Econ 714 Quantitative Macro, Fall of 2006

 Department of Economics, University of
Pennsylvania. 429 Mc Neil, Ph. 8987767
 Tuesdays 5:00 pm to 8:00. In McNeil building 410.
 Off Hours: Wed 14:00 to 15:00. (or by appointment).

http://www.ssc.upenn.edu/~vr0j/71406/
The last and bonus class will be held on Tuesday Dec 12 at
5:00 in 410 McNeil. This is, usual time, usual place.
A combination of circumstances deemed this the best outcome.
What are we doing? A
class by class expost
diary.

December 12.
So I will talk about
demographics and we will relax a bit and try to think what
is interesting. There may be a beer after. This time not on
my purse.

December 5.
We had a variety
of presentations ranging from
Michel's on MPI to Leonardo who talked on a problem
with inventory management to Shooshan whose interests are on
a model of bankruptcy to Ben who talked about the findings
of the fundamental question in RBC research. I interrupted
enough times to remind people of what presentations are
about and to make it clear that the customer (the listener)
is always right. Because there was no time I did not talk
about demographics.

November 28.
Daniel went over
Chebychev approximations and
problems in highdimensional interpolation which yields
to sparse grids and Smolyak polynomials and their
implementation (incl. Krueger/Kubler OLG model). We talked a
little about demographics how to model population
distributions of one sex and exogenous laws of motion.

November 21.
We went go over
models where a policy has to be chosen. We talked about the
time consistent Markov Perfect Equilibria. We looked at the
specific example of optimal public expenditures under a
period by period balanced budget constraint without leisure
and with an income tax schedule. We also talked briefly
about solving equations using different strategies.

November 10.
We went again over how
to deal with economies with aggregate uncertainty and
probability measures as state variables using the Krusell
and Smith notion of equilibria with approximating agents. We
discussed in particular economies with bonds and leisure in
the utility function when prices are not known functions of
moments of the distribution. We then used some of these
ideas to discuss economies with housing. We had a brief talk
about Tchebyshev approximations and we started talking about
how to deal with economies with game theoretic elements to
it, such as economies where the government is set to
(possibly optimally) choose policy.

November 7.
We went over what to do
when the aggregate state of the economy is a probability
measure. We talked about QuasiAggregation (KrusellSmith)
and how to substitute RE Eq with Eq with approximating
agents. We looked at Aiyagari economies with aggregate
shocks, Aiyagari economies with aggregate shocks and labor
choice, and Aiyagari economies with aggregate shocks and a
bond market.

October 26.
We did a longer session
on homeworks in the conference room. Then we talked about
how to model houses. We spend a lot of time discussing how
to model traditional mortgages versus home equity lines of credit.

October 17.
We talked about what
features should a housing model have.

October 10.
Se Kyu talked about how
to access the most commenly used data sets (CPS, PSID, CEX
and SCF). He added some nice economics about how to think
of labor share and employment dynamics. He
used these notes

October 3.
We talked about the
transition problem and the three possible ways of dealing
with it. We talked small variations of the growth model
for business cycle properties: labor hoarding and
lotteries. We saw a good set of homeworks (splines and
value function iteration).

Sept 26.
We raised the issue of what is an
approximation. We talked about the metric, the family of
functions and cited both piecewise linear and splines as
typical approximations and what are there main properties.
Various nice notes on these issues can be found
on
Makoto's page. We then moved to the discussion of
loglinearization of Euler equations. A place to look at
it is
Harald Uhlig's page. Another place is
Makoto's slides. There was a further set of
presentations that were also quite good, making me
very happy about the quality of this
class.

Sept 19.
We continued talking about the question
of what is the contribution of technology shocks. We put
special emphasis on the care with which to address the
mapping of National Accounting in the US and in the
model. We talked about how to calibrate the model. We also
attacked the issue of how to solve a model with
discretization of the state space. There was a first set
of homeworks presented. While it may seem otherwise,
students did a pretty good job considering it was the
first time. I expect next week to be considerably
better.

