investigation. The discrete state space approach can be viewed as
a projection method with step functions as a basis. Similarly, the first order perturbation
method is nothing more than a simple linearization around steady state. In addition, the
parameterized expectations approach can be easily transformed into a projection method.
Moreover, since all the functional equations for rational expectations models imply the
existence of some integrals, the quadrature approximation may make an appearance in
almost every methodology.
39
Judd (1998) contains an example of a partially revealing rational expectations problem
which cannot be solved by discretizing the state space, but which can be approximated by
more general projection methods.
47
Several studies have tried to asses the performance of these numerical methods.
However, even for relatively simple models their performance may vary greatly
40
.
Despite all of their sophistication, none of these methods can consistently outperform the
others.
Even comparing the methods is not a walk in the park. Several authors including
Judd (1992), Den Haan and Marcet (1994), Collard and Juillard (2001), Santos (2000)
and Reiter (2000) proposed different criteria for evaluating the performance of numerical
solutions. Unfortunately, each criterion has its caveats and it has to be applied selectively,
based on the specificity of the model under investigation. Therefore, one has to choose
carefully the proper methodology when in need of numerical solutions.
40
See the studies by Taylor and Uhlig (1990), Judd (1992), Rust (1997), Christiano and
Fischer (2000), Santos (2000), Collard and Juillard (2001), Fair (2003), Schmitt-Grohé
and Uribe (2004).
48
Chapter II. Using Scenario Aggregation Method to Solve a Finite
Horizon Life Cycle Model of Consumption
II.1. Introduction
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 horizon
41
. 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
41
In the next chapter I will focus more on the length of the span over which the decision
making process takes place.
49
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 life-
cycle 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.
II.2. A Simple Life-Cycle Model with Precautionary Saving
I consider the following version of a life-cycle model. Suppose an individual
agent is faced with the following intertemporal optimization problem:
max E
9 |
t
F
t
(c
t
) | I
0
? T
?
(2.2.1)
{c
t
}
T
=0
t
_t=0
e
?
where F
t
is a utility function which has the typical properties assumed in the literature,
i.e. it is twice differentiable, it is increasing with consumption and exhibits negative
second derivative. The information set I
0
contains the level of consumption, assets, labor
income and interest rate for period zero and all previous periods.
Maximization is subject to the following transition equation:
A
t =
(1+ r
t
) A
t
÷
1
+ y
t
÷ c
t
, t = 0,1,...,T ÷1, (2.2.2)
A
t
> ÷b, with A
-1
, A
T
given (2.2.3)
50
where A
t
represents the level of assets at the beginning of period t , y
t
the labor income
at time t , and c
t
represents consumption in period t . The initial and terminal conditions,
A
÷
1
and A
T
, are given. Uncertainty is introduced in the model through the labor income.
The realizations of the labor income are described by the following process:
y
t
= y
t
÷
1
+ ç
t
, t = 1,...,T , with y
o
given (2.2.4)
and ç
t
being drawn from a normal distribution, ç
t
~
N
(0,o2
y
) . For now, I will not make
any particular assumption about the process generating the interest rate, r
t
. Therefore, to
summarize the model, a representative consumer derives utility in period t from
consuming c
t
, discounts future utility at a rate | and wants, in period zero, to maximize
his present discounted value of future utilities for a horizon of T +1 periods. At the
beginning of each period t the consumer receives a stochastic labor income y
t
, and
based on the return on his assets A
t
÷
1
, from the beginning of period t ÷1 to the beginning
of period t , he chooses the consumption level c
t
, and thus determines the level of assets
A
t
according to equation (2.2.2).
Of particular importance in this problem is the random variable ç
t
. In the
standard formulation of the problem, ç
t
is assumed to be distributed normally with mean
zero and some variance o2
y
. Instead of making the standard assumption, if I assume that
ç
t
's sample space has only a few elements, then the optimization problem (2.2.1) -
(2.2.4) is a perfect candidate for being solved using the scenario aggregation method. Let
me assume for the moment that the sample space is given
by
{e
1
,e
2
,...,e
n
} with the
associated probabilities { p
1
, p
2
,..., p
n
} . If S is the set of all scenarios then its cardinal is
51
given by n
T
. It is obvious that as the sample space for the forcing variable increases, the
number of scenarios increases proportional with power T . Therefore, applying the
scenario aggregation method to find an approximate solution for this problem may only
be feasible when T and n are relatively small. In the next chapter, I will present a
solution for T relatively large.
II.3. The Concept of Scenarios
II.3.1. The Problem
In this section, I formally introduce a multistage optimization problem and then,
in the following sections, I will present the idea of scenario aggregation and how it can be
applied to such a problem.
The multistage stochastic optimization problem consists of minimizing an
objective function, F : R
m
C R subject to some constraints, which usually describe the
dynamic links between stages.
The objective function F is time separable and is given by a sum of functions,
T
F
=
9 F
t
with each function F
t
, F
t
: R
m
C R corresponding to stage t of the
t =0
optimization problem. These functions depend on a set of variables u
t
, which in turn
represent the decisions that need to be made at each stage t . For simplicity I assume that
u
t
is a m
u
·1 vector, with m
u
independent of t , that is, the same number of decisions is
to be made at each stage.
52
If U (t) represents the set of all feasible actions at stage t , then u
t
has to be part
of the set U (t) , that is, u
t
¸ U (t), t = 0,...,T , U (t) [ R
m
u . The temporal dimension of
the problem is characterized by stages t and state variables X (t) .
The link between stages is given by:
x
t
+
1
= G
t
(x
t
,u
t
,u
t
+
1
) .
Hence, the problem can be formulated as:
min
E
?
¿ F
t
( x
t
,u
t
) | I
0
? T
subject to:
??
t =0
x
t
+
1
= G
t
(x
t
,u
t
,u
t
+
1
,ç
t
)
?
?
(2.3.1)
(2.3.2)
where I
0
is the information set at time t = 0 and ç
t
is the forcing variable.
In the next few sections, I will present the concept of scenarios as well as possible
decomposition methods along with the idea of scenario aggregation.
II.3.2. Scenarios and the Event Tree
In this section, I present an intuitive description for the concept of scenarios. A
formal description is presented in Appendix, section A1. Suppose the world can be
described at each point in time by the vector of state variables x
t
. In the case of a
multistage optimization problem, let u
t
denote the control variable and let ç
t
be the
forcing variable. I assume that an agent makes decisions reflected in the control variable
u
t
. For simplicity let ç
t
be a random variable witch can take two values ç
a
and ç
b
with
probabilities p
a
and 1÷ p
a
.
53
If the horizon has T +1 time periods and {ç
a
,ç
b
} is the set of possible
realizations for ç
t
then the sequence
ç
s
=
(ç
0
s
,ç
1
s
,K,ç
T
s
)
is called a scenario
42
. From now on, for notation simplification, I will refer to a scenario
s simply by ç
s
or by the index s . Given that the set of all realizations for ç
t
is finite,
one can define an event
tree
{N,
A
} characterized by the set of nodes N and the set of
arcs A . In this representation, the nodes of the tree are decision points and the arcs are
realizations of the forcing variables. The arcs join nodes from consecutive levels such
that a node n
t
j
at level t is linked to N
t
+
1
nodes, n
k
+
1
, k = 1,..., N
t
+
1
at level t +1. In t
Figure 1 I represent such a tree for a span of T = 3 periods. As mentioned above, the
forcing variable takes only two
values,
{ç
a
,ç
b
} and hence the tree has 15 nodes. The arcs
that join nodes from consecutive levels represent realizations of the forcing variable and
are labeled accordingly.
The set of nodes N can be divided into subsets corresponding to each level
(period). Suppose that at time t there are N
t
nodes. For example, for t = 1, there are two
nodes, node2 and node3. The arcs reaching these two nodes belong each to several
scenarios s . The bundle of scenarios that go through one node plays a very important
role in the decomposition as well as in the aggregation process. The term equivalence
class has been used in the literature to describe the set of scenarios going through a
particular node.
42
Other definitions of scenarios can be found in Helgason and Wallace (1991a, 1991b )
and Rosa and Ruszczynski (1994).
54
node1, t=0
çt=ça çt=çb
node2, t=1 node3, t=1
çt=ça çt=çb çt=ça çt=çb
node4, t=2 node5, t=2 node6, t=2 node7, t=2
çt=ça çt=çb çt=ça çt=çb çt=ça çt=çb çt=ça çt=çb
node8, t=3 node9, t=3 node10, t=3 node11, t=3 node12, t=3 node13, t=3 node14, t=3 node15, t=3
Figure 1 Event tree
By definition, an equivalence class at time t is the set of all scenarios having the
first t +1 realizations common. As mentioned in the above description of the event tree,
at time t there are N
t
nodes. Every node is associated with an equivalence class. Then,
the number of distinct equivalence classes at time t is also N
t
.
In Figure 2 one can see that for t = 1 there are two nodes and consequently two
equivalence classes, {s
1
, s
2
, s
3
, s
4
} and {s
5
, s
6
, s
7
, s
8
} . The number of elements of an
equivalence class is given by the number of leaves stemming from the node associated
with it. In this example, the number of leaves stemming from both nodes is four, which is
also the number of scenarios belonging to each class.
55
node1, t=0
(s1,s2,s3,s4,s5,s6,s7,s8)
çt=ça çt=çb
node2, t=1 node3, t=1
(s1,s2,s3,s4) (s5,s6,s7,s8)
çt=ça çt=çb çt=ça çt=çb
node4, t=2 node5, t=2 node6, t=2 node7, t=2
(s1,s2) (s3,s4) (s5,s6) (s7,s8)
çt=ça çt=çb çt=ça çt=çb çt=ça çt=çb çt=ça çt=çb
node8, t=3 node9, t=3 node10, t=3 node11, t=3 node12, t=3 node13, t=3 node14, t=3 node15, t=3
(s1) (s2) (s3) (s4) (s5) (s6) (s7) (s8)
Figure 2 Equivalence classes
The transition from a state at time t to one at time t + 1 is governed by the control
variable u
t
but is also dependent on the realization of the forcing variable, that is, on a
particular scenario s . Since scenarios will be viewed in terms of a stochastic vector ç
with stochastic components ç
0
s
,ç
1
s
,K,ç
T
s
, it is natural to attach probabilities to each
scenario. I denote the probability of a particular realization of a scenario, s , with
p(s) = prob(ç
s
) .
Let us consider the case of the event trees represented in Figure 1 and Figure 2 and
assume the probability of realization ç
a
is prob(ç
t
= ç
a
) = p
a
while the probability of
realization ç
b
, is prob(ç
t
= ç
b
) = p
b
, with p
a
+ p
b
= 1. Then, due to independence
56
across time, one can compute the probability of realization for scenario s
1
,
prob(ç
s
= s
1
) = p
3
a . Similarly, the probability of realization for scenario s
2
is
prob(ç
s
= s
2
) = p
2
p
b
, or prob(ç
s
= s
2
) = p
2
(1÷ p
a
) .
a a
Further on, I define
43
the probabilities associated with a scenario conditional upon
belonging to a certain equivalence class at time t . For example, the probability associated
with scenario s
1
, conditional on s
1
belonging to equivalence
class
{s
1
, s
2
, s
3
, s
4
} is given
by prob ( s
1
| s
1 e
{s
1
, s
2
, s
3
, s
4
}) = p
2
a
II.4. Scenario Aggregation
44
In this section, I will show how a solution can be obtained by using special
decomposition methods, which exploit the structure of the problem by splitting it into
manageable pieces, and then aggregate their solutions. In the multistage stochastic
optimization literature, there are two groups of methods that have been discussed: primal
decomposition methods that work with subproblems that are assigned to time stages
45
and dual methods, in which subproblems correspond to scenarios
46
. Most of the methods,
regardless of which group belong to, use the general theory of augmented Lagrangian
decomposition. In this chapter I will concentrate on a methodology that belongs to the
second group and has been derived from the work of Rockafellar and Wets (1991).
43
For a more formal definition, see the Appendix, section A1.
44
Section A2 in the Appendix offers a more formal description of scenario aggregation.
45
46
See the work of Birge (1985), Ruszczynski (1986, 1993), Van Slyke and Wets (1969).
See the work of Mulvey and Ruszczynski (1992), Rockafellar and Wets (1991),
Ruszczynski (1989), Wets (1988).
57
Let us assume for a moment that the original problem can be decomposed into
subproblems, each corresponding to a scenario. Then the subproblems can be described
as:
T
u ŒU [R
mu
min 9 F
t
x
s
t ,u
s
t ,
t t
t=1
( )
s ŒS
(2.4.1)
where u
st
and x
st
are the control and the state variable respectively, conditional on the
realization of scenario s while S is a finite, relatively small set of scenarios. Moreover,
suppose that each individual subproblem can be solved relatively easy. The question then
becomes how to blend the individual solutions into a global optimal solution. Let the
term policy
47
describe a set of chosen control variables for each scenario and indexed by
the time dimension.
The policy function has to satisfy certain constraints if two different scenarios s
and s ' are indistinguishable at time t on information available about them at the time.
Then u
s
t = u
s
t
'
, that is, a policy can not require different actions at time t relative to
scenarios s and s ' if there is no way to tell at time t which of the two scenarios will be
followed. In the literature, this constraint is sometimes referred to as the non-
anticipativity constraint. Going back to Figure 2, for t = 1, if the realization of ç
t
is ç
a
,
the decision maker will find himself at the decision point node2. There are four scenarios
that pass through node2 and the non-anticipativity constraint requires that only one
decision be made at that point since the four scenarios are indistinguishable. A policy is
47
A formal description of the policy function is presented in Appendix.
58
defined as implementable if it satisfies the non-anticipativity constraint, that is, u
t
must
be the same for all scenarios that have common past and present
48
.
