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Macroeconomic Theory II

General Information

Lectures: Tuesday and Thursday 9:30-11:30 in K-MEC 5-80

Office Hours: Wednesday 1:30-2:30 in my office, Room 712, Department of Economics, 19 W. 4th Street, on the 7th floor. You can always contact me by phone at extension X2-9771 or by email to arrange an alternative time, if that does not suit you.

Teaching Assistant and Tutorials: Diego Daruich. Office hours: Monday 4:00-5:00 pm in Room 720 in the Department of Economics. Again, contact Diego by e-mail to arrange an alternative time, if this does not work for you. Tutorials are on Friday 9:30-11:30 in 517.

Homework: There will be weekly problem sets that are required for a passing grade. The problem sets are handed out on Thursdays and are due the next Thursday at the beginning of the class. You are allowed to cooperate with other students, but every student has to hand in his/her own uniquely written assignment.

Examinations: There will be a final exam on Friday, May 16, 10-12 in room 517

Summary and Objectives 

Syllabus: Download it here.

Summary: This last section of the core macro sequence is devoted to studying economies where agents are heterogeneous. These models are helpful to analyze a wide range of questions pertaining to business cycles, income distribution, asset pricing, consumption insurance, labor supply, the aggregate and redistributive effects of policies, etc. We will start with some "aggregation theorems" to show that in some cases a representative agent still exists. Next, we will move towards economies with "incomplete markets" where agents can only borrow and save through a risk-free bond. We begin by characterizing in detail the individual problem. Next, we proceed to the description of the stationary equilibrium. Then, we study an incomplete-markets model with aggregate shocks. The last set of classes are devoted to defining economies where there is default in equilibrium, and economies with heterogeneous firms. We may add one or two new topics, depending on the speed at which we settle.

Objectives: The aim of this course is to learn: 1) this important class of macroeconomic models, and 2) how to solve numerically for the equilibrium of these economies, a necessary step to use these models for quantitative research.

Textbooks, Reading Material, and Website 

Textbooks: The main textbook is Recursive Macroeconomic Theory, by Lars Ljungqvist and Tom Sargent (LS), MIT Press, third edition, 2012. You will also use Recursive Methods in Economic Dynamics, by Stokey, Lucas, and Prescott (SLP), Harvard University Press, 1989.

Background readings: Two useful background readings for this course are:
Read them once at the beginning of the course, in that order. You'll probably find many of the sections hard to follow. Read them again at the end of the course, and you will see the light...

Finally, all the papers listed below, class by class, are required, with the exception of those marked with (XR) which are eXtra Readings, just for your own benefit, if you're interested in the topic.

Course website: Check regularly this web-page where announcements, readings, and notes will be regularly posted in PDF form to be downloaded.

Course Outline

In what follows, I will outline the topics that we cover during the course, as we make progress.

March 25: Aggregation
We defined aggregation as a property of an economic model where the evolution of the aggregate equilibrium quantities and prices does not depend on the distribution of individual endowments. We briefly discussed aggregation of CRS production functions, and aggregation of preferences when every agent is the same along every dimension. We studied conditions under which Gorman (or demand) aggregation holds. We studied the equilibrium of the growth model with complete markets, quasi-homothetic utility, and household heterogeneity in endowments. In the presentation, we followed Chatterjee's article. We showed that a "representative agent" exists. The dynamics of aggregate quantities and prices are independent of the distribution of wealth, and are the same as in the representative agent economy you studied earlier. This is a stark example of Gorman aggregation.
March 27: Aggregation continued
We showed that in SS of the growth model with complete markets, heterogeneous endowments and homothetic preferences, the wealth distribution is indeterminate, but given an initial distribution the equilibrium dynamics are unique. We then covered the Negishi method, and derived the general aggregation with complete markets result by Constantinides (1982). The lecture notes contain an application based on the papers by Maliar-Maliar (2001, 2003). Finally, we discussed the two approaches one can take to modelling market incompleteness.

Lecture notes             Homework 1               Solution to Homework 1

In the first recitation on March 28th, Dario will teach some basic concepts of measure theory from SLP, chapters 7, 8.1,11.1,11.2 and 12.4. We need them from class 5 onwards. Note that the first recitation is in KMC 3-80 at the usual time because we need 517 for the Open House.

Notes on measure theory

April 1: Full Insurance and the Permanent Income Hypothesis
We we  talked about the dynamics of individual consumption in complete markets and empirical tests of full insurance. We then described the budget constraint of an agent who is cut off from all insurance markets and can only save/borrow with a non-state contingent asset. We introduced the strict version of the PIH, quadratic utility and B*R=1. We showed the martingale property of consumption, certainty equivalence, we showed that consumption equals the annuitized value of financial and human wealth, and we toyed around with a special case (permanent-transitory income shocks). We showed that with panel or repeated cross-section data we can identify time-varying variances of the income shocks. Finally, we showed that, whether ad-hoc borrowing constraints bind or not in the PIH depends on the income process.