Sept 12.
We talked about the class, and what
it is about. We went over the syllabus and the
requirements and we suffer the ignominy of not having a
working projector. We then started with a discussion of
how to measure the contribution of productivity changes
to output and labor fluctuations.
Course Description
This course should be thought of as the third course of the
Macro sequence, it follows naturally after 702 and 704. Its
purpose is to learn the map from models to data i.e. to answer
quantitative questions that we are interested in (in the
process of doing so, some interesting theoretical questions
arise). We will develop tools by stating general questions,
and then discussing how to approach its answer.
The tools that we will be developing beyond those already
covered in the first year can be grouped into:
 Theoretical tools. Not all the necessary tools
have been acquired in the first year. We will look at
representative agent models, models with a continuum of
agents represented with measures, overlapping generations
models, as well as models where agents form households. We
will look at models where equilibria are optima and where
they are not. We will look at stationary and
nonstationary equilibria. We will look at models without
perfect commitment and without perfect information.

Empirical tools. A necessary condition to be able
to do applied theory is to be able to characterize
some properties of the world. This involves the
capability of accessing some data and of understanding
the way it is organized as well as the principles that
guide the construction of the main sources. This
requires some knowledge of NIPA and of the way data
are organized,

Computational Tools. Students should be able to
construct and characterize the properties of the
equlibrium allocations of artificial model economies.

Calibration. We will spend a lot of
time thinking of how a model is related to the
data. This is I think the more important part of the
learning process. We will discuss this in much
detail.
This is a Ph.D. course not a Masters course. As such
students are not expected to learn what other people have
discovered, but the tools that are needed in order to
discover things by themselves. Because of this reason the
active work of the students is crucial to achieve the
objective of mastering the tools that are described
above. This is a course to learn to do things, and,
therefore, it requires to do some things.
Every class except the first one we will devote the first
twenty minutes or so to students presentations of
homeworks. I expect professional competence in this regard.
Course Requirements
Students will place the solution to the homeworks and to other
requirements in electronic form in
/work/econ/econ714_Fall_2006 in subdirectories that each
student should have under her own name. The solutions should
be in all be placed in a subdirectory with the students' name
and the homeworks name
(/work/econ/econ714_Fall_2003/Diestefano/H1_sept_16 for
example). Please copnnect
to
http://www.ssc.upenn.edu/acct_request.html and fill out
the online account request form so that an account for you for
the SSC/Econ UNIX server can be created; once you fill this
form out, Chris Couples will create accounts for you; you'll
need to go to McNeil 304 to set your password after you fill
the form out, and once this is done, you'll be able to access
your class folder, at /work/econ/econ714_Fall_2006, on the
server lambic.ssc.upenn.edu. email
sschelp@ssc.upenn.edu to become members of
the econ/econ714_Fall_2006 group in order to have
access.
There are various types of requirements that are a necessary
part of the course, all of which have to be fulfilled.

Regular Homeworks.There are two
types of homeworks. The ones that are set up in this page
and that are due each Tuesday and are set usually well in
advance, and those that I ask in class (that I may try to
post in this page soon thereafter but are anyway due
whenever I say in class). Homeworks have due dates that are
enforced by the date of the file. As of now there are
homeworks due next Tuesday.

Class Presentations Every student
will make at least two class presentations with at least
one being of a subset of a homework. The first
presentation should take no more than 15 minutes and it
will be absolutely professional. Every second wasted,
every statement not planned, every bad thing will be
highlighted. The second presentation (that will depend on
class size and interests) maybe on a paper or on another
homework.

Referee Report. I will assign a
paper to each of you as we go along to write a referee
report and perhaps also to present the paper in no more
than 20 minutes. The refere report should be no longer
than five pages and should contain a clear and concise
exposition of the main points of the article as well as a
critical evaluation of the article's contributions.

Wikipedia Article. This is
something that should be done by the end of the
course. The moment you post it email me and place a copy
in your directory. Think of a topic of the course no
matter how silly.
This course believes drastically in Learning by Doing. To learn
the material that we cover requires that students do all the
homeworks in a timely manner. Given the way to collect the
homeworks, timeliness is automatically recorded. I will look at
what is done weekly.
Seventh and last Homework
batch. Due Tuesday Dec 6 th.

The end of the Aiyagari Economy
Solve for the stationary distribution that you
found. Describe some of its inequality properties
using efficient communication methods.

Obtaining the GEE
Use a symbolic package (say Mathematica or
Maple (which works with Matlab)) to obtain the GEE
of a benevolent government in a neoclassical
growth model who can use labor taxes to optimally
choose the level of government expenditures (which
provide separable utility) when households care
about leisure and there is a period by period
balanced budget constraint.
Sixth Homework batch. Due Tuesday
Nov 7 th.