In addition, a policy has to be admissible. A policy is admissible if it always
satisfies the constraints imposed by the definition of the problem. It is clear that not all
admissible policies are also implementable.
By definition, a contingent policy
49
is the solution, u
s
, to a scenario subproblem.
It is obvious that a contingent policy is always admissible but not necessarily
implementable. Therefore, the goal is to find a policy that is both admissible and
implementable. Such a policy is referred to as a feasible policy. One way to create a
feasible policy from a set on contingent policies is to assign weights (or probabilities) to
each scenario and then aggregate the contingent policies according to these weights.
The question that the scenario aggregation methodology answers is how to obtain
the optimal solution U from a collection of implementable policies U . In this chapter, I ˆ
will present a version of the progressive hedging algorithm originally developed by
Rockafellar and Wets (1991).
48
For certain problems the non-anticipativity constraint can also be defined in terms of
the state variable, that is, x
t
(e ) must be the same for all scenarios that have common past
and present.
49
I borrow this term from Rockafeller and Wets (1991).
59
II.5. The Progressive Hedging Algorithm
The algorithm is based on the principle of progressive hedging
50
which consists of
starting with an implementable policy and creating sequences of improved policies in an
attempt to reach the optimal policy.
Let us go back to the definition of an implementable policy. By computing
u
s
t =
ˆ
¿
}
p s
?
{s
t
}
i
u
s
t
'
= E u
s
t
'
| {s
t
}
i
s?e
{s
t
i
( )
( )
for all s
e
{s
t
}
i
(2.5.1)
for all scenarios s e S and all periods t = 1,...,T , one creates a starting collection of
implementable policies, denoted by U
0
. In equation (2.5.1) E represents the expectation ˆ
operator. Therefore, in order to obtain an initial collection of implementable policies one
should first compute some contingent policies for each scenario and then apply the
expectation operator for each period t and each scenario s conditional on it belonging to
the corresponding equivalence class, {s
t
}
i
.
The progressive hedging algorithm finds a path from U
0
, the set of ˆ
implementable policies, to U , the set of optimal policies, by solving a sequence of
problems in which the scenarios subproblems are not the original ones, but a modified
version of those by including some penalty terms. The algorithm is an iterative process
starting from U
0
and computing at each iteration k a collection of contingent policies ˆ
U
k
which are then aggregated into a collection of implementable policies U
k
that are ˆ
supposed to converge to the optimal solution U . The contingent policies U
k
are found as
optimal solutions to the modified scenario subproblems:
50
This term was coined by Rockafellar and Wets (1991). The idea is based on the theory
of the proximal point algorithm in nonlinear programming.
60
min F
s
( x
s
,u
s
) + w
s
u
s
+
1
µ u
s
÷ u
s
ˆ
2
2
(2.5.2)
where ? is the ordinary Euclidian norm, µ is a penalty parameter and w
s
is an
information price
51
. The use of µ is justified by the fact that the new contingent policy
should not depart too much from the implementable policy found in the previous
iteration. The modified scenario subproblems (2.5.2) have the form of an augmented
Lagrangian.
In the next subsection, I present a detailed description of the progressive hedging
algorithm, which uses subproblems in the form of an augmented Lagrangian as shown
above.
II.5.1. Description of the Progressive Hedging Algorithm
The optimal solution of the problem described by equations (2.3.1) - (2.3.2), U ,
represents the best response an optimizing agent can come up with in the presence of
uncertainty. An advantage of this algorithm is that one does not necessarily need to solve
subproblems (2.5.2) exactly. A good approximation
52
of the solution is enough in
allowing one to solve for the solution of the global problem.
Let U
k
denote a collection of admissible policies and W
k
a collection of
information prices corresponding to iteration k . The progressive hedging algorithm, as
designed by Rockafellar and Wets (1991), consists of the following steps:
51
52
I borrow this term from Rockafellar and Wets (1991).
One can envision transforming the scenario subproblems into quadratic problems by
using second order Taylor approximations.
61
Step 0. Choose a value for µ , W
0
and for U
0
. The value of µ may remain
constant throughout the algorithm but it can also be adjusted from iteration to iteration
53
.
Changing the value of µ may improve the speed of convergence. Throughout this
chapter, I will consider µ as being constant. U
0
can be composed of the contingent
policies u
s
(0
)
= u
1
s(0
)
,u
1
s(0
)
,...,u
T
s(0
)
obtained from solving all the scenarios subproblems,
( )
whether modified or not. W
0
can be initialized to zero, W
0
= 0 . Calculate the collection
of implementable policies, U
0
= JU
0
, where J is the aggregation operator
54
. ˆ
Step 1. For every scenario s e S , solve the subproblem:
min ¿
?
F
t
s
( x
s
t ,u
s
t ) + w
s
t u
s
t +
1
µ u
s
t ÷ u
s
t ? T
?
t=1 ?
2
ˆ 2?
?
(2.5.3)
For iteration k +1 , let u
s
(k+1
)
= u
1
s(k+1
)
,u
s
2(k+1
)
,...,u
T
s(k+1
)
denote the solution to the
( )
subproblem corresponding to scenario s . This contingent policy is admissible but not
necessarily implementable. Let U
k
+
1
be the collection of all contingent policies u
s
(k+1
)
.
Step2. Calculate the collection of implementable policies, U
k
+
1
= JU
k
+
1
. While ˆ
these policies are implementable, they are not necessarily admissible in some cases
55
. If
the policies obtained are deemed a good approximation, the algorithm can stop. A
stopping criterion should be employed in this step.
53
See Rockafeller and Wets (1991) and Helgason and Wallace (1991a, 1991b) for a
discussion on the values of µ . Rosa and Ruszczynski (1994) also provide some
algorithm for updating similar penalty parameters.
54
55
See the appendix for more details on the aggregation operator.
Contingent policies are always admissible. If the domain of admissible policies is
convex then any linear combination of the contingent policies will also belong to that
domain. As noted above, by definition, the aggregation operator is linear. Therefore, for a
convex problem the implementable policies computed in step 1 are also admissible.
62
Step3. Update the collection of information prices W
k
+
1
by the following rule:
W
k
+
1
= W
k
+ µ U
k
÷U
k
(
For each scenario s e S rule (2.5.4) translates into:
ˆ
)
(2.5.4)
w
s
t (k+1
)
= w
s
t (k
)
+ µ u
s
t (k
)
÷ u
s
t(k)
(
ˆ
)
for t = 1,...,T
(2.5.5)
This updating rule is derived from the augmented Lagrangian theory. In principle, the
rule can be changed with something else as long as the decomposition properties are not
altered.
Step 4. Reassign k := k +1 and go back to step one.
Next, I investigate how this methodology can be applied to a type of dynamic
programming problem closed to what is often employed by economists for their models.
II.6. Using Scenario Aggregation to Solve a Finite Horizon Life Cycle Model
In this section, I will take a closer look at the viability of scenario aggregation in
approximating a rational expectations model. I choose a standard finite horizon life cycle
model that has an analytical solution, which will be used as a benchmark for the
performance of the scenario aggregation method.
I start by presenting an algorithm for solving the problem given by (2.2.1) -
(2.2.4) under the assumption that the length of the horizon, T , and the number of
realizations of the forcing variable, n , are relatively small. The algorithm used is similar
to that developed by Rockefeller and Wets (1991). As mentioned above, the idea is to
split the problem into many smaller problems based on scenario decomposition and solve
those problems iteratively imposing the non-anticipativity constraint. For computational
convenience, I will reformulate the problem (2.2.1) - (2.2.4) as a minimization rather than
63
maximization. Hence, for each scenario s e S , represented by the sequence of
realizations y
s
= y
s
0 , y
1
s
,K, y
T
s
, the problem becomes:
( )
min
?
¿ |
t
?÷F
t
(c
s
t ) + w
s
t c
s
t +
1
µ (c
s
t ÷ c
s
t )
?
?
?
T
?
2
??
(2.6.1)
subject to
c
t
? t =0 ?
2
??
A
ts
= 1+ r
t
s
A
t
s÷
1
+ y
s
t ÷ c
s
t , t = 0,1,...,T
( ) (2.6.2)
Expressing c
st
and c
st
as a function of A
t
s
and A
t
s
, the augmented Lagrangian function,
for a fixed scenario s , becomes:
L
=
¿ |
t
÷F
t
?
(1+ r
t
s
) A
t
s÷
1
+ y
s
t ÷ A
ts
? + w
s
t
?
(1+ r
t
s
) A
t
s÷
1
+ y
s
t ÷ A
ts
? +
T
t =0
{
? ? ? ? (2.6.3)
+
1
µ ?(1+ r
t
s
) A
t
s÷
1
+ y
s
t ÷ A
ts
÷
(1+ r
t
s
) A
t
s÷
1
+ y
s
t ÷ A
ts
? ?
2?
( )
2
??
?
All the underlined variables in the above equations represent implementable policies or
states derived from applying implementable policies.
Before going through the steps of the algorithm, I will make a few assumptions
about the functional form of the utility function as well as about the interest rate. First, it
is assumed that preferences are described by a negative exponential utility function.
Hence:
F
t
(c
t
) = ÷ 1 exp (÷u c
t
)
u
(2.6.4)
where u is the risk aversion coefficient. Secondly, the interest rate, r
t
, is taken to be
constant. Finally, the distribution of the forcing variable is approximated by a discrete
counterpart. The realizations as well as the associated probabilities are obtained using a
Gauss-Hermite quadrature and matching the moments up to order two. The number of
points used to approximate the original distribution determines the number of scenarios.
64
By decomposing the original problem into scenarios, the subproblems become
deterministic versions of the original model.
II.6.1. The Algorithm
Given the assumptions made in the previous section, problem (2.6.1) becomes:
min
?
¿
?
? T |
t
?
1 exp ÷u c
s
+ w
s
c
s
+
1
µ c
s
÷ c
s
2 ??
c
t
? t =0
?u
(
t
) t
t
2 ( t
t
)
?
???
(2.6.5)
Consequently the Lagrangian for scenario s is:
L
=
¿ |
t
? 1 exp ?÷
u
((1+
r
) A
t
s÷
1
+ y
s
t ÷ A
t
s
)? + w
s
t
?
(1+
r
) A
t
s÷
1
+ y
s
t ÷ A
ts
? + T
t =0
?
?
u
? ? ? ? (2.6.6)
+
1
µ ?(1+
r
) A
t
s÷
1
+ y
s
t ÷ A
ts
÷
((1+
r
) A
t
÷
1
+ y
s
t ÷ A
t
)? ? 2
2? ??
?
Since the consumption variable was replaced by a function of the asset level, the
algorithm will be presented in terms of solving for the level of assets.
Step 0. Initialization: Set w
st
= 0 for all stages t and scenarios s . Choose a value
for µ that remains constant throughout the algorithm, let it be µ = 5 . Later on, in this
chapter, I will discuss the impact the value of µ has on the convergence process. At this
point, one needs a first set of policies. The convergence process, and implicitly the speed
of the algorithm, is impacted by the choice of the first set of policies.
One suggestion made in the literature by Helgason and Wallace (1991a, 1991b) is
to use the solution to the deterministic version of the model. This would amount to using
the certainty equivalence solution in this case. I will first implement the algorithm using
as starting point the certainty equivalence solution and then I will take advantage of the
fact that for certain specifications of the model each scenario subproblem has an exact
solution. I will then compare the convergence properties of the algorithm in these two
cases.
65
Let
{c
ceq
}
t
=
0
denote the solution to the deterministic problem. Then, using the T
t
transition equation (2.6.2) one can compute the level of assets for each scenario s ,
A
s
(0
)
= A
0
s(0
)
, A
1
s(0
)
,..., A
T
s(÷01
)
. Next, it becomes possible to compute the implementable
{ }
states A
(
0
)
= A
(
00
)
, A
1
(0
)
,...,
A
T
(0÷)1{
}
as a weighted average of A
0
s(0
)
corresponding to all
scenarios s , using as weights the probabilities of realization for each scenario.
Alternatively, one can compute the first set of contingent policies by solving a
deterministic life cycle consumption model for each scenario s :
mi
sn
¿ |
t
1 exp ÷u
?
(1+
r
) A
t
s÷
1
+ y
s
t ÷ A
ts
?
A
t
T
t =0
u
{
?
?
}
(2.6.7)
with A
(
÷s1
)
and A
T
(s
)
given. As before, let A
s
(0
)
= A
0
s(0
)
, A
1
s(0
)
,..., A
T
s(÷01
)
denote the solution
{ }
to this problem. This solution is admissible but not implementable. The implementable
solution for each period t , A
t
0
, is computed as the weighted average of all the contingent
solutions for period t , A
t
s(0
)
, with the weights being given by the probability of
realization for each particular scenario s .
Step 1. For every scenario s e S , solve the subproblem:
mi
sn
¿ |
t
? 1 exp ÷u
?
(1+
r
) A
t
s÷
1
+ y
s
t ÷ A
ts
? +
A
t
T
t =0
?
u
?
{
?
?}
+W
ts ?
(1+
r
) A
t
s÷
1
+ y
s
t ÷ A
ts
? +
? ?
(2.6.8)
+
1
µ (1+
r
) A
t
s÷
1
+ y
s
t ÷ A
ts
÷
?
(1+
r
) A
t
÷
1
+ y
s
t ÷ A
t
? ?
2
{
?
2
??
}
?
A detailed description of how the solution is computed can be found in the Appendix.
The advantage of the scenario aggregation method is that the solution to problem (2.6.8)
does not have to be computed exactly.
