Econ 714 Quantitative Macro, Fall of 2006

José-Víctor Ríos-Rull:,

Last modified: Sun Dec 10 16:39:08 Eastern Standard Time 2006

  • Department of Economics, University of Pennsylvania. 429 Mc Neil, Ph. 898-7767
  • Tuesdays 5:00 pm to 8:00. In McNeil building 410.
  • Off Hours: Wed 14:00 to 15:00. (or by appointment).

  • 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 ex-post diary.

    1. 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.

    2. 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.

    3. November 28.

      Daniel went over Chebychev approximations and problems in high-dimensional 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.

    4. 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.

    5. 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.

    6. November 7.

      We went over what to do when the aggregate state of the economy is a probability measure. We talked about Quasi-Aggregation (Krusell-Smith) 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.

    7. 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.

    8. October 17.

      We talked about what features should a housing model have.

    9. 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

    10. 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).

    11. 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 log-linearization 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.

    12. 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.

    13. 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:

    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 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 email 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.

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

    2. 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.
    1. 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.

    2. The beginning of the Aiyagari Economy Solve for the decision rule of the problem in Aiyagari (94) or in Castaneda, Diaz-Gimenez, and Rios-Rull (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.
    1. Read data sets: II. The Panel-Data 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.

    2. 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 age-sex 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.
    1. 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(zt )   Kt q   Nt1 - q  
      The shock z follows the univariate process that you estimated in homework 9.

      Per period Preferences are log ct + a (1-nt).

      This means that there are a total of four parameters to pick (b ,a , q , d).
      1. 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.
      2. Define recursive competitive equilibria.
      3. Compute a loglinear approximation to the decision rules and plot them.
      4. Generate 500 samples of length 200 each and compute the standard business cycles statistics as found in the Cooley volume.
      5. Discuss the findings.
      6. Discuss optimality (and compute the optimal allocation if needed).
      7. Give an answer to the main question here. What is the contribution of the shock to movements in output and in labor.
      8. Solve this model with either spline approximations to the value function or to the decision rules and with piecewise approximations to the decision rules.
      9. 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.

    1. 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
      1. As a system of 400 equations and unknowns.
      2. By guessing period 1's capital and checking that in 200 periods it is sufficiently close to the new steady state.
      3. 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.

    2. 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.

    3. Read data sets: I. The Cross-Section 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.
    1. 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.

      1. 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 10-5 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
      2. Brute force successive iterations of the Bellman equation. Verify the code with the example of the previous question.
      3. Successive iterations of the Bellman equation taking into account only monotonicity of the decision rule.
      4. Successive iterations of the Bellman equation taking into account only concavity of the value function.
      5. Successive iterations of the Bellman equation taking into account both concavity of the value function and monotonicity of the decision rule.
      6. Use Policy iterations (waiting until converged) to solve the problem.
      7. Use policy iterations with 5, 10, 20 and 50 steps in between policy reassessments.
      8. 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.
      9. 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.
      10. Do any of the previous with a labor choice that is continuous. The coefficient of leisure is .34.

    2. 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.

    3. 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.
    1. 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 1-4 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.
      1. Solve the value function by successive approximations.
      2. Generate a sample of length 200 and plot it.
      3. Compute the Stationary Distribution of individual States.
      4. Compute using both methods average consumption and autocorrelation of consumption.

    2. Closed Form Solution

      Use log utility, Cobb-Douglas 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.
    1. Data manipulation.

      1.A Fetch and plot US quarterly GDP Investment plus durables plus net exports, non-durables 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.

    2. 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.

    3. 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).

    4. 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 Cobb-Douglas ( 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.

    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.