Design a quantitative question
State a real business cycle question and pose and
solve the model designed to answer it. Get an
answer. Remember we are looking for clarity first and
interest second. You have to use a global solution
method. Either you pose an approximation to the
Euler equation (or law of motion) using piecewise
liner decision rules or Chebyshev polynomials or a
spline approzimation to the value function. Use 10
pieces and compare the performance to that with 20
pieces. If you do not have an interesting question
to ask and want to repeat that of problem 10, then
you have to compare the solution methods. Look at
the paper of Fernandez Villaverde et al in
computer methods for inspiration if you want.

The beginning of the Aiyagari Economy
Solve for the decision rule of the problem in Aiyagari
(94) or in Castaneda, DiazGimenez, and RiosRull
(2003) JPE paper with leisure. Make sure that you
allow the shock to affect pref also and not only
income. Use any global continuous method of your
choice (preferably Schumaker splines on values or some
polynomial on decisions). This is like the
previous problem but the range of asset holdings
should go from zero to 25 times the highest income
endowment. The intervals or pieces should increase
with wealth.
Fifth Homework batch. Due Tuesday
October 23th.

Read data sets: II. The PanelData data sets
Use now either the NLSY or the PSID. Get again your own
group of people the {i,g}. See how many of them were
married in the initial period and 10 years later and
build a transition matrix with the number of children in
the household. Again separate them by education
groups. Four this time. Report an additional feature of
this group that you may find interesting.

Read data sets: III. A new data Sets. The IPUM or
Census
Recent news reported that most Americans now do not live
in traditional families. Using the most recent wave of
Census data (find out what this is) the proportions of
people in agesex group living in each type of
household. Find out what the census uses as partitions
for types of households. Report an additional feature
that you may find interesting.
Fourth Homework batch. Due Tuesday
October 16th.

Solving the baseline model and measuring the
contribution of productivity shocks to output and
employment fluctuations.
Pose now the standard stochastic business cycle model
with two inputs of production capital, and labor.
F(z,K,N) = exp(z_{t} )
K_{t}^{
q
} N_{t}^{1

q
}


The shock z follows the univariate process that
you estimated in homework 9.
Per period Preferences are log c_{t}
+ a
(1n_{t}).
This means that there are a total of four parameters
to pick
(b ,a
, q
, d).
 Pick values for those parameters so that
the steady state of the economy of a quarterly
version of this model matches some statistics
that you your self specify.
 Define recursive competitive equilibria.
 Compute a loglinear approximation to the
decision rules and plot them.
 Generate 500 samples of length 200 each and
compute the standard business cycles
statistics as found in the Cooley volume.
 Discuss the findings.
 Discuss optimality (and compute the optimal
allocation if needed).
 Give an answer to the main question here. What
is the contribution of the shock to movements in
output and in labor.
 Solve this model with either spline
approximations to the value function or to the
decision rules and with piecewise approximations to
the decision rules.
 Do a variation of this model that you find
interesting and repeat the exercise. Say with
either labor hoarding or lotteries and comment the
answer.

The first transition problem.
Take the steady state of the previous problem and assume
that overnight productivity doubles. Compute the
transition and report the values for 200 periods in the
following ways
 As a system of 400 equations and unknowns.
 By guessing period 1's capital and checking that in
200 periods it is sufficiently close to the new steady state.
 By guessing period's 199 capital stock and checking
that 200 period's earlier it had a stock of capital
equal to the steady state of the economy without the
productivity change.

The second transition problem.
Consider now the problem of agent that lives 65
periods has an endowment of 1 unit of labor every the
first 50 periods of his life and has standard
preferences. Solve his maximization problem when the
interest rate is 4\% using the three methods.

Read data sets: I. The CrossSection data set
Problem.
Denote by i,g your own age plus 12 and your
gender. Now take the CPS or the CEX and find out the
average number of hours worked by people of type g, i
plus/minus 2. Sort this group into quartiles by education,
and report the average time worked and the average wage
for those that work, the average time worked per person,
the fraction that work, the fraction married, the number
of children among those married and the number of children
per parent. This for the years 1980 and the latest
available. Report an additional feature of this group
that you may find interesting.
As an added bonus go back to 1970 through 1979.
Third Homework batch. Due Tuesday Oct 2nd.