66
Let A
s
(k
)
= A
0
s(k
)
, A
1
s(k
)
,...,
A
T
s(÷k1) {
} denote the contingent solution to this problem,
where k denotes the iteration. Based on this solution I also compute the consumption
path for each scenario, c
s
(k
)
. This solution is admissible but not implementable and
therefore the next step is to compute the implementable solution based on the contingent
solutions A
s
(k
)
.
Step 2. First, compute the implementable states A
k
. As it was mentioned in step
0, A
t
k
is computed as the weighted average of all the contingent solutions for period t ,
A
t
s(k
)
, with the weights being given by the probability of realization for each particular
scenario s . Since the space of the solutions for the problem being solved is convex, the
implementable solution is also admissible. At this point, if solution A
k
is considered
good enough, the algorithm can stop and A
k
becomes officially the solution of the
problem described by (2.2.1) - (2.2.4). In order to make a decision on the viability of A
k
as the optimal solution, one needs to define a stopping criterion. Based on the value of
A
k
I compute the implementable consumption path c
k
and then use the following error
sequence
56
:
c(k
)
=
¿ |t
?
( c
(
tk
)
÷ c
(
tk÷1) )2 + A
(
tk
)
÷ A
(
tk÷1) ?
T
t =0
?
?
(
)
?
2
?
(2.6.9)
where k is the iteration number. The termination criterion is c (k
)
< d where d is
arbitrarily chosen. In the next section, I will discuss the importance of the stopping
criterion in determining the accuracy of the method.
56
This is similar to what Helgason and Wallace (1990a) proposed. Later on in this
chapter we will discuss the impact the choice of the value for d has on the results.
67
Step 3. For t = 0,1,...,T and all scenarios s update the information prices:
w
s
t (k+1
)
= w
s
t (k
)
+ µ ?(1+
r
) A
t
s÷(k
)
÷ A
t
(÷k1
)
÷ A
t
s(k
)
÷ A
t
(k
)
? for t = 1,...,T
?
(
1
)( )
?
Step 4. Reassign k := k +1 and go back to step one.
II.6.2. Simulation Results
In this section, I present a brief picture of the results obtained by the
implementation of the scenario aggregation method compared to the analytical solution.
These results show that the numerical approximation obtained through scenario
aggregation is close to the analytical solution for certain parameterizations of the model.
In order to asses the accuracy of the scenario aggregation method I will use several
criteria put forward in the literature. First, I compare the decision rule, i.e. the
consumption path obtained through scenario aggregation with the values obtained from
the analytical solution. In this context, I use two relative criteria similar to what Collard
and Juillard (2001) use. One, E
a
, gives the average departure from the analytical solution R
and is defined as:
a
T +
1
¿
c
*t1
T
c
*
t ÷ c
t
E= R
t =0
(2.6.10)
The other, E
m
, represents the maximal relative error and is defined as: R
E = max ? c
t
÷*c
t
m
?*
?
?
T
(2.6.11)
R
?c
?
? ?
t
?
t
=0 ?
where c
*
t is the analytical solution and c
t
is the value obtained through scenario
aggregation. Alternatively, since the problem is ultimately solved in terms of the level of
assets, the two criteria could also be expressed using the level of assets:
68
E
=
¿t*a
?
t
??T
÷1
R
1
T
÷
1
A
*
÷ A
t
, E
m
= max ? A
*
÷ A
t
T
t
=
0
A
t
R
? A
*
?
?
?
t
?
t
=0 ?
where A
*
is given by the analytical solution and A
t
by the scenario aggregation. Even t
though the scenario aggregation methodology does not use the Euler equation in
obtaining the solution, I will use the Euler equation based criteria proposed by Judd
(1998) as an alternative for determining the accuracy of the approximation. The criterion
is defined as a one period optimization error relative to the decision rule. The measure is
obtained by dividing the current residual of the Euler equation to the value of next
period's decision function. Subsequently, two different norms are applied to the error
term: one, E
a
, gives the average and the other, E
m
, supplies the maximum. Judd (1998)
E E
labeled these criteria as measures of bounded rationality.
The simulations were done using the following common set of parameter values:
the discount factor | = 0.96 ; the initial and terminal values for the level of assets
A
÷
1
= 500 and A
T
= 1000 ; the income generating process has a starting value of
y
0
= 200 . In addition, the interest rate is assumed deterministic. I used two values for the
interest rate, r = 0.04 and r = 0.06 . The distribution of the forcing variable was
approximated by a 3 point discrete distribution. As I mentioned in the description of the
progressive hedging algorithm, a few factors can influence the performance of the
scenario aggregation method. Let us first look at how the starting values and stopping
criterion influence the results.
69
II.6.2.1. Starting Values and Stopping Criterion
As I mentioned above, the starting values and the stopping criterion are very
important elements in the implementation of the algorithm. I consider for the moment
that the starting values are given by the certainty equivalence solution of the life cycle
consumption model. I analyze the case where the value for the coefficient of risk aversion
is u = 0.01, the variance for the income process is o2
y
= 100 and the interest rate is
r = 0.06 . The stopping criterion is given by the sequence c
(
k
)
as defined in (2.6.9) and I
arbitrarily choose d = 0.004 . Therefore when c
(
k
)
becomes smaller than d = 0.004 I stop
and declare the solution obtained in iteration k as the solution to the problem described
by (2.2.1) - (2.2.4). In Table 1 I provide the values for the accuracy measures discussed
above, using the level of assets, as opposed to the level of consumption. One can see that
the approximation to the analytical solution obtained by stopping when c
(
k
)
is smaller
than the arbitrarily chosen d is very good.
Table 1. Accuracy measures for d=.004
u
0.01
o
2
y
E
a
R
E
m
R
E
a
E
E
m
E
100 0.001445515 0.002392885 0.000005019 0.000008735
70
The results presented in Table 1 are obtained after 159 iterations. Next, I will look
at the behavior of the sequence c
(
k ) for the case presented above.
3.5
3
2.5
2
1.5
1
0.5
0
Evolution for the c(k)sequence
5
4
3
2
1
0
x 10
-3
Evolution for the c(k)sequence (zoom)
0 50 100 150 200 250 300 350 150 200 250 300
Iteration k Iteration k
Evolution of the value for Evolution of the value for
the objective function the objective function (zoom)
-111.5
-112
-112.5
-113
-113.5
-114
-114.5
-111.876
-111.878
-111.88
-111.882
-111.884
-111.886
-111.888
-115 -111.89
0 50 100 150 200 250 300 350 150 200 250 300 350
Iteration k Iteration k
Figure 3. Evolution of the c
(
k ) sequence and the value of the objective for u = .01 and o2
y
= 100
One can see in Figure 3 that the value for sequence c
(
k
)
continues to decrease
until iteration 250 when it attains the minimum value. At the same time, the value of the
objective continues to increase until iteration 266 when it attains its maximum. It is worth
noting that the value of the objective is computed as in equation (2.6.12). Based on these
observations one may elect to choose as stopping criterion the point where c
(
k
)
attains its
minimum or when the objective function attains its maximum as opposed to an arbitrary
value d . Next, I look at how close is the approximation to the analytical solution when
71
v
a
l
u
e
o
f
c
( k
) s
e
q
u
e
n
c
e
v
a
l
u
e
o
f
c
( k
) s
e
q
u
e
n
c
e
V
a
l
u
e
o
f
t
h
e
o
b
j
e
c
t
i
v
e
f
u
n
c
t
i
o
n
V
a
l
u
e
o
f
t
h
e
o
b
j
e
c
t
i
v
e
f
u
n
c
t
i
o
n
using these criteria. In Table 2 one can see that there is not much difference between the
last two criteria when compared to the analytical solution. The only difference is that the
value of the expected utility is marginally higher in the second case.
Table 2. Accuracy measures for various stopping criteria
u
0.01
o
2
y
E
a
R
E
m
R
E
a
E
E
m
E
Stopping criterion
100 0.001445515 0.002392885 0.000005019 0.000008735 Arbitrary d = 0.004
100 0.002137894 0.002691210 0.000007190 0.000013733 Minimum of c
(
k)
100 0.002137894 0.002691210 0.000007190 0.000013733 Maximum objective
A somewhat interesting result is that the ad-hoc stopping criterion d = 0.004
leads to a better approximation of the analytical solution. This is explained by the fact
that the progressive hedging algorithm leads to the solution that would be obtained
through the aggregation of the exact solutions for every scenario. Here the starting point
is the certainty equivalent solution and the path to convergence, at some point, is very
close to the analytical solution.
II.6.3. The Role of the Penalty Parameter
In the implementation of the progressive hedging algorithm, I chose the penalty
parameter to be constant. Its role is to keep the contingent solution for each iteration close
to the previous implementable policy. However, its value also has an impact on the speed
of convergence. I will now consider the previous parameterization of the model and I am
72
going to change the value of the penalty parameter to see how it changes the speed of
convergence. In Figure 4 one can see that as µ increases so does the number of iterations
needed to achieve convergence. While a higher value of the penalty parameter helps the
convergence of contingent policies to the implementable policy, it also slows the global
convergence process, requiring more iterations.
Evolution for the c(k)sequence Evolution for the c(k)sequence
-3 for µ = 0.1 -5x 10 x 10 for µ = 0.5
5 3
2.54
2
3
1.5
2
1
10.5
0 0
150 200 250 300 1000 1200 1400 1600 1800 2000 2200
Iteration k Iteration k
Evolution for the c(k)sequence Evolution for the c(k)sequence
x 10
-9
for µ = 2 for µ = 5 x 10
-11
4 8
3.5 7
3 6
2.5 5
2 4
1.5 3
1 2
0.5 1
0 0
7000 7500 8000 8500 9000 9500 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8
Iteration k Iteration k x 10
4
Figure 4. Convergence for different values of the penalty parameter.
For µ = 0.1 , 250 iterations are needed to achieve convergence, while for µ = 0.5 ,
1780 iterations are needed. For higher values, such as µ = 5 , the number of iterations
needed to achieve convergence increases to over 25000 iterations.
73
v
a
l
u
e
o
f
c
( k
) s
e
q
u
e
n
c
e
v
a
l
u
e
o
f
c
( k
) s
e
q
u
e
n
c
e
V
a
l
u
e
o
f
t
h
e
o
b
j
e
c
t
i
v
e
f
u
n
c
t
i
o
n
V
a
l
u
e
o
f
t
h
e
o
b
j
e
c
t
i
v
e
f
u
n
c
t
i
o
n
II.6.4. More simulations
In this section I investigate how close the scenario aggregation solution is to the
analytical solution for various parameters. Table 3 shows the values for the four criteria
enumerated above for different values of the coefficient of risk aversion and of the
variance of the random variable entering the income process. All the simulations whose
results are presented in Table 3 were done using a three point approximation of the
distribution of the random variable entering the income process. The relative measures
are computed using the level of assets.
Table 3. Accuracy measures for various parameters when interest rate r=0.04
u
o2y 0.01 0.05 0.1
E
a
R
E
m
R
E
a
E
E
m
E
E
a
R
E
m
R
E
a
E
E
m
E
E
a
R
E
m
R
E
a
E
E
m
E
1 .0000 .0000 .0000 .0000 .0001 .0001 .0000 .0000 .0002 .0003 .0000 .0000
4 .0000 .0001 .0000 .0000 .0004 .0005 .0000 .0000 .0009 .0011 .0000 .0000
25 .0005 .0007 .0000 .0000 .0029 .0037 .0000 .0000 .0058 .0074 .0000 .0000
100 .0023 .0029 .0000 .0000 .0116 .0147 .0000 .0000 .0230 .0290 .0000 .0000
For lower values of the coefficient of risk aversion the approximation is relatively good.
As the coefficient of risk aversion increases in tandem with the variance of the income
process, the accuracy suffers when looking at relative measures. The Euler equation
measure still indicates a very good approximation.
Let us now look at how this approximation affects the value of the original
objective, i.e. the expected discounted utility over the lifetime horizon. Table 4 shows the
74
ratio of the expected utilities for the whole horizon with the scenario aggregation as the
as the denominator and the analytical solution as the numerator.
Table 4. The ratio of lifetime expected utilities
F
as
F
sc
u
o2y 0.01 0.03 0.05 0.1
1 1.00000 1.00000 1.00000 1.00003
4 1.00000 1.00001 1.00003 1.00058
25 1.00000 1.00002 1.00141 1.02027
100 1.00003 1.00051 1.02273 1.39364
The discounted utilities are computed as in the original formulation of the problem:
F
sc
= 1
N
¿1 ?
?
¿
0
??
u
(
t
)
?
??? ?÷
T |
t
? 1 exp ÷u c
i
??
(2.6.12)
and
N
i= t=
F
as
= 1
N
?÷T |
t
? 1 exp ÷u c
i
* ?? (2.6.13)
N
¿1 ?
?
¿
0
??
u
(
t
)
?
???
i= t=
where N is the number of simulations, F
sc
is the discounted utility obtained with
scenario aggregation and
F
as
is the discounted utility obtained with the analytical
solution. In this formulation, both quantities are negative so their ratio is positive. Note
however that the initial formulation of the problem using the objective function specified
in (2.6.12) and (2.6.13) was a maximization. Therefore, higher ratio in Table 4 means that
the solution obtained through scenario aggregation leads to higher discounted lifetime
utility than the analytical solution. I simulate 2000 realizations of the income process and
then I average the discounted utilities over this sample. The result shows that the solution
75
obtained through scenario aggregation leads to higher overall expected utility as the
coefficient of risk aversion increases. This is explained by the fact that the level of
consumption in the first few periods is higher in the case of scenario aggregation. In the
context of a short horizon, this leads to higher levels of discounted utility.
II.7. Final Remarks
The results show that scenario aggregation can be used to provide a good
approximation to the solution of a life-cycle model for certain values of the parameters.