    1. 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?
    2. 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.
      • 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].
    3. 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.
    4. 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.
    5. 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 [ Ríos-RullRíos-Rull2001] or a general taxation question [ Fullerton and RogersFullerton and Rogers1993], [ Conesa and GarrigaConesa and Garriga2004].
    6. Economies with measures of things, countries, people, firms, that have idiosyncratic shocks. Most of this chapter follows the first half of [ Ríos-RullRíos-Rull1998].
      • 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.
    7. Now we can use these models to talk about a variety of issues.
    8. 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 Ríos-RullRegalia and Ríos-Rull1998], [ Bethencourt and Ríos-RullBethencourt and Ríos-Rull2003], [ Ríos-Rull and SeitzRíos-Rull and Seitz2003], [ Ríos-Rull and WongRíos-Rull and Wong2003], [ K. and ColesK. and Coles1997],
    9. Durable goods and housing. [ Diaz and Luengo-PradoDiaz and Luengo-Prado2004], [ Fernández-Villaverde and KruegerFernández-Villaverde and Krueger2003], [ Ortalo-Magne and RadyOrtalo-Magne and Rady2003].
    10. Bankruptcy. Agents have the option to file for bankrutpcy or not. [ Chatterjee, Corbae, Nakajima, and Ríos-RullChatterjee et al.2004].
    11. Bankruptcy and private information. [ Chatterjee, Corbae, and Rios-RullChatterjee et al.2004].
    12. Positive theory of Policy models.
    13. Political Economy models. [ Ríos-RullRíos-Rull1998], [ Krusell, Quadrini, and Ríos-RullKrusell et al.1996].
    14. Other technical issues:


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    Aiyagari, S. R., J. Greenwood, and N. Güner (2000). On the state of the union. Journal of Political Economy  108 (2), 213-244.
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    Auerbach, A. J. and L. J. Kotlikoff (1987). Dynamic Fiscal Policy. New York: Cambridge University Press.
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    Bethencourt, C. and J.-V. Ríos-Rull (2003). The living arrangement of elderly widows. Mimeo, University of Pennsylvania.
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    Cagetti, M. and M. De Nardi (2003). Entrepreneurship, frictions and wealth. FRB Mpls Staff Report 322.
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    Castañeda, A., J. Díaz-Giménez, and J. V. Ríos-Rull (1998). A general equilibrium analysis of progressive income taxation: Quantifying the trade-offs. Mimeo, University of Pennsylvania.
    [ Ceria and Ríos-RullCeria and Ríos-Rull1992]
    Ceria, S. and J. V. Ríos-Rull (1992). On the existence uniqueness and computability of non-optimal recursive equilibria in linear quadratic economies. Unpublished Manuscript, Carnegie Mellon University.
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    Chang, R. (1995). Report on political party negotiations, income distribution and endogenous growth. Unpublished manuscript, Federal Reserve Bank of Atlanta.
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    [ Chatterjee, Corbae, Nakajima, and Ríos-RullChatterjee et al.2004]
    Chatterjee, S., D. Corbae, M. Nakajima, and J.-V. Ríos-Rull (2004). A quantitative theory of unsecured consumer credit with risk of default. Unpublished Manuscript, CAERP.
    [ Chatterjee, Corbae, and Rios-RullChatterjee et al.2004]
    Chatterjee, S., D. Corbae, and J.-V. Rios-Rull (2004). A competitive theory of credit scoring. Mimeo, University of Pennsylvania, CAERP.
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    Christiano, L. J. and J. D. Fisher (1994). Algorithms for solving dynamic models with occasionally binding constraints. Federal Reserve Bank of Minneapolis Staff Report 171.
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    [ Conesa and KruegerConesa and Krueger2002]
    Conesa, J. C. and D. Krueger (2002). Optimal progressivity of the income tax code. Mimeo, CRES, Universidad de Barcelona.
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    Cubeddu, L. M., M. Nakajima, and J.-V. Ríos-Rull (2001). Marital risk and capital accumulation. Mimeo, University of Pennsylvania.
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    Cubeddu, L. M. and J.-V. Ríos-Rull (2003). Families as shocks. Journal of the European Economic Association  1 (2-3), 671-682. (Papers and Proceedings of the Seventeenth Annual Congress of the European Economic Association, Venice, 21-24 August 2002).
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    De Nardi, M. (2004). Wealth inequality and intergenerational links. res  71 (3), 743-768.
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    Diaz, A. and M.-J. Luengo-Prado (2004). Precautionary savings and wealth distribution with durable goods. andiaz/pdfs/research/durables.pdf.
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    Díaz-Gimenez, J., V. Quadrini, and J.-V. Ríos-Rull (1997, Spring). Dimensions of inequality: Facts on the U.S. distribution of earnings, income and wealth. Federal Reserve Bank of Minneapolis Quarterly Review   21, 3-21.
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    Fernández-Villaverde, J. and D. Krueger (2003). Consumption over the life cycle: Some facts from consumer expenditure survey data. CAERP Working papers No. 2003-07.
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    Fernandez-Villaverde, J. and A. Tsyvinski (2003). Optimal fiscal policy in a business cycle model without commitment. Mimeo, University of Pennsylvania.
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