Value function iteration Harder
You have to write computer code to do the
following. For the most basic stochastic growth model with
parameters
a=.36,
b=.985,
d=.0225,
q=.64,
s=2.,
, and for a stochastic version of it with a two
state Markov shock that multiplies the production
function. The two values of the Markov shock are 1.01
and 1/1.01.
 Compute the steady state of the deterministic
economy so that output is one (this may require to
add a TFP parameter to normalize output to
1.). Solve it for these parameters and for those
that apply to the previous question.
Write code to solve the problem of the agent
when you discretize around the state state in intervals
of .01% from the deterministic steady state between .85
and 1/.85 of the steady state. The accuracy to use is
105 of the value function. You should write different
versions of this code that differ in what are the tricks
that you use to solve it. For each of these versions you
are supposed to record the time needed to achieve the
desired precision with each of the algorithms described
as well as the number of iterations that is needed.
Use as initial value of the value function to start
iterating the solution to the previous problem
The set that you have to do is
 Brute force successive iterations of the Bellman
equation. Verify the code with the example of the
previous question.
 Successive iterations of the Bellman equation
taking into account only monotonicity of the
decision rule.
 Successive iterations of the Bellman equation
taking into account only concavity of the
value function.
 Successive iterations of the Bellman equation taking
into account both concavity of the value function and
monotonicity of the decision rule.
 Use Policy iterations (waiting until converged)
to solve the problem.
 Use policy iterations with 5, 10, 20 and 50 steps
in between policy reassessments.
 Use interpolation. Solve the problem solving for the
optimal policy at intervals first of 10\% of the state,
then 1\% then .1\% and then .01\%. Use to solve the
intermediate problems the method that has performed best
among the previous ones.
 Pick one the previous methods and compare the time
and number of iterations it takes to find the solution
starting from a value function equal to zero versus a
more inteligent choice of the value function.
 Do any of the previous with a labor choice that
is continuous. The coefficient of leisure is
.34.

Approximate with splines
Take the function of problem 3 and
approximate with a cubic spline with 10, 20 and 50
equally spaced points. Plot them all and use the
splines to find zeroes of the function. Approximate
them also with a Schumaker spline.
Splines Module
and Driver for
Schumacker splines with and without derivative information
written by Makoto Nakajima from the algorithm described in
Judd's book. These routines are particularly well written in
terms of containing sufficient comments to be used by others
or by the writer after some time of having written
them. Take them as models of style.
Leonardo Rezende has a couple of nice matlab scripts
appropriate for this issue.

Calculation of Series of Output and Capital and
Labor Income, Capital Input and Labor Share.
Using the logic that we looked at in the first class
about how to map the stochastic growth model to the US NIPA
(that can be found in the Cooley and Prescott chapter in the
Cooley 1996 book) compute series for output and for labor
share and compute a series for the Solow residual. Estimate
a univariate process for it.
Second Homework batch. Due Tuesday
September 26th.

Value function iteration Easy
You have to write computer code to do the following. For
the simple storage problem with log utility and Markov
endowment of 14 coconuts, and at most 1 regrigerator in
storage with parameter
b=.95,
(use [.4 .3 .2 .1  .2 .4 .3 .1  .1 .3 .4 .2  .1 .2
.3 .4] as the Markov matrix.
 Solve the value function by successive
approximations.
 Generate a sample of length 200 and plot it.
 Compute the Stationary Distribution of
individual States.
 Compute using both methods average consumption
and autocorrelation of consumption.

Closed Form Solution
Use log utility, CobbDouglas production with full
depreciation to solve analytically the value function
and decision rule of the social planner.
First Homework batch. Due
Tuesday September 19th. Name them sept_19_h1, sept_19_h2 and
so on.

Data manipulation.
1.A Fetch and plot US quarterly GDP
Investment plus durables plus net exports,
nondurables plus services, and aggregate hours both
from CPS and from the firm survey (see Cooley
chapter 1, page 30). Store it in pdf, eps, and emf
or wmf formats.
1.B HP filter and plot US quarterly (log)
GDP. Store it in postcript or pdf. Compute the same
table as in the Cooley book for those 4 variables using
data up to 2003:4 or later.
1.C Calculate a linear trend and decompose log
GDP in the linear trend the hp trend and the hp
residual.
1.D Plot the growth rates together with the hp
residual and comment the differences.
1.E Compute a VAR of those 4 variables and plot
the impulse responses. Make sure that you explicitly
state what are the identifying assumptions that you
make.

Interpolation.
Write a routine that linearly interpolates. Apply it by
storing the value of exp (x) between 0 and 1. in intervals
of .1 and assessing the value by interpolation in
intervals of .05. Plot the function and what results from
using approximation.