There are a few remarks to be made regarding the convergence. As pointed out earlier in
this chapter the value of µ has an impact on the speed of convergence. Higher values of
µ lead to faster convergence of the contingent policies towards an implementable policy
but that also means that the overall convergence is slower and hence it impacts the
accuracy if an ad-hoc stopping criterion is used. Therefore, one needs to choose carefully
the values of the ad-hoc parameters. On the other hand, if the scenario problems have an
exact solution then the final implementable policy can be obtained through a simple
weighted average with the weights being the probabilities of realization for each scenario.
76
Chapter III. Impact of Bounded Rationality
57
on the Magnitude of
Precautionary Saving
III.1. Introduction
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 supporters
58
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 rationality.
57
The concept of bounded rationality in this chapter should be understood as a set of
assumptions that departs from the usual rational expectation paradigm. Its meaning will
become clear later in the chapter when the underlying assumptions are spelled out.
58
Sargent (1993) for example, identifies several areas in which bounded rationality can
potentially help, such as equilibrium selection in the case of multiple possible equilibria
and behavior under "regime changes".
77
One 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, simulation-
based models, artificial intelligence based methodologies such as neural networks and
genetic algorithms, evolutionary models drawing their roots from biology, behavioral
78
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 thumb
59
. 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.
59
Some of the examples are Gali et al. (2004), Allen and Carroll (2001), Lettau and
Uhlig (1999) and Ingram (1990).
79
III.2. Empirical Results on Precautionary Saving
There seems to be little agreement in the empirical literature on precautionary
saving, especially when it comes to its relationship to uncertainty. Skinner (1988) found
that saving was lower than average for certain groups
60
of households that are perceived
to have higher than average income uncertainty. In the same camp, Guiso, Jappelli and
Terlizzese (1992), using data from the 1989 Italian Survey of Household Income and
Wealth, found little correlation between the level of future income uncertainty and the
level of consumption
61
. In addition, Dynan (1993), using data from the Consumer
Expenditure Survey, estimated the coefficient of relative prudence and found it to be "too
small to be consistent with widely accepted beliefs about risk aversion".
On the other hand, Dardanoni (1991) basing his analysis on the 1984 cross-
section of the UK FES (Family Expenditure Survey) suggested that the majority of
saving in the sample arises for precautionary motives. He found that average
consumption across occupation and industry groups was negatively related to the within
group variance of income. Carroll (1994) found that income uncertainty was statistically
important in regressions of current consumption on current income, future income and
uncertainty. Using UK FES data, Merrigan and Normandin (1996) estimated a model
where expected consumption growth is a function of expected squared consumption
growth and demographic variables and their results, based on the period 1968-1986,
60
61
Specifically, the groups identified were farmers and self-employed.
In fact the study on Italian consumers did find that consumption was marginally lower
while wealth was marginally higher for those who were facing higher income uncertainty in
the near future.
80
indicate that precautionary saving is an important part of household behavior. Miles
(1997), using several years of cross-sections of the UK micro data and regressing
consumption on several proxies for permanent income and uncertainty, found that, for
each cross-section, the latter variable played a statistically significant role in determining
consumption. In a study trying to measure the impact of income uncertainty on household
wealth, Carroll and Samwick (1997), using the Panel Study of Income Dynamics, found
that about a third of the wealth is attributable to greater uncertainty. Later on, Banks et al.
(2001), exploiting not only the cross-sectional, but also the time-series dimension of their
data set, find that section specific income uncertainty as opposed to aggregate income
uncertainty plays a role in precautionary saving. Finally, Guariglia (2001) finds that
various measures of income uncertainty have a statistically significant effect on savings
decisions.
In this chapter, I am going to show that, by introducing bounded rationality in a
standard life cycle model, one can increase the richness of the possible results. Even if
the setup of the model would imply the existence of precautionary savings, under certain
parameter values and rules followed by consumers, the precautionary saving is apparently
almost inexistent. As opposed to most of the literature
62
studying precautionary savings, I
introduce uncertainty in the interest rate, beside income uncertainty. In this context, the
size of precautionary saving no longer depends exclusively on income uncertainty.
62
A notable exception is Binder et al. (2000).
81
III.3. The Model
I start this section by presenting the formulation of a standard finite horizon life-
cycle consumption model. Then I will introduce a form of bounded rationality
63
and
investigate the path for consumption and savings.
Consider the finite-horizon life-cycle model under negative exponential utility.
Suppose an individual agent is faced with the following intertemporal optimization
problem:
ma
T
x E
?
¿ ÷|
t
1 exp (÷u c
t
) | I
0
? T
subject to
{c
t
}
t
=0
??
t =0
u ?
?
(3.3.1)
A
t =
(1+ r
t
) A
t
÷
1
+ y
t
÷ c
t
, t = 0,1,...,T ÷1, (3.3.2)
A
t
> ÷b, with A
-1
, A
T
given (3.3.3)
where u is the coefficient of risk aversion, A
t
represents the level of assets at the
beginning of period t , y
t
the labor income at time t , and c
t
represents consumption in
period t . The initial and terminal conditions, A
÷
1
and A
T
are given. The information set
I
0
contains the level of consumption, assets, labor income and interest rate for period
zero and all previous periods. The labor income is assumed to follow an arithmetic
random walk:
63
As it was already mentioned above, the approach in defining bounded rationality in this
chapter has some similarities to the approach followed by Lettau and Uhlig (1999) in the
sense that several rules are used to account for the inability of the boundedly rational
agent to optimize over long horizons.
82
y
t
= y
t
÷
1
+ ç
t
, t = 1,...,T , with y
o
given (3.3.4)
and ç
t
being drawn from a normal distribution, ç
t
~
N
(0,o2
y
) . When the interest rate is
deterministic, this problem has an analytical solution
64
. However, if the interest rate is
stochastic, the solution of this finite horizon life cycle model becomes more complicated
and it can not be computed analytically. For now, I will not make any particular
assumption about the process generating the interest rate. Therefore, to summarize the
model, a representative consumer derives utility in period t from consuming c
t
,
discounts future utility at a rate | and wants, in period zero, to maximize his present
discounted value of future utilities for a horizon of T +1 periods. At the beginning of
each period t the consumer receives a stochastic labor income y
t
, finds out the return r
t
on his assets A
t
÷
1
, from the beginning of period t ÷1 to the beginning of period t , and, by
choosing c
t
, determines the level of assets A
t
according to equation (3.3.2).
Now, I introduce a rationally bounded agent in the following way. First, I assume
that the agent does not have either the resources or the sophistication to be able to
optimize over a long horizon. For example, if the agent enters the labor force at time zero
and faces the problem described by (3.3.1) - (3.3.4) over a time span extending until his
retirement, let it be period T , the assumption is that the agent does not have the ability to
optimally choose, at time zero, a consumption plan over that span. Instead, he focuses on
choosing a consumption plan over a shorter horizon, let it be T
h
+1 periods.
Secondly, because of his limited ability to process large amounts of information
he repeats this process every period in order to take advantage of any new available
64
See the appendix for a detailed description of the analytical solution.
83
information. This idea of a shorter and shifting optimization horizon is similar to the
approach taken by Prucha and Nadiri
65
(1984, 1986, and 1991). Now, the question is how
an individual who lacks sophistication, can optimally
66
choose a consumption plan even
for a short time span. In order to model the decision process I make use of the scenario
aggregation method. Under this assumption, the agent evaluates several possible paths
based on the realization of the forcing variables specified in the model. By assigning
probabilities to each of the possible paths, the agent is in the position to aggregate the
scenarios (paths), i.e., to compute the expected value for his decision.
In order to be able to use the scenario aggregation method, the forcing variables
need to have a discrete distribution but in the model presented above, they are described
as being drawn from a normal distribution. This leads to the third element that can be
brought under the umbrella of bounded rationality. Since the agent has limited
computational ability, the distribution of the forcing variable is approximated by a
discrete distribution with the same mean and variance as the original distribution. This
approximation does not necessarily have to be viewed as a bounded rationality element
since similar approaches have been employed repeatedly in numerical solutions using
state space discretization
67
.
Given the assumptions made about the abilities of the rationally bounded
representative agent, I will now go through the details of solving the problem described
65
In their work, a finite and shifting optimization horizon is used to approximate an
infinite horizon model.
66
67
Optimality here means the best possible solution given the level of ability.
Tauchen, among others, used this kind of approximation on various occasions, such as
Tauchen (1990), Tauchen and Hussey (1991).
84
by equations (3.3.1) - (3.3.4). Hence, at every point in time, t , the agent solves the
problem:
ma}xh
E
?
¿ ÷|
t
1 exp (÷u c
t
+
t
) | I
t
?
for
t = 0,1,...,T ÷ T
h
T
h
or
{c
t
+tT=0t
?
?
t
=0
u ?
?
(3.3.5)
ma}x
÷
t
E
?
¿ ÷|
t
1 exp (÷u c
t
+
t
) | I
t
? for t = T ÷ T
h
+1,...,T ÷1 T ÷t
subject to
{c
t
+t
T
=0t
??
t
=
0
u ?
?
(3.3.6)
A
t
+
t
=
(1+ r
t
+
t
) A
t
+t÷
1
+ y
t
+
t
÷ c
t
+
t
,
(3.3.7)
t = 0,1,...,T ÷1, t = 0,..., min (T
h
,T ÷
t
)
with A
-1
, A
t
-1, A
t
+T
h
and A
T
given (3.3.8)
where A
t
+
t
represents the level of assets at the beginning of period t +t , y
t
+
t
the labor
income at time t +t , and c
t
+
t
represents consumption in period t +t . The initial and
terminal conditions, A
-1
, A
t
÷
1
, A
t
+T
h
and A
T
are given. The information set I
t
contains the
level of consumption, assets, labor income and interest rate for period t and all previous
periods. The labor income is assumed to follow an arithmetic random walk:
y
t
+
t
= y
t
+t ÷
1
+ ç
t
b+
t
,
t = 1,...,T , t = 0,..., min (T
h
,T ÷
t
)
with y
0
given
(3.3.9)
ç
t
b+
t
being drawn from a discrete distribution,
D
(0,o2
y
) with a small number of
realizations.
In making the above assumptions, the belief is that they would better describe the
way individuals make decisions in real life. It is often the case that plans are made for
shorter horizons, but not entirely forgetting about the big picture.
85
Recalling the results of Skinner (1988) who found that saving was lower than
average for farmers and self employed, groups that are otherwise perceived to have
higher than average income uncertainty, one can assume that planning for those groups
does not follow the recipe given by the standard life cycle model. Given the high level of
uncertainty, I believe it would be more appropriate to model these consumers as if they
plan their consumption path only for a short period of time and then reevaluate. This
would be consistent with the fact that farmers change their crop on a cycle of several
years and may be influenced by the fluctuations in the commodities markets and other
government regulations. Similarly, some among the self employed are likely to have
short term contracts and are more prone to reevaluate their strategy on a high frequency
basis. Therefore, the model above seems like a good description on how the decision
making process works. The only detail that remains to be decided is how the consumer
chooses the short horizon terminal condition, that is, the level of assets, or the wealth. For
this purpose, I propose three different rules and I investigate their effect on the saving
behavior.
So far, no assumption has been made about the process governing the realizations
of the interest rate. From now on, I assume that the interest rate is also described by an
arithmetic random walk:
r
t
= r
t
÷
1
+u
t
, t = 1,...,T , with r
o
given (3.3.10)
Since in this formulation the problem does not have an analytical solution, the classical
approach would be to employ numerical methods in order to describe the path of
consumption, even for a very short horizon. In order to find the solution corresponding to
the model incorporating the bounded rationality assumption I will use the scenario
86
aggregation
68
methodology. Then I will compare this solution with the numerical
solution
69
that would result from the rational expectation version of the model when
optimizing over the whole T period horizon.
III.3.1. Rule 1
Under rule 1, the consumer considers several possible scenarios for a short
horizon and assumes that for later periods certainty equivalence holds. In this context, he
makes a decision for the current period and moves on to the next period when he
observes the realization of the forcing variables. Then he repeats the process by making a
decision based on considering all the relevant scenarios for the near future and assuming
certainty equivalence for the distant future. Hence, the decision making process takes
place every period. More precisely, when optimizing in period t , the consumer considers
all the scenarios in the tree event determined by the realizations of the forcing variable
for the first T
h
periods. From period t + T
h
he considers that certainty equivalence holds
for the remaining T ÷ t ÷ T
h
periods. This translates specifically to considering that
income and interest rate are frozen for each existing scenario for the remaining T ÷ t ÷ T
h
periods. To be more specific, for time t = 0 , the consumer considers all the scenarios
available in the event tree for the first T
h
periods and assumes certainty equivalence for
68
Since an analytical solution can be obtained when income follows an arithmetic
random walk and interest rate is deterministic, it is not necessary to discretize both
forcing variables, but only the interest rate. This approach reduces considerably the
computational burden. A short description on the methodology used along with the
solution for one scenario with deterministic, interest rate is presented in the appendix.
More details on the scenario aggregation methodology can be found in the second
chapter.
69
The numerical solution is obtained using projection methods and is due to Binder et al.
(2000).
87
the remaining T ÷ T
h
periods. When it advances to period t = 1, he optimizes again
considering all the scenarios available in the tree event for periods 1, 2,...,T
h
+1 and
assumes certainty equivalence for the remaining T ÷ T
h
÷1 periods.
In fact, this rule can be considered as an extension to the scenario aggregation
method in order to avoid the dimensionality curse. One may recall that due to its
structure, the number of scenarios in the scenario aggregation method increases
exponentially with the number of periods. In effect, this rule is limiting the number of
scenarios considered and it is consistent with a rationally bounded decision maker who
can only consider a limited and, most likely, low number of possible scenarios.