April 3: Precautionary saving and the income fluctuation problem (LS 17)
We have introduced the notion of precautionary saving (additional saving in the presence of uncertainty). We have related it to the convexity of marginal utility (prudence) and to the presence of borrowing constraints potentially binding in the future. We have defined a natural borrowing limit for the stochastic case. We have derived necessary conditions on the interest rate so that the optimal individual consumption sequence is bounded above, in the deterministic case and in the stochastic case. We have also shown, somewhat heuristically, that when income shocks are iid and BR<1 if absolute risk aversion declines monotonically with consumption, then the consumption sequence is bounded.
Lecture Notes        Derivation of Sibley's result in the multiperiod (finite-horizon) problem

Homework 2  (note that the first problem is 17.2 in L-S_v3)       Solution to Homework 2

April 8: Numerical Techniques to Solve the Stochastic Consumption-Saving Problem
We have discussed how to discretize an AR1 process with the Tauchen method and the Rouwenhorst method. We have described in great detail a method to solve the income fluctuation problem based on iterating over the Euler equation and linearly interpolating the decision rule outside grid points.

Lecture Notes

Homework 3    due May 8.

April 10: The Neoclassical Growth Model with Incomplete Markets I (LS 18.1-18.14)
We  have described the neoclassical growth model populated by a continuum of agents who face idiosyncratic labor income risk and trade only a risk-free asset (i.e., the model in Aiyagari 1994). We defined a stationary RCE and proved existence of an equilibrium. We also explained that the endowment economy of Huggett 1993 is a special case.

Lecture Notes

April 15: Some Applications
We have explained how to calibrate the model and and compute the steady-state equilibrium. Then we have illustrated how to use this class of models to analyze questions related to precautionary saving and wealth inequality. In particular, we outlined a model with entrepreneurs and workers and argued that it can generate a more skewed wealth distribution, since entrepreneurs have access to a higher return on their investment.
Lecture Notes

April 17: Optimal Ramsey Taxation with Incomplete Markets. Constrained Efficiency

We began by studying the optimal level of government redistribution and the optimal quantity of government debt in the model. We then discussed the difference between the first-best allocation and the constrained-efficient (second-best) allocation in the Aiyagari model. We argued that the constrained planner, through saving decisions, will manipulate prices in order to raise wages (if the income of the poor is labor intensive), hence redistributing from the lucky-rich to the unlucky-poor.
Homework 4            Solution to Homework 4

April 22: Transitional Dynamics
We defined a RCE of an economy undergoing a transition between two steady-states due to a tax reform, and studied how to compute the transitional dynamics by means of a shooting algorithm. We learned how to measure welfare changes from the tax reform.

April 24: Adding Aggregate Risk: A Near-Aggregation Result (LS 18.15)
We have extended the standard incomplete markets model to incorporate aggregate fluctuations in productivity. We have explained how to solve for the equilibrium of this model by approximating the law of motion for the distribution. We have explained the intuition for the "near aggregation" result: saving policies are linear for the rich, and the rich hold the bulk of the capital stock, so they determine its evolution.

Lecture Notes        Homework 5      Solution to Homework 5

April 29: Micro and Macro Labor Supply Elasticity
We derived the expression for the Frisch elasticity of labor supply and discussed issues in estimation of this magnitude from micro data. We explained there is a tension between the small micro estimates and the large values used by macroeconomists. We presented the Hansen-Rogerson indivisible model, where the micro elasticity is small and the macro elasticity (i.e., the elasticity of aggregate hours to the average wage) is infinity. We argued that even if one relaxes the lottery/full insurance assumption and designs an economy with indivisible labor at the household level and incomplete markets, we still find that the aggregate elasticity is much higher than the micro elasticity.

Lecture Notes

May 1: Lifecycle Economies

Wage inequality rises over the life-cycle. So does consumption inequality, but by much less. Hours inequality is flat. We argued that the complete-markets model (with separable utility) is unable to reproduce these facts. We studied an overlapping-generations version of the neoclassical growth model with incomplete market and we argued it can go a long way in matching the facts. 
Lecture Notes          Homework 6           Solution to Homework 6

May 6: Aggregate shocks in lifecycle economies and default
I have explained why the near-aggregation result of Krusell-Smith does not carry out to life cycle economies. Next, we have studied an incomplete-market economy where agents face borrowing constraints that are tight enough so that they never have the incentive to default in equilibrium. Then, we have formalized a model where agents can default and the financial sector takes into account the default probability and increases the prices of loans accordingly.

Lecture Notes 

May 8: Industry Dynamics
We studied the equilibrium of an industry with firms facing shocks to their productivity level, and with endogenous firm entry and exit. We analyzed the impact of firing costs on the average productivity of the industry.

Final Exam: Friday May 16, 10-12 in room 517

Sample finals from past years: 2005, 2006, 2007, 2008, 2010, 2012     (in the missing years I did not teach the course)