Solving Equations of one unknown.
(Parts of Homework 1 of Chapter 5 of Judd's book.) Solve
sin 2
p
x
2x=0 using bisection between x
_{
0
}
=5
and x
_{
1
}
=5
(If this interval is a bad one change it).

Production Function manipulation.
Compute labor factor
shares with a CES production function
Y=[
q
K
^{
r
}
+(1
q
)N
^{
r
}
)]
^{
1/
r
}
when K=N=1, and K=2, N=1. Are they the same?
What about with CobbDouglas (
r
=1).
Note that Labor share = w*N/Y, and that under competition w=(dY/dN).
Class schedules.
There are regular lectures on every foreseeable Tuesday.
What about knowledge of Computers?
This is not a course in computer languages so students are
responsible to learn to write computer programs. Students are
also responsible for learning their way around McNeil
computational facilities. I do not expect anybody to have a
computer at home or anything like that. It is better to work
in McNeill's computer room because you can talk to each other.
There
are three general classes of computer languages.
 Fortran
90. This is the best and more powerfull computer
language. Among economists nobody prefers C. It
is a little bit hard at the beginning (you have to
declare variables and the like) but all students
tell it is well worth to learn it as soon as
possible. A very good introduction to fortran can
be
found HERE.
 Matlab, Gauss, Scilab and Octave. These are
very popular packages in economics. They are
relatively easy to learn and code writing is
easy. They generate a lot slower code than F90
(about 100 times) but they are probably a good
choice to solve some problems. They may have an
interface with f90 but I have never seen it
working. Matlab is growing at the expense of
gauss.
 Mathematica and Maple. These are packages
capable of doing symbolic manipulation of
equations. Occasionally they can also be used to
do numeric calculations. It does not hurt to know
them.
Students
should be able to write code in F90 in addition to
matlab or gauss and to stata. Most students tell me in
later years that I should have enforced harder the
learning of F90, but I am willing to consider
exceptions. If somebody has a serious reason not to
use F90, please come and talk to me. At least one
homework should be answered in f90.
Look
at Tips
for Doing Computational Work in Economics by Tony Smith for
insights.
Grading Rules and Empirical
Requirement.
To satisfactorily complete the course, students have to do all
the requirements well. To pass the empirical requirement,
students have to write a project that will just require a
description (not completion) of an independent paper. Such
proposal consists on defining a question and describing all the
steps involved in arriving to a solution.
For those that do not register but take the course, I recommend
that they do the homeworks. We learn to solve problems by facing
them. Learning jointly with others greatly speeds the
process. The deadline for the Empirical Requirement is the last
day of class.
What about textbooks?
In addition to the standard macro books ([
Stokey and LucasStokey and Lucas1989],
[
HarrisHarris1987], [
Ljungqvist and SargentLjungqvist and Sargent2000])
I find that there are a few books of interest.
 [ Cooley and PrescottCooley and Prescott1995]. It is now dated but it contains some
important lines of attack on business cycles. The computational
techniques are a little bit obsolete, but the questions less so.
 [ JuddJudd1998]. is a general computational textbook with special
attention to economists. While it is short on some details that we
care about (complicated equilibrium considerations, multidimensional
value functions, multidimensional interpolation) it is a very useful
book for many topics.
 [ Marimon and ScottMarimon and Scott1998]. It has a bunch of chapters that deal with specific
problems. I find the continuous time chapter nice as well as some
scattered other chapters.
 [ Miranda and FacklerMiranda and Fackler2002]. It is quite a nice book. Like others
it is too irrelevant in some places and too easy in others. It is
designed for matlab which is a pity, but it has a nice
implementation (via a <a
href="http://www4.ncsu.edu/unity/users/p/pfackler/www/compecon/toolbox.html"
> downloadable toolbox</a>) of function global approximations.
 [ Heer and MauÃnerHeer and MauÃner2004]. This is a nice book with a lot more
economics than the others. Its consideration
of the theory is closest to what we do. It has also many
examples. Parts can be found <A
HREF="http://www.unibamberg.de/sowi/economics/heer/engindex.htm»
here.</a>
 [ Press, Teukolski, Vetterling, and FlanneryPress et al.1992]. The classic book for numerical