Following are some graphical representations of the simulations for rule 1. Each
graph contains the values for the coefficient of risk aversion, u . The graphs also contain
the numerical solution and, for comparison purposes, the evolution of assets if the
solution were computed in the case of certainty equivalence. I first consider a group of 12
cases varying certain parameters of the model. For all simulations in this group, the total
number of periods considered is T = 40 and the optimizing horizon is T
h
= 6 . The
starting level of income is y
0
= 200 , the initial level of assets is A
÷
1
= 500 while the
terminal value is A
T
= 1000 . The discount factor is | = 0.96 , the starting value for the
interest rate, r
0
= 0.06 while the standard deviation for the interest process is given by
o
r
= 0.0025. I use a discrete distribution with three possible realizations to approximate
the original distribution of the forcing variable and that implies that in each period t , for
t s T ÷ T
h
= 34 , the optimization process goes over 3
T
h = 729 scenarios. For periods
34 = T ÷ T
h
< t s T ÷1 = 39 the number of scenarios considered decreases to 3
T
÷
t
. The
88
parameters that are changing in the simulations are the variance for the income process
and the coefficient of risk aversion. I consider all cases obtained combining three values
for the standard deviation of income, o
y e
{1, 5, 10} and four values for the coefficient
of risk aversion, u
e
{0.005, 0.01, 0.05,
0.1
} . The results presented in this section as well
as for the rest of the chapter are based on 1000 simulations. This means that for both the
income generating process and the interest rate generating process, I consider 1000
realizations for each period. The decision to use only 1000 realizations was based on the
observation that the sample drawn provided a good representation of the arithmetic
random walk process assumed in the model. Specifically, both the mean and the standard
deviation of the sample were close to their theoretical values.
Some general results have emerged from all these simulations. First, the path for
the level of assets for the solution obtained in the bounded rationality case always lies
below the path for the level of assets for the numeric al solution obtained in the rational
expectation case. Consequently, the consumption path in the bounded rationality case
starts with values of consumption higher than in the rational expectations case.
Eventually the paths cross and the consumption level in the rational expectations case
ends up being higher toward the end of the horizon.
89
360
340
320
300
280
260
240
220
200
180
160
ConsumptionPathfor u =0.005
numerical solution
boundedrationality
360
340
320
300
280
260
240
220
200
180
160
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numerical solutionbounded
rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
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Time t (periods)
ConsumptionPathfor u =0.05
numerical solution
boundedrationality
360
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Time t (periods)
ConsumptionPathfor u =0.1
numerical solutionbounded
rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 5. Consumption paths for o
y
= 1, r
0
= 0.06 and o
r
= 0.0025.
One can see in Figure 5 that consumption is increasing over time for both
solutions, with the steepest path corresponding to the lowest value of the coefficient of
risk aversion.
When looking at the asset path for the same value of the standard deviation of the
income process, one notices in Figure 6 that the level of saving in the certainty
equivalence case is mostly higher than the level of saving obtained in the bounded
rationality case as well as under the rational expectations assumption.
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Level of Assets for u =0.01
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numerical solution
boundedrationality
certaintyequivalence
400
numerical solution
boundedrationality
certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
1600
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Level of Assets for u =0.05
1600
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Time t (periods)
Level of Assets for u =0.1
400
numerical solution
boundedrationality
certaintyequivalence
400
numerical solution
boundedrationality
certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 6. Asset paths for o
y
= 1, r
0
= 0.06 and o
r
= 0.0025.
While for lower levels of the coefficient of risk aversion u
e
{0.005,
0.01
} , the
asset path obtained assuming certainty equivalence crosses under the other two paths in
the later part of the horizon, the same is not true for higher values of the coefficient of
risk aversion, u
e
{0.05,
0.1
} .
It is not only the relative position of the three paths that changes in the context of
an increasing coefficient of risk aversion, but also the absolute size of the level of
savings. Moreover, the shape of the paths for both the rational expectation and bounded
rationality case changes from concave to convex.
I present now a new set of simulations with the standard deviation of income
being increased to o
y
= 5 . One can see in Figure 7 that the consumption paths for
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u
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{0.005,
0.01
} are not much different from those presented in Figure 5 while for
higher values of the risk aversion coefficient, u
e
{0.05,
0.1
} , the consumption paths are
steeper than in the previous case.
Looking now at the level of savings, one notices in Figure 8 a similar change to
that observed in the case of consumption. While not much has changed for the lower
values of the coefficient for risk aversion, the asset paths for higher values of the risk
aversion coefficient, u
e
{0.05,
0.1
} , have changed, effectively becoming concave, as
opposed to convex in the previous case. Besides the concavity change, one can observe
that for u = 0.1 the level of assets resulting from the numerical approximation of the
rational expectations model is higher than in the case of certainty equivalence for the
bigger part of the lifetime horizon.
360
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Consumption Path for u=0.005
numerical solution
bounded rationality
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Consumption Path for u=0.01
numerical solution
bounded rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
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Time t (periods)
Consumption Path for u=0.05
numerical solution
bounded rationality
360
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260
240
220
200
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160
Time t (periods)
Consumption Path for u=0.1
numerical solution
bounded rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 7. Consumption paths for o
y
= 5 , r
0
= 0.06 and o
r
= 0.0025.
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Level of Assets for u=0.005
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Level of Assets for u=0.01
numerical solution numerical solution
400 bounded rationality 400 bounded rationality
certainty equivalence certainty equivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
1600
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Time t (periods)
Level of Assets for u=0.05
1600
1400
1200
1000
800
600
Time t (periods)
Level of Assets for u=0.1
numerical solution numerical solution
400 bounded rationality 400 bounded rationality
certainty equivalence certainty equivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 8. Asset paths for o
y
= 5 , r
0
= 0.06 and o
r
= 0.0025.
By raising the variance of the income again, one can see in Figure 9 that the path
for consumption becomes a lot steeper for u
e
{0.05,
0.1
} . On the other hand, there
seems to be little change in the consumption pattern for u = 0.005 .
On the savings front, the level of precautionary saving increases tremendously for
the highest coefficient of risk aversion, u = 0.1, and quite substantially for u = 0.05 .
Consequently, in these two cases, the level of savings for the rational expectation model,
as well as the bounded rationality version, becomes noticeably higher than what certainty
equivalence produces. Yet, the level of savings continues to be higher for the much lower
coefficient of risk aversion, u = 0.005 , when compared with the savings pattern for
u = 0.01 and u = 0.05 .
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Consumption Path for u=0.005
numerical solution
boundedrationality
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Consumption Path for u=0.01
numerical solution bounded
rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
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Time t (periods)
Consumption Path for u=0.05
numerical solution
boundedrationality
400
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200
150
Time t (periods)
Consumption Path for u=0.1
numerical solution bounded
rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 9. Consumption paths for o
y
= 10 , r
0
= 0.06 and o
r
= 0.0025.
Another interesting observation is that if one compares the level of savings from
the panel corresponding to u = 0.05 and o
y
= 10 in Figure 10, to the level of savings
from the panel corresponding to u = 0.005 and o
y
= 1 in Figure 6, the two are almost the
same, if not the later higher. This is to say that for values of coefficient of risk aversion
and of standard deviation for income ten times as high as the ones in Figure 6, the level
of precautionary saving is almost unchanged.
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1800 1800
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1200 1200
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800 800
600 600numerical
solution numerical
solution
boundedrationality boundedrationality400 400
certaintyequivalence certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Level of Assets for u =0.05 Level of Assets for u =0.1
1800 1800
1600 1600
1400 1400
1200 1200
1000 1000
800 800
600 600
numerical solution numerical solution
400 boundedrationality 400 boundedrationality
certaintyequivalence certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 10. Asset paths for o
y
= 10 , r
0
= 0.06 and o
r
= 0.0025.
As a general observation, it seems that the level of precautionary saving derived
from the rational expectation model is consistently higher, even if not by high margins,
than the level of savings obtained in the case of bounded rationality. For consumption,
the paths can be steeper or flatter but the general allure remains the same. The rationally
bounded consumer tends to start with a higher consumption while after a few periods the
unboundedly rational consumer tends to take over and continue to consume more until
the end of the horizon.
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III.3.2. Rule 2
Under rule 2, the consumer considers all the relevant scenarios for the immediate
short horizon and then, for the later periods, he only takes in account what I call the
extreme cases. Rule 2 is similar to rule 1 in the way the decision maker emphasizes the
importance of scenarios only for the short term horizon. The difference is that under rule
2, rather than assuming certainty equivalence for the later periods, the consumer
considers the extreme case scenarios as a way of hedging against uncertainty in the
distant future. More precisely, when optimizing in period t , the consumer considers all
the scenarios in the event tree determined by the realizations of the forcing variable for
the first T
h
periods but then he becomes selective and only considers the extreme cases
70
for the remaining T ÷ t ÷ T
h
periods. To be more specific, for time t = 0 , the consumer
considers all the scenarios available in the event tree for the first T
h
periods and only the
extreme cases for the remaining T ÷ T
h
periods. When it advances to period t = 1, he
optimizes again considering all the scenarios available in the tree event for periods
1, 2,...,T
h
+1 and only the extreme cases for the remaining T ÷ T
h
÷1 periods.
In fact, this rule can also be considered as an extension to the scenario
aggregation method in an attempt order to avoid the dimensionality curse. One may recall
that due to its structure, the number of scenarios in the scenario aggregation method
increases exponentially with the number of periods. This rule is in fact limiting the
number of scenarios considered by trying to keep intact the possible variation in the
forcing variable. As opposed to rule 1 where from time t + T
h
the assumption is that the
70
The notion of extreme cases covers scenarios for which the realization of the forcing
variable remains the same. For more details see section 0 in the appendix.
96
forcing variable keeps its unconditional mean value, that is, zero, until the end of the
horizon, this rule expands the number of scenarios by adding all the extreme case
scenarios stemming from the nodes existent at time t + T
h
. This expansion can also be
seen as the equivalent of placing more weight on the tails of the original distribution of
the forcing variable. This rule is consistent with a rationally bounded decision maker who
can only consider a limited and, most likely, low number of possible scenarios but wants
to account for the variance in the forcing variable in the later periods of the optimization
horizon.
Following are some graphical representations of the simulations for rule 2. The
graphs depicting the consumption paths contain the bounded rationality solution as well
as the numerical solution. For comparison purposes, the graph panels containing the
evolution of assets display the savings pattern resulting from the solution obtained in the
case of certainty equivalence on top of the solutions for the rational expectations and the
bounded rationality models.
As in the case of rule 1, one can see in Figure 11 that consumption is increasing
over time for both solutions, with the steepest path corresponding to the lowest value of
the coefficient of risk aversion.
As opposed to the previous rule, the rationally bounded consumer does not always
start with a higher level of consumption. In fact, in this panel, for u = 0.05 and u = 0.1,
the solution of the rational expectations model has higher starting values for
consumption.
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ConsumptionPathfor u =0.005
numerical solution
boundedrationality
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ConsumptionPathfor u =0.01
numerical solution
boundedrationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
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220
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160
Time t (periods)
ConsumptionPathfor u =0.05
numerical solution
boundedrationality
360
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280
260
240
220
200
180
160
Time t (periods)
ConsumptionPathfor u =0.1
numerical solution
boundedrationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 11. Consumption paths for o
y
= 1, r
0
= 0.06 and o
r
= 0.0025.
Looking at the asset paths for the same value of the standard deviation of the
income process, one can notice in Figure 12 that the level of saving in the certainty
equivalence case is mostly higher than the level of saving obtained in the bounded
rationality case as well as under the rational expectations assumption. While for lower
levels of the coefficient of risk aversion u
e
{0.005,
0.01
} , the asset path obtained
assuming certainty equivalence crosses under the other two paths in the later part of the
horizon, the same is not true for higher values of the coefficient of risk aversion. For
u
e
{0.05,
0.1
} there is only one period, the one next to last, when the level of savings
under certainty equivalence is lower than in the other two cases.
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Level of Assets for u =0.005
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Level of Assets for u =0.01
numerical solution numerical solution
400 boundedrationality 400 boundedrationality
certaintyequivalence certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
1600
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800
600
Time t (periods)
Level of Assets for u =0.05
1600
1400
1200
1000
800
600
Time t (periods)
Level of Assets for u =0.1
numerical solution numerical solution
400 boundedrationality 400 boundedrationality
certaintyequivalence certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 12. Asset paths for o
y
= 1, r
0
= 0.06 and o
r
= 0.0025.
As it was the case with rule 1, an increase in the coefficient of risk aversion
results in a decrease of the absolute size of the level of savings. Moreover, the shape of
the paths for both the rational expectation and bounded rationality cases changes from
concave to convex. As opposed to rule 1, for u
e
{0.05,
0.1
} the level of savings under
bounded rationality is higher than under rational expectations.
The next set of simulations has the standard deviation of income increased to
o
y
= 5 . The consumption paths for u
e
{0.005,
0.01
} in Figure 13 are not much different
from those presented in Figure 11 while for higher values of the risk aversion coefficient,
u
e
{0.05,
0.1
} , the consumption paths are steeper than in the previous case.
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numerical solution
boundedrationality
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ConsumptionPathfor u =0.01
numerical solutionbounded
rationality
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Time t (periods)
ConsumptionPathfor u =0.05
numerical solution
boundedrationality
360
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180
160
Time t (periods)
ConsumptionPathfor u =0.1
numerical solutionbounded
rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 13. Consumption paths for o
y
= 5 , r
0
= 0.06 and o
r
= 0.0025.
For the level of savings, the change is similar to that observed in the case of
consumption. In Figure 14 one can see that, while not much has changed for the lower
values of the coefficient for risk aversion, the asset paths for higher values of the risk
aversion coefficient, u
e
{0.05,
0.1
} , have changed, effectively becoming concave, as
opposed to convex in the previous case. Besides the concavity change, one can observe
that for u = 0.1 the level of assets resulting from the numerical approximation of the
rational expectations model is higher than in the case of certainty equivalence for the
bigger part of the lifetime horizon.
