Gianluca Violante, NYU


"I have never let my schooling interfere with my education" (Mark Twain)

Macroeconomic Theory II (G31.1026)

Heterogeneity in Macroeconomics


Professor Gianluca Violante
Room 712, Department of Economics
19 W 4th street
Tel. 212-992-9771

General Information Examination Summary and Objectives Readings Course Outline


Download the preliminary syllabus here

General Information 

Lecture Times and Location: Monday and Wednesday 9:55-11:55 in Room 517. The last class is on May 2nd.

Office Hours: Monday 12:00-1:00 in my office, Room 712. You can always contact me by phone at extension 29771 or
by email   to arrange an alternative time.

TA and Tutorials: The TA is Dan Greenwald who can be reached by e-mail. He sits in room 721. Weekly tutorials are held on Friday 9:30-11:30 am in Room 517.

Homework: There will be weekly problem sets that are required for a passing grade. The problem sets are handed out on Wednesdays and are due the next Wednesday 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.


There will be a written final examination on May 7th, 9:30-11:30, room 517.

Summary and Objectives 

Summary: This last section of the course is devoted to studying economies where agents are heterogeneous. These models are helpful to analyze a wide range of questions pertaining to income distribution, asset pricing, financial markets, 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.

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

Reading Material

Textbook and Readings: The main textbook is Recursive Macroeconomic Theory, by Lars Ljungqvist and Tom Sargent, MIT Press, second edition, 2004 (denoted by LS below). You will also use Recursive Methods in Economic Dynamics, by Stokey, Lucas and Prescott, Harvard University Press, 1989. In the recitation on March 26th, Dan 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.

Background readings: Two useful background readings for this course are

Read them once at the beginning of the course, 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.

All the papers listed below 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 Outline 

March 19: Aggregation (LS, 8.5.3)
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 quantities. 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 and household heterogeneity in endowments, where agents have (quasi) homothetic utility. 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 21: Aggregation (continued)
We discovered that in SS of the growth model with complete markets and heterogeneous endowments the wealth distribution is indeterminate, but given an initial distribution the equilibrium dynamics are unique. We then covered the Negishi method, and derived the aggregation with complete markets result by Constantinides (1982). The lecture notes contain an application based on the papers by Maliar-Maliar (2001, 2003).

Lecture notes (updated)              Homework 1               Dan's solution to Homework 1

Remember that Friday there is the Matlab recitation

March 26: 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 discussed the ad-hoc and micro-founded approaches to modeling market incompleteness. 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 beta*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.

March 28: Precautionary Saving and the Income Fluctuation Problem (LS, 16.5-16.8, 17.3-17.5)
We have shown that borrowing constraints bind with probability one in the PIH model if income is iid. 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                       Homework 2           Dan's solution to Homework 2

Derivation of Sibley's result in the multiperiod (finite-horizon) problem (thanks Dan!)

Recall that Friday there is the measure theory recitation              

April 2: 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

April 5: The Neoclassical Growth Model with Incomplete Markets I (LS 17.1-17.2, 17.6-17.12)
We have kept discussing numerical solution methods for the income fluctuation problem. We have seen the endogenous grid method and learned how to simulate from the model. We have briefly discussed how to gauge the accuracy of the solution. Next, we  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.

     Lecture Notes  (updated)          Dan's measure theory notes     

    Computational Assignment  (due 5/2)     

April 9: Stationary equilibrium in the Incomplete Markets Models (continued)
We have discussed conditions for existence and uniqueness of the the equilibrium in Aiyagari's model. We have explained how to calibrate the model and and compute the stead-state equilibrium. Then we have illustrated how to use this class of models to analyze questions related to precautionary saving and wealth inequality.

April 12: Stationary equilibrium in the Incomplete Markets Models (continued)
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. Next we analyzed the model with endogenous labor supply, and explained how to compute the stationary equilibrium. We also argued that a Ramsey planner would choose a positive level of taxation in this model, by trading-off distortions and redistribution. We started thinking about constrained efficiency in this class of models.

     Lecture Notes                      Homework 4

     No recitation this Friday
     Important: in the computational assignment, set \beta=0.90 throughout the computation.

April 16: Constrained Efficiency in the Aiyagari model & Transitional Dynamics
We began by discussing 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.  Next 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.

Lecture Notes on constrained efficiency
Lecture Notes on transitional dynamics

April 18: Adding Aggregate Risk: A Near-Aggregation Result (LS 17.14.2)
We have defined different welfare criteria to study the effects of policy reforms (conditionl and ex-ante welfare). Then 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

April 23: 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

April 25: Lifecycle Economies
Earnings 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. We quickly explained why the near-aggregation result of Krusell-Smith does not carry out to life cycle economies.

Lecture Notes                  Homework 6

April 30: Default
We first studied an economy with collateral. 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 2: 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. We thought about how to close the model in general equilibrium.

Lecture Notes  

May 7: FINAL EXAM, 9:55-11:55 am in Room 517

Final Exam 2005 
Final Exam 2006   
Final Exam 2007     
Final Exam 2008
Final Exam 2010