analysis. Very useful.
Syllabus
 Introduction. What is a Model?
 A measurement tool. How big is bla bla ?
 A device to assess the implications of changes. What
happens if bla bla bla?
 First Question. How big is the contribution of
productivity shocks to aggregate fluctuations: the most basic
structure.
 Review of the theory. The optimal growth model. Using
dynamic programming to solve for the optimal allocation.
The second welfare theorem. A Recursive version of the
second welfare theorem.
 Computation of the model. This will involve the
review of more than one method. And some homeworks. I
will put now the list of things about solving an optimal
decision model through computers that we may see in the
whole course. We will not see all this time though.
 Value function iterations.
 Euler Equation Methods. General Logic.
 Calibration of the model. This is the most important part of
the chapter. So far calibration has been a tainted word with too
many meanings. We will introduce a very disciplined approach to
restrict the model quantitatively. [ Cooley and PrescottCooley and Prescott1995].
 Another question. What are the implications of increasing wage
inequality? ([ Krueger and PerriKrueger and Perri2000], [ Krueger and PerriKrueger and Perri1999],
[ Heathcote, Storesletten, and ViolanteHeathcote et al.2004]).
 How to measure wage dispersion? Temporary versus
permanent changes.
 Transition, deterministic evolution over time. Convergence to
a new steady state.
 Extensions to the basic question. Does feature bla bla matter?
We will review a few of them to learn about other classes of
economies, which means both a new set of calibration and
computational issues.
 OLG Models. We will look at the basic
[ Auerbach and KotlikoffAuerbach and Kotlikoff1987] model in the context of, say, a social
security question [ Conesa and KruegerConesa and Krueger2002], an aging question
[ RíosRullRíosRull2001] or a general taxation question
[ Fullerton and RogersFullerton and Rogers1993], [ Conesa and GarrigaConesa and Garriga2004].
 Economies with measures of
things, countries, people, firms, that have idiosyncratic
shocks. Most of this chapter follows the first half of
[ RíosRullRíosRull1998].
 Theory. A steady state.
 Theory. A recursive equilibria.
 Computational. The household problem.
 Computational. The stationary distribution of
households.
 Computational. The measure as a state variable.
 Now we can use these models to talk about a variety of
issues.
 The size of precautionary savings. [ AiyagariAiyagari1995].
 Wealth and income inequality. [ Castañeda, DíazGiménez, and RíosRullCastañeda et al.1998],
[ Quadrini and RíosRullQuadrini and RíosRull1997], [ DíazGimenez, Quadrini, and RíosRullDíazGimenez et al.1997], [ De NardiDe Nardi2004].
 Entrpreneurship, creation and destruction of firms.
[ QuadriniQuadrini2000], [ Cooley and QuadriniCooley and Quadrini1998],
[ Meh and QuadriniMeh and Quadrini2004], [ Cagetti and De NardiCagetti and De Nardi2003], [ TerajimaTerajima2004].
 Business cycles behavior. [ Krusell and SmithKrusell and Smith1997],
[ Krusell and SmithKrusell and Smith1998], [ Nakajima and RíosRullNakajima and RíosRull2003].
 Distributional implications of fiscal
policies. [ Castañeda, DíazGiménez, and RíosRullCastañeda et al.1998], [ Fullerton and RogersFullerton and Rogers1993],
[ Cagetti and De NardiCagetti and De Nardi2004], [ Conesa and KruegerConesa and Krueger2002].
 Family economics. Households form and are destroyed. In doing so
many activities change, work effort, child bearing, children's
education and so on. We start looking at quantitative general
equilibrium models that are starting to look at these issues.
[ Aiyagari, Greenwood, and GünerAiyagari et al.2000], [ KnowlesKnowles1998] and
[ Regalia and RíosRullRegalia and RíosRull1998], [ Bethencourt and RíosRullBethencourt and RíosRull2003],
[ RíosRull and SeitzRíosRull and Seitz2003], [ RíosRull and WongRíosRull and Wong2003],
[ K. and ColesK. and Coles1997],
 Durable goods and housing. [ Diaz and LuengoPradoDiaz and LuengoPrado2004],
[ FernándezVillaverde and KruegerFernándezVillaverde and Krueger2003], [ OrtaloMagne and RadyOrtaloMagne and Rady2003].
 Bankruptcy. Agents have the option to file for bankrutpcy or
not. [ Chatterjee, Corbae, Nakajima, and RíosRullChatterjee et al.2004].
 Bankruptcy and private information. [ Chatterjee, Corbae, and RiosRullChatterjee et al.2004].
 Positive theory of Policy models.
 Political Economy models. [ RíosRullRíosRull1998], [ Krusell, Quadrini, and RíosRullKrusell et al.1996].
 Other technical issues:
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