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Level of Assets for u =0.01
numerical solution numerical solution
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certaintyequivalence certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
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600
Time t (periods)
Level of Assets for u =0.05
1600
1400
1200
1000
800
600
Time t (periods)
Level of Assets for u =0.1
numerical solution numerical solution
400 boundedrationality 400 boundedrationality
certaintyequivalence certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 14. Asset paths for o
y
= 5 , r
0
= 0.06 and o
r
= 0.0025.
For a yet higher variance of income, one can notice in Figure 15 that the path for
consumption becomes a lot steeper for u
e
{0.05,
0.1
} . On the other hand, there seems to
be little change in the consumption pattern for u = 0.005 . On the savings front, the level
of precautionary saving increases tremendously for the highest value of the coefficient of
risk aversion considered here, u = 0.1, and quite substantially for u = 0.05 . As it can be
easily seen in Figure 16, in these two cases, the level of savings for the rational
expectation model, as well as the bounded rationality version, becomes noticeably higher
than what certainty equivalence produces.
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Consumption Path for u=0.005
numerical solution
bounded rationality
400
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Consumption Path for u=0.01
numerical solution
bounded rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
400
350
300
250
200
150
Time t (periods)
Consumption Path for u=0.05
numerical solution
bounded rationality
400
350
300
250
200
150
Time t (periods)
Consumption Path for u=0.1
numerical solution
bounded rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 15. Consumption paths for o
y
= 10 , r
0
= 0.06 and o
r
= 0.0025.
Yet, the level of savings continues to be higher for the much lower coefficient of
risk aversion, u = 0.005 , when compared with the savings pattern for u = 0.01 and
u = 0.05 .
As in the case of rule 1, comparing the level of savings from the panel
corresponding to u = 0.05 and o
y
= 10 in Figure 16, to the level of savings from the
panel corresponding to u = 0.005 and o
y
= 1 in Figure 12, leads to the observation that
the two are almost the same. This is to say that for values of the coefficient of risk
aversion and of standard deviation for income ten times as high as the ones in Figure 12,
the level of precautionary saving is almost unchanged.
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1800 1800
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400 boundedrationality 400 boundedrationality
certaintyequivalence certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Level of Assets for u =0.05 Level of Assets for u =0.1
1800 1800
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1400 1400
1200 1200
1000 1000
800 800
600 600
numerical solution numerical solution
400 boundedrationality 400 boundedrationality
certaintyequivalence certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 16. Asset paths for o
y
= 10 , r
0
= 0.06 and o
r
= 0.0025.
As in the case of rule 1 the level of savings under bounded rationality is fairly
close to the level of precautionary saving derived from the rational expectation model.
However, in contrast to rule 1, the relative size depends on the parameters of the model
and hence the level of precautionary saving derived from the rational expectation model
is no longer consistently higher when compared to the level of savings obtained in the
case of bounded rationality. Consequently, the rationally bounded consumer no longer
starts consistently with a higher consumption level.
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III.3.3. Rule 3
In this section, I will consider a simpler rule than the previous two, meaning that
the level of wealth A
t
+T
h
is chosen such that, given the number of periods left until time
T , a constant growth rate would insure that the final level of wealth is A
T
.
Following are some graphical representations of the simulations for rule 3. All the
graphs contain a representation of the numerical solution and, for comparison purposes,
the graphs detailing the evolution for the level of assets also contain the certainty
equivalent solution.
The simulations for rule 3 use the same values of the parameters as in the
previous two sections. Consequently, the numerical solution for the rational expectations
model exhibits the same characteristics as discussed before. Therefore, when presenting
the results in this section I will concentrate on the solution derived from assuming
bounded rationality.
As one can see in Figure 17, the consumption paths have kept their upward slope
but for lower values of the coefficient of risk aversion, the difference between the rational
expectation and bounded rationality solutions is considerably higher than for the previous
two rules. The difference can be clearly seen in the picture, with the rationally bounded
consumer consuming more in the beginning while the unboundedly rational consumers
consumes more from the 12
th
period until the end of the horizon. On the other hand, for
higher values of the coefficient of risk aversion, consumption paths are almost
indistinguishable.
104
360
340
320
300
280
260
240
220
200
180
160
Consumption Path for u=0.005
numerical solution
boundedrationality
360
340
320
300
280
260
240
220
200
180
160
Consumption Path for u=0.01
numerical solution bounded
rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
360
340
320
300
280
260
240
220
200
180
160
Time t (periods)
Consumption Path for u=0.05
numerical solution
boundedrationality
360
340
320
300
280
260
240
220
200
180
160
Time t (periods)
Consumption Path for u=0.1
numerical solution bounded
rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 17. Consumption paths for o
y
= 1, r
0
= 0.06 and o
r
= 0.0025.
Looking at the asset paths in Figure 18 one will notice that, for low values of the
coefficient of risk aversion, the bounded rationality assumption leads to much lower
levels of precautionary saving than in the case of rational expectations or certainty
equivalence. However, the surprising result is that for higher values of the coefficient of
risk aversion, there is almost no difference between the level of savings under rational
expectations and bounded rationality.
By increasing the standard deviation of income to o
y
= 5 , one can see in Figure
19 a clear difference between the consumption paths for bounded rationality and rational
expectations for all levels of risk aversion. As before, the two consumption paths have an
upward slope with the rational expectation solution being the steeper one.
105
C
o
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p
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1600
1400
1200
1000
800
600
Level of Assets for u =0.005
1600
1400
1200
1000
800
600
Level of Assets for u =0.01
numerical solution numerical solution
400 boundedrationality 400 boundedrationality
certaintyequivalence certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
1600
1400
1200
1000
800
600
Time t (periods)
Level of Assets for u =0.05
1600
1400
1200
1000
800
600
Time t (periods)
Level of Assets for u =0.1
400
numerical solution
boundedrationality
certaintyequivalence
400
numerical solution
boundedrationality
certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 18. Asset paths for o
y
= 1, r
0
= 0.06 and o
r
= 0.0025.
360
340
320
300
280
260
240
220
200
180
160
Consumption Path for u=0.005
numerical solution
bounded rationality
360
340
320
300
280
260
240
220
200
180
160
Consumption Path for u=0.01
numerical solution
bounded rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
360
340
320
300
280
260
240
220
200
180
160
Time t (periods)
Consumption Path for u=0.05
numerical solution
bounded rationality
360
340
320
300
280
260
240
220
200
180
160
Time t (periods)
Consumption Path for u=0.1
numerical solution
bounded rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 19. Consumption paths for o
y
= 5 , r
0
= 0.06 and o
r
= 0.0025.
106
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The asset paths represented in Figure 20 show clearly a higher level of
precautionary saving in the case of rational expectations. The path corresponding to
certainty equivalence produces higher levels of saving than the bounded rationality path.
1600
1400
1200
1000
800
600
Level of Assets for u =0.005
1600
1400
1200
1000
800
600
Level of Assets for u =0.01
numerical solution numerical solution
400 bounded rationality 400 bounded rationality
certainty equivalence certainty equivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
1600
1400
1200
1000
800
600
Time t (periods)
Level of Assets for u =0.05
1600
1400
1200
1000
800
600
Time t (periods)
Level of Assets for u =0.1
numerical solution numerical solution
400 bounded rationality 400 bounded rationality
certainty equivalence certainty equivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 20. Asset paths for o
y
= 5 , r
0
= 0.06 and o
r
= 0.0025.
Increasing again the standard deviation for income to o
y
= 10 , one will notice in
Figure 21 that there is not much change in the paths for consumption at low levels of risk
aversion. However, the slope of consumption for u = 0.1 increases quite a lot.
107
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400
350
300
250
200
150
Consumption Path for u=0.005
numerical solution
bounded rationality
400
350
300
250
200
150
Consumption Path for u=0.01
numerical solution bounded
rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
400
350
300
250
200
150
Time t (periods)
Consumption Path for u=0.05
numerical solution
bounded rationality
400
350
300
250
200
150
Time t (periods)
Consumption Path for u=0.1
numerical solution bounded
rationality
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Time t (periods) Time t (periods)
Figure 21. Consumption paths for o
y
= 10 , r
0
= 0.06 and o
r
= 0.0025.
On the saving side, one can see in Figure 22 that for the highest coefficient of risk
aversion, the rational expectations solution provides a much higher level of savings,
while the rationally bounded consumer still saves less than in the case of certainty
equivalence for u = 0.01.
While the level of precautionary saving depends heavily on the parameter values
of the model for the unboundedly rational consumer, the same can not be said for the
rationally bounded consumer in the case of rule 3. The asset path for the rationally
bounded consumer is barely concave and increasing the variance of income does not
seem to create the same type of changes as the ones observed for the fully rational
consumer. This behavior is the result of optimizing for only short periods of time coupled
with the fact that the intermediary asset level targets are chosen assuming a constant
growth rate.
108
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Level of Assets for u=0.005 Level of Assets for u=0.01
1800 1800
1600 1600
1400 1400
1200 1200
1000 1000
800 800
600 600
numerical solution numerical solution
400 boundedrationality 400 boundedrationality
certaintyequivalence certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
1800
1600
1400
1200
1000
800
600
Timet (periods)
Level of Assets for u=0.05
1800
1600
1400
1200
1000
800
600
Timet (periods)
Level of Assets for u=0.1
numerical solution numerical solution
400 boundedrationality 400 boundedrationality
certaintyequivalence certaintyequivalence
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Timet (periods) Timet (periods)
Figure 22. Asset paths for o
y
= 10 , r
0
= 0.06 and o
r
= 0.0025.
In conclusion, in the case of rule 3, the rule employed by the rationally bounded
consumer for the accumulation of assets is overshadowing the precautionary motives
embedded in the functional specification of the model.
109
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III.4. Final Remarks
The level of precautionary saving under bounded rationality depends quite heavily
on the behavior assumptions. While in many of the simulations presented in this chapter
the level of precautionary saving chosen on average by the rationally bounded consumer
is below that resulting from a rational expectations model, there a few parameterizations
of the model, under rule 2, for which the rationally bounded consumer saves more.
The simulations also show that for low coefficients of risk aversion, variation in
income uncertainty does not affect much the level of saving. If one adds to this
observation the possibility that self selection exists (individuals with high risk aversion
choose occupations with low income uncertainty), it is easy to see why some empirical
studies would find relatively low levels of precautionary saving.
Another interesting result is that under rule 3, where the rationally bounded
consumer follows some form of financial planning, there is not much difference for asset
paths across various levels of risk aversion and income uncertainty. This result is
consistent with the observation made by Lusardi (1997) that the saving rates do not
change much across occupations.
Most of the studies looking to asses the importance of precautionary saving, or the
impact of income uncertainty on precautionary saving, have assumed that interest rate
uncertainty does not play an important role in the decision making process. For the model
discussed in this chapter, the assumption of a constant interest rate would result in an
asset path that is constant regardless of the realizations for the income process. By
introducing uncertainty in the interest rate process, that is no longer the case. The
dynamic of the asset path is especially influenced by the realization of the interest rate
110
process for lower levels of risk aversion. Therefore, the empirical literature should also
consider the impact of interest rate uncertainty when studying the importance of
precautionary motives on the level of saving.
While the results presented in this chapter point to an important role for the
bounded rationality in the decision making process, it would be difficult to test the
model's validity in a standard empirical setting. The problem is that the results depend
heavily on the rules adopted as well as on the parameterization of the model and it would
be difficult to distinguish between the effects of the general assumptions corresponding to
bounded rationality and those specific to a particular rule. Therefore, a more appropriate
framework for testing the validity of the model would be an experimental setting. In such
a framework, one can potentially "calibrate" the model by identifying the level of risk
aversion and the level of patience for each subject. Once these parameters are determined
it becomes easier to test hypotheses regarding the decision making process. There have
been several studies in the field of experimental economics investigating consumption
behavior under uncertainty (Hey and Dardanoni (1988), Ballinger et al. (2003) and
Carbone and Hey (2004)) that concluded that actual behavior differs significantly from
what is considered optimal. While these studies provide some insights in the decision
making process, they do not test for any particular alternative to the optimal behavior
corresponding to an unboundedly rational individual. Therefore a future area of research
is the design of an experimental framework that could test the hypotheses regarding the
decision making process advanced in this chapter.
111
Appendices
Appendix A. Technical notes to chapter 2
Appendix A1. Definitions for Scenarios, Equivalence Classes and Associated Probabilities
Suppose the world that can be described at each point in time by the vector of
state variables x
t
, and let u
t
denote the control variable while ç
t
is the forcing variable.
Suppose ç
t
is a random variable, with the underlying probability space
71
(O, E,
P
) . ç
t
is
defined as ç
t
: O C R where O is countable and finite. If the horizon has T +1 time
periods and ç
t
(
e
) is a realization of ç
t
for the event e ŒO in time period t , then the
sequence
ç
s
(
e
)
=
(ç
0
s
(
e
),ç
1
s
(
e
),K,ç
T
s
(
e
))
is called a scenario
72
. From now on, for notation simplification, I will refer to a scenario
s simply by ç
s
or by the index s and, in vector form, by ç
s
= ç
0
s
,ç
1
s
,K,ç
T
s
.
( )
Let S (e ) denote the set of all scenarios. Given that O is finite, the set S (e ) is
also finite. Therefore, one can define an event tree {N,
A
} characterized by the set of
nodes N and the set of arcs A . In this representation, the nodes of the tree are decision
points and the arcs are realizations of the forcing variables. The arcs join nodes from
71
72
O is the sample space, E is the sigma field and P is the probability measure.
Other definitions of scenarios can be found in Helgason and Wallace (1991a, 1991b)
and Rosa and Ruszczynski (1994).
112
consecutive levels such that a node n
it
at level t is linked to N
t
+
1
nodes n
k
+
1
, k = 1,..., N
t
+1 t
at level t +1.
The set of nodes N can be divided into subsets corresponding to each level
(period). Suppose that at time t there are N
t
nodes. The arcs reaching the nodes
n
it
, i = 1,K, N
t
belong each to several scenarios ç
q
(e ), q = 1,..., L
t
where L
t
represents
the number of leaves stemming from a node at level t . The bundle of scenarios that go
through one node plays a very important role in the decomposition as well as in the
aggregation process. The term equivalence class has been used in the literature to
describe the set of scenarios going through a particular node.
By definition, the equivalence class {s
t
}
i
, i = 1,K, N
t
is the set of all scenarios
having the first t +1 coordinates, ç
0
,K,ç
t
common. This means that for two scenarios
ç
j
=
(ç
0
j
,ç
1
j
,K,ç
t
j÷
1
,ç
t
j
,...,ç
T
j
) and ç
k
=
(ç
0
k
,ç
1
k
,K,ç
t
k÷
1
,ç
t
k
,...ç
T
k
) that belong to the
equivalence class
{
s
} , i =1,K, N t
i
t
the first t +1 elements are common, that is,
ç
lj
= ç
l
k for l = 0,...,t . Formally,
{
s
}
=
{ç
t
i
k
|ç
lk
= ç
l
i
for l = 0,...,t }
As mentioned in the above description of the event tree, at time t there are N
t
nodes. Then, the number of distinct equivalence classes
{
s
}
t
i
is also N
t
, that is,
i = 1,K, N
t
. Every node n
it
, i = 1,K, N
t
is associated with an equivalence
class
{s
t
}
i
.
The number of elements of the
set
{s
t
}
i
is given by the number of leaves stemming from
node i , level (stage) t .
113
Since scenarios are viewed in terms of a stochastic vector ç with stochastic
components ç
0
s
,ç
1
s
,K,ç
T
s
, it is natural to attach probabilities to each scenario. I denote
the probability of a particular realization of a scenario, s , with
p(s) = prob(ç
s
) .
These probabilities are non-negative numbers and sum to one. Formally,
p(s) > 0 and
9 p(s) =1. I assume that for each scenario ç
sŒS
s
the stochastic components
ç
0
s
,ç
1
s
,K,ç
T
s
are independent. Then
(
)
T
p(s) = prob ç
s
(
e
)
=
÷ prob ç
t
s
(
e
) ( ) (A.1.1)
t =0
Further on, I define the probabilities associated with a scenario conditional upon
belonging to a certain equivalence
class
{s
t
}
i
at time t :
p s s Πs
t
( {
}
) =
prob
(ç i
s
ç
s
Œ
{s
t
}
i
= )
p
(p{(ss)
}
), t
i
where p s
t
({
}
) i
is the probability mass of all scenarios belonging to the class
t
{
s
} .
t
i
Under the assumptions outlined above, p s
t s
({
}
)
=
÷
prob
(ç t
(
e
)) .
Therefore, the
conditional probability is easily computed as
i
t=0
p
s
{s
t
} = prob ç
s
ç
s
e
{s
t
}
i
=
(
i
)
( )
T
[
prob
(ç
t
(
e
)) s
t =t +1
The transition from the state at time t to that at time t + 1 is governed by the control
variable u
t
but is also dependent on the realization of the forcing variable, that is, on a
particular scenario s .
114
Appendix A2. Description of the Scenario Aggregation Theory
The idea is to show how a solution can be obtained by using special
decomposition methods that exploit the structure of the problem by splitting it into
manageable pieces and coordinate their solution.
Let us assume for a moment that the original problem can be decomposed into
subproblems, each corresponding to a scenario. Then the subproblems can be described
as:
T
min 9 F
t
x
s
t ,u
s
t ,
u ŒU [R
mu
( )
s ŒS
(A.2.1)
t t
t=1
where u
st
and x
st
are the control and the state variable respectively, conditional on the
realization of scenario s while S is a finite, relatively small set of scenarios.
Formally, by definition, a policy is a function or a mapping U : S ÷ R
m
assigning
to each scenario s e S a sequence of controls U (s)
=
(u
s
0 ,u
1
s
,K,u
s
t ,K,u
T
s
) , where u
st
denotes the decision to be made at time t if the scenario happens to be s . Similarly, the
state variable at each stage is associated with a particular scenario s . I use the notation
x
st
to show the link between the state variable and scenario s at time t . One can think of
T
the mappings U : S ÷ R
m
as a set of time linked mappings U
t
: S ÷ R
m
t with m
=
¿ m
t
.
t =1
The policy function has to satisfy certain constraints if two different scenarios s
and s ' are indistinguishable at time t on information available about them at time t .
Then u
s
t = u
s
t
'
, that is, a policy can not require different actions at time t relative to
scenarios s and s ' if there is no way to tell at time t which of the two scenarios will be
115
followed. This constraint is referred to as the non-anticipativity constraint. One way to
model this constraint is to introduce an information structure by bundling scenarios into
equivalence classes
73
as defined above. In this way, the scenario set S is partitioned at
each time t into a finite number of disjoint
sets,
{s
t
}
i
. Let the collection of all scenario
equivalence classes at time t be denoted by B
t
, where B
t
=
U{s
t
}
i
. In most cases i
partition B
t
+
1
is a refinement of partition B
t
, that is, every equivalence
class
{s
t
}
i
e B
t
is
a union of some equivalence classes
{
s
} t
+1
j
e B
t
+
1
. Formally,
{
s
}
t
i
=
U
{
s
} t
+1
j =1...m
i
j
.
Looking back to the event tree representation discussed in the previous section, m
i
represents the number of nodes n
t
j+
1
at level t +1 that are linked to the same node n
it
.
A policy is defined as implementable if it satisfies the non-anticipativity
constraint, that is, u
t
(e ) must be the same for all scenarios that have common past and
present
74
. In other words, a policy is implementable if for all t = 0,K,T the t
th
element
is common to all scenarios in the same class
{
s
} , i.e. if t
i
u
t
(ç
i
) = u
t
(ç
k
) whenever
{
s
}
=
{
s
}
t i t k
.
Let E be the space of all mappings U : S ÷ R
n
with components U
t
: S ÷ R
n
t .
Then the subspace
73
74
Some authors, such as Rockaffeler and Wets (1991), use the term scenario bundle.
For certain problems the non-anticipativity constraint can also be defined in terms of
the state variable, that is, x
t
(e ) must be the same for all scenarios that have common past
and present.
116
H = U e E |U
t
is constant on each class {s
t
}
i
e B
t
, for t = 1,...,T {
identifies the policies that meet the non-anticipativity constraint.
}
A policy is admissible if it always satisfies the constraints imposed by the
definition of the problem. It is clear that not all admissible policies are also
implementable. By definition, a contingent policy is the solution, u
s
, to a scenario
subproblem. It is obvious that a contingent policy is always admissible but not
necessarily implementable. Therefore, the goal is to find a policy that is both admissible
and implementable. Such a policy is referred to as a feasible policy.
One way to create a feasible policy from a set on contingent policies is to assign
weights (or probabilities) to each scenario and then blend the contingent policies
according to these weights. Specifically, if the probabilities associated with each scenario
are defined as in (A.2.1), one calculates for every period t and for every equivalence
class
{s
t
}
i
e B
t
the new policy u
t
by computing the expected value:
u
t
({
s
}
)
=
¿
}
p
(s
?
{
s
}
)u (s?)
t i
s?e
{
s
t
i
t i t
(A.2.2)
Then one defines the new policy for all scenarios s that belong to the equivalence class
{
s
} e B
t
i
t
as:
u
s
t = u
t
ˆ
({
s
}
) for all s
e
{
s
}
t i t i
(A.2.3)
Based on its definition, u
st
is implementable. The operator J :U ÷ U defined by (A.2.2)
ˆ
and (A.2.3) is called the aggregation operator.
Let us
rewrite equation
(2.4.1) as:
ˆ
117
min F
s
( x
s
,u
s
), seS (A.2.4)
u
t
eU
t
_R
mu
T
by defining the functional F
s
( x
s
,u
s
)
=
¿ F
t
( x
t
(s),u
t
(s
)
) .
t =1
Then the overall problem can be reformulated as:
min ¿ p
s
F
s
( x
s
,u
s
) over all U e E I H
seS
(A.2.5)
Let us assume for a moment that u
s
is an implementable policy obtained as in (A.2.3) ˆ
from contingent policies u
s
and u
s
is the optimal policy for the particular scenario s of
the problem described by (A.2.5). Let U and U be the collections of policies u
s
and
ˆ ˆ
u
s
respectively. One can easily see that U represents the optimal policy for the problem
described by (A.2.5). The question that the scenario aggregation methodology answers is
how to obtain the optimal solution U from a collection of implementable policies U . ˆ
Appendix A3. Solution to a Scenario Subproblem
In order to take advantage of the fact that scenario aggregation does not require
the computation of an exact solution for each scenario, I transform the Lagrangian (2.6.8)
by replacing the utility function with a first order Taylor series expansion around the
solution obtained in the previous iteration. Hence:
e
÷
uc
t
= e
÷
uc
t
s
s(k÷1)
?1÷u c
st
÷ c
s
t (k÷1
)
?
?
( )
?
From the transition equation, consumption can be expressed as:
c
s
t
=
(1+
r
) A
t
s÷
1
+ y
s
t ÷ A
t
s
118
Then
e
÷
uc
t
= e
÷
uc
t
s
s(k÷1)
{1÷u ??(1+
r
)
(A
s
and
t ÷1
÷ A
t
s÷(k÷1
)
÷ A
ts
÷ A
t
s(k÷1
)
? .For iteration
scenario s the Lagrangian becomes:
1
)(
)
}
?
(
k
)
min ¿ |
?
T
t
?e
÷
uC
s
t(k÷1) ?
?
u ?1÷
u
(1+
r
) A
t
s÷
1
÷ A
t
s÷(k÷1
)
+u A
ts
÷ A
t
s(k÷1
)
? +W
ts ?
(1+
r
) A
t
s÷
1
+ y
s
t ÷ A
ts
? +
t =0
?
?
(
1
)( )
? ? ?
+
1
µ ?(1+
r
) A
t
s÷
1
÷ A
(
t÷k1÷1
)
÷ A
ts
÷ A
(
tk÷1
)
?
2?
( )( )
?
2
}
Then, the first order condition with respect to A
t
s
is given by:
|
t
e
÷
uc
(
{
s k÷1)
t
÷W
t
s(k
)
÷ µ ?(1+
r
) A
t
s÷
1
÷ A
(
t÷k1÷1
)
÷ A
ts
÷ A
(
tk÷1
)
? +
?
( )(
)
} ?
|t+
1 ÷
(1+
r
) e
÷
uc{
s(k÷1)
t+1
+
(1+
r
)W
t
s+(1k
)
+
µ
(1+
r
)
?
(1+
r
) A
ts
÷ A
(
tk÷1
)
÷ A
t
s+
1
÷ A
(
t+k1
)
? = 0
Rearranging the terms leads to:
?
( )(
1
?
)
}
1 ?e
÷
uc
s
t(k÷1)
÷
(1+
r
) | e
÷
uc
s
t(+k1÷1) ÷W
s
(k
)
+ (1+
r
) |W
s
(k
)
? +
µ
?
?
t
t +
1
? ?
+
(1+
r
) A
(
t÷k1÷1
)
÷ A
(
tk÷1
)
÷ (1+
r
)2 | A
(
tk÷1
)
+ (1+
r
) | A
(
t+k1÷1
)
= (A.3.1)
=
(1+
r
) A
t
s÷
1
÷ A
ts ÷
(1+
r
)2 | A
ts +
(1+
r
) | A
t
s+1
Let
I
s
t (k
)
= 1 ?e
÷
uc
t
÷
(1+
r
) | e
÷
uc
t
+
1
÷W
t
s(k
)
+ (1+
r
) |W
t
s+(1k
)
?
s(k÷1) s(k÷1)
µ
?
?
Then the first order condition with respect to A
t
s
can be written as:
?
?
(A.3.2)
I
s
t (k
)
+ (1+
r
) A
(
t÷k1÷1
)
÷ ?1
+
(1+
r
)2 | ? A
(
tk÷1
)
+ (1+
r
) | A
(
t+k1÷1
)
=
? ?
=
(1+
r
) A
t
s÷
1
÷ ?1
+
(1+
r
)2 | ? A
ts +
(1+
r
) | A
t
s+1
? ?
For t = T ÷1 the first order condition becomes:
I
s
T(÷k1
)
+ (1+
r
) A
T
(k÷÷21
)
÷ ?1
+
(1+
r
)2 | ? A
T
(k÷÷11
)
+ (1+
r
) | A
T
(k÷1
)
=
? ?
(A.3.3)
=
(1+
r
) A
T
s÷
2
÷ ?1
+
(1+
r
)2 | ? A
T
s÷
1
+
(1+
r
) | A
T
? ?
119
Noting that A
T
(k÷1
)
= A
T
= A
T
equation (A.3.3) can be written as:
I
s
T(÷k1
)
+ (1+
r
) A
T
(k÷÷21
)
÷ ?1
+
(1+
r
)2 | ? A
T
(k÷÷11
)
= (1+
r
) A
T
s÷
2
÷ ?1
+
(1+
r
)2 | ? A
T
s÷1
?
Similarly, for t = 0 one obtains:
? ? ?
I
s
0(k
)
+ (1+
r
) A
(
÷k1÷1
)
÷ ?1
+
(1+
r
)2 | ? A
(
0k÷1
)
+ (1+
r
) | A
1
(k÷1
)
=
? ?
(A.3.4)
(1+
r
) A
÷
s
1
÷ ??1
+
(1+
r
)2 | ?? A
0s +
(1+
r
) | A
1
s
Again, noting that A
÷
1
is given, A
(
÷k1÷1
)
= A
÷
s
1
so equation (A.3.4) becomes:
I
s
0(k
)
÷ ?1
+
(1+
r
)2 | ? A
(
0k÷1
)
+ (1+
r
) | A
1
(k÷1
)
= ÷ ?1
+
(1+
r
)2 | ? A
0s +
(1+
r
) | A
1
s
? ? ? ?
Rewriting the system of equations in matrix form, leads to:
?÷?1
+
(1+
r
)2 |?
??
?
?
(1+
r
)
?
(1+
r
)|
÷?1
+
(1+
r
)2 |?
0 0
0
K
0
0
0
0
?
?
s
?? A
0
?
?
?
0
?
(1+
r
)
?
(1+
r
)|
÷?1
+
(1+
r
)2 |? (1+
r
) |
K
0
?? A
s
1 ?
?? s ?
? ? ? K
0
?? A
2
? =
?
M M M M M M M
?? M ?
?? s ?
?
?
??A
T
÷
1
? ?
?
?
?
0 0 0 0
K
(1+
r
)
÷?1
+
(1+
r
) |
?
?
2
??
?
I
s
0(k
)
÷?1
+
(1+
r
)2 |?
A
(
0k÷1
)
+(1+
r
) |A
(
1k÷1)
?
? ?
? ? ? ?
?I
s
(k
)
+(1+
r
) A
(
k÷1
)
÷?1
+
(1+
r
)2 |? A
(
k÷1
)
+(1+
r
) |A
(
k÷1
)
? ?1
=?
?
?
0
?
M
?1
2
?
? ? ?
?
I
T
÷
1
+
(1+
r
) A
T
÷
2
?
( )
?
T÷1
?
s(k)
(k÷1
)
÷?1+ 1+r 2 |? A
(
k÷1)
?
?
120
Appendix B. Technical notes to chapter 3
Appendix B1. Analytical Solution for a Scenario with Deterministic Interest Rate
Consider the problem described by (3.3.1) - (3.3.4). Solving the period-by-period
budget constraint (3.3.2) for c
t
, t = T ÷ 1 and t = T , and substituting back into the utility
function, the period T ÷ 1 optimization problem is given by:
e exp ÷u
(1+ r
T
÷
1
) A
T
÷
2
+ y
T
÷
1
÷ A
T
÷
1
?
max ¹÷
c
{
e
}
?÷
A
T
÷1
©
¹
u
_ exp ÷u
(1+ r
T
) A
T
÷
1
+ y
T
÷
A
T
?
(B.1.1)
{
e
}
¹
E·| u
?
I
ˆ÷ T÷
1
˜ ?
subject to
·
.
A
T
÷
1
> ÷b
+?
˜¹
(B.1.2)
Taking derivatives with respect to A
T
÷
1
, the Euler equation for (B.1.1) is given by:
exp ÷
u
(1+ r
T
÷
1
) A
T
÷
2
÷u y
T
÷
1
+u A
T
÷
1
?
e
e
¹
?
exp ÷
u
(1+ r
T
÷
1
) A
T
÷
2
÷u y
T
÷
1
÷u b? ,
= max c
e ? ÷
¹
?
(B.1.3)
¹
|
(1+ r
T
) E exp ÷
u
(1+ r
T
) A
T
÷
1
÷u y
T
+u A
T
? I
T
÷1 ©
{
e ?
}
? u
2
o
2
y
?
¹
?
Note that y
T
= y
T
÷
1
+ ç
T
while E ?
exp
(÷uç
T
) | I
T
÷
1
? = exp ?
? ?
? 2 ? and hence solving ?
? ?
(B.1.3) for the optimal wealth level at the beginning of period T ÷ 1 yields:
e*
A
= max c
¹÷b
,
(1+ r
T
÷
1
) A
T
÷
2
+ I*
T
+ A
T
?
÷
.
T ÷1
e
?¹
?
(B.1.4)
©
¹
(2 + r
T
)
¹?
where I*
T
= I + log
|
(1+ r
T
)? /u , and I = uo 2
y
/ 2 .
{
e
?}
Going now to period T ÷ 2 , the optimization problem is given by
121
? exp ÷u
?
(1+ r
T
÷
2
) A
T
÷
3
+ y
T
÷
2
÷ A
T
÷
2
?
{
?
}
{
*
?÷
? ÷ | E ? exp ÷u
?
(1+ r
T
÷
1
) A
T
÷
2
+ y
T
÷
1
÷ A
T
÷
1
? +
max ?
? ? ?
}
A
T
÷2
?
?
u
exp ?÷
u
((1+ r
T
) A
*
÷
1
+ y
T
÷ A
T
)? T
?
?
u
+| ?
?
I ?? ???
subject to
u
T ÷2
?
?
?
?
(B.1.5)
A
T
÷
2
> ÷b (B.1.6)
Taking derivatives with respect to A
T
÷
2
, and noting that
E exp ÷u A
*
÷
1
I
T
÷
2
? = exp ÷u A
*
÷
1
,
e
(
T
)
?
(
T
)
the Euler equation for (B.1.5) is given by:
exp ?÷
u
(1+ r
T
÷
2
) A
T
÷
3
÷u y
T
÷
2
+u A
T
÷
2
?
?
?
?
exp ?÷
u
(1+ r
T
÷
2
) A
T
÷
3
÷u y
T
÷
2
÷ub? ,
=
max ? ?
?
*
?
?
?
?
(
B.
1.
7)
?
?|
(1+ r
T
÷
1
) exp ?÷
u
(1+ r
T
÷
1
) A
T
÷
2
+u A
T
÷
1
? E ?exp (÷u y
T
÷
1
) I
T
÷
2
?? ?
? ? ??
Since y
T
÷
1
= y
T
÷
2
+ ç
T
÷
1
, (B.1.7) can be rewritten as:
exp ?÷
u
(1+ r
T
÷
2
) A
T
÷
3
÷u y
T
÷
2
+u A
T
÷
2
?
?
?
?
exp ?÷
u
(1+ r
T
÷
2
) A
T
÷
3
÷u y
T
÷
2
÷ub? ,
= max ?
? ?
?
?
?
*
? ?
?|
(1+ r
T
÷
1
) exp ?÷
u
(1+ r
T
÷
1
) A
T
÷
2
÷u y
T
÷
2
+u A
T
÷
1
? E ?exp (÷uç
T
÷
1
) I
T
÷
2
??
? ? ? ??
Assuming that liquidity constraint is not binding, solving (B.1.7) for A
T
÷
2
yields:
ln
|
(1+ r
T
÷
1
)? uo 2y
(1+ r
T
÷
2
) A
T
÷
3
÷ A
T
÷
2
= ÷
e u ?
+
(1+ r
T
÷
1
) A
T
÷
2
÷
A
*
÷
1
÷ T
2
(B.1.8)
Using the notation from above, equation (B.1.8) can be written as:
I*T ÷
1
= ÷ A
*
÷
1
+
(2 + r
T
÷
1
) A
T
÷
2
÷
(1+ r
T
÷
2
) A
T
÷3 T
Similarly, for period t , the equivalent of equation (B.1.9) is given by:
122
(B.1.9)
I*t+
1
= ÷ A
*
+
1
+
(2 + r
t
+
1
) A
t
÷
(1+ r
t
) A
t
÷1 t
(B.1.10)
It is clear that the optimal wealth level at the beginning of period t does not depend on
labor income received at the beginning of the period. This result is not general, but is
rather specific to the life-cycle model with a negative exponential utility function and
labor income following an arithmetic random walk process.
Solving for the beginning-of-period wealth levels from t = 0 to t = T ÷1 means
solving the system of linear equations:
_ A
0
ˆ
_
(1+ r
0
) A
÷
1
+ I*
1
ˆ
· A
1
˜ ·
I*
2
˜
· ˜ · ˜
· A
2
˜ · I*3 ˜
D·
·
M
˜ =
·
˜·
· A
T
÷
3
˜ ·
M
I*T ÷2
˜
˜
˜
(B.1.11)
·A ˜ ·
˜
· T ÷
2
˜ · I
T
÷1
*
˜
· A
T
÷1 ˜ ·
. +. A
T
+ I*
T
+˜
where D is a tridiagonal coefficient matrix,
_ 2 + r
1
÷1 0 L 0 0 0ˆ
·
÷
(1+ r
1
) 2 + r
2
÷1 L 0 0 0
˜
· ˜
D= · M M M M M M˜ (B.1.12)
·
0 0 0
· L
÷
(1+ r
T
÷
2
) 2 + r
T
÷1 ÷1
˜
˜
·0 .
0 0
L
0
÷
(1+ r
T
÷
1
)
2 + r
T
˜ +
Once the values for wealth levels are computed, the consumption levels follow.
The solution presented in this section is in fact the solution for a scenario obtained by
discretizing the distribution of the forcing variable for the interest rate. Since an
analytical solution can be obtained when income follows an arithmetic random walk and
interest rate is deterministic, it is no longer necessary to discretize both forcing variables,
but only the interest rate. This approach reduces considerably the computational burden.
123
For different labor income processes, a dual discretization is necessary, that is, for both
forcing variables.
Appendix B2. Details on the Assumptions in Rule 1
In period t the consumer wants to solve the optimization problem given by:
ma
T
x
E
_9
T ÷| t ÷
t
_ 1 ˆ
exp
÷u c |
I
?
subject to
{c
t
}
t
=t
et=t
·
u
˜
.+
(
t
) t ??
(B.2.1)
A
t =
(1+ r
t
) A
t
÷
1
+ y
t
÷ c
t
, t = t,t +1,...,T ,
(B.2.2)
with A
t
÷
1
, A
T
given t = 0,1,...,T ÷1,
y
t
= y
t
÷
1
+ ç
t
, t = t +1,...,T ,
(B.2.3)
with y
t
given t = 0,1,...,T ÷1,
r
t
= r
t
÷
1
+u
t
,
with r
t
given
t = t +1,...,T ,
t = 0,1,...,T ÷1,
(B.2.4)
The assumption is that the forcing variable u
t
has three possible realizations,
{u
a
,u
b
,u
c
} . The set of its realizations determines the event tree and consequently the set
of scenarios. For T
h
periods the number of all scenarios is 3
T
h . The consumer considers
all the possible scenarios from period t to period t + T
h
. From there on it assumes that
for every leaf the scenario will be determined by u
t
taking its unconditional mean, that
is, zero. For example, if the short optimizing horizon is given by T
h
= 4 and the sequence
of realizations for u
t
up to period t + 4 , for a particular scenario,
is
{u
a
,u
c
,u
b
,u
c
} , the
assumption made by consumer is that for this particular scenario the realizations of u
t
for
the rest of the periods will be 0 , that is, the whole scenario
is
{u
a
,u
c
,u
b
,u
c
, 0, 0,...,
0
} .
124
This process is repeated as the consumer advances to period t +1 and goes again
through the optimization procedure. The number of scenarios considered remains the
same unless T ÷ t < T
h
, which is to say that there are fewer than T
h
periods left until the
terminal period.
Appendix B3. Details on the Assumptions in Rule 2
In period t the consumer wants to solve the optimization problem given by:
ma
T
x
E
9 ÷|t÷
t
_ 1 ˆ
exp
(÷u c
t
) | I
t
? T
?
subject to
{c
t
}
t
=t
e_
t=t
·
u
˜
.+
?
(B.3.1)
A
t =
(1+ r
t
) A
t
÷
1
+ y
t
÷ c
t
, t = t,t +1,...,T ,
(B.3.2)
with A
t
÷
1
, A
T
given t = 0,1,...,T ÷1,
y
t
= y
t
÷
1
+ ç
t
, t = t +1,...,T ,
(B.3.3)
with y
t
given t = 0,1,...,T ÷1,
r
t
= r
t
÷
1
+u
t
,
with r
t
given
t = t +1,...,T ,
t = 0,1,...,T ÷1,
(B.3.4)
The assumption is that the forcing variable u
t
has three possible realizations,
{u
a
,u
b
,u
c
} . The set of its realizations determines the event tree and consequently the set
of scenarios. For T
h
periods the number of all scenarios is 3
T
h . The consumer considers
all the possible scenarios from period t to period t + T
h
. From there on it assumes that
for every leaf only three more scenarios emerge, with u
t
taking only one of the three
values
{u
a
,u
b
,u
c
} every period until the end of the horizon. For example, if the short
optimizing horizon is given by T
h
= 4 and the sequence of realizations for u
t
up to
125
period t + 4 , for a particular scenario, is {u
a
,u
c
,u
b
,u
c
} , the assumption made by
consumer is that only three more scenarios will stem from the leaf corresponding to
scenario
{u
a
,u
c
,u
b
,u
c
} . These three scenarios are given
by
{u
a
,u
c
,u
b
,u
c
,u
a
,u
a
,...,u
a
} ,
{u
a
,u
c
,u
b
,u
c
,u
b
,u
b
,...,u
b
}
and
{u
a
,u
c
,u
b
,u
c
,u
c
,u
c
,...,u
c
} . Effectively, the total number
of scenarios considered is 3
T
h +
1
as opposed to 3
T
÷
t
which would represent the total
number of scenarios for the horizon from period t to period T .
This whole process is repeated as the consumer advances to period t +1 and goes
again through the optimization procedure. The number of scenarios considered remains
the same unless T ÷ t < T
h
, which is to say that there are fewer than T
h
periods left until
the terminal period.
126
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