*"The era of closed-form solutions for their own sake should be over. Newer generations get similar intuitions from computer-generated examples than from functional expressions"*, Jose-Victor Rios-Rull, JME (2008).

Quantitative Macroeconomics follows the first year PhD macro sequence: Econ 501 and 502. The goal of this course is to equip you with a wide set of tools to (i) solve macroeconomic models with heterogenous agents and (ii) relate these models to data to answer quantitative questions. You will learn to do so by doing. That is, this course will require intensive computational work by students.

First, we will discuss numerical methods to solve for the equilibrium allocations of representative agent models. Second, we will study how to solve heterogeneous agents economies in infinite horizon, lifecycle environments, and with overlapping generations. In macroeconomic models, the presence of heterogeneity requires taking good care of distributions and aggregate consistency. We will discuss carefully how to do this in both stationary and nonstationary environments such as business cycles or development processes of structural transformation.

The grade will be some weighted average of regular homeworks, and an original research project to be presented in class.

We meet Tuesdays 17:30-20:30 in Seigle Hall 303.

**Tue Jan 17th:**We went over the syllabus [.pdf] and discussed what you are expected to do. We also refreshed our knowledge on discrete dynamic programming. Slides: [Dynamic Programming]**Tue Jan 24th:**We have introduced some sources of aggregate data and tools to analyze them. We discussed the properties of deterministic and stochastic trends used to stationarize time series. Then have defined the cycle using the Hodrick-Prescott filter; we have discussed the differences with the Baxter-King filter. We went over the NIPA accounts, and BLS. Then we computed the business cycle facts of the U.S. economy including volatility, persistence, co-movements with output, and phase shifts. We have discussed in detail how to construct the labor share of income; extended to durables and government capital. We studied the construction of the productivity residual, and seen the differences in these residual when investment-specific technical change is incorporated in the accounting exercise. Last, we have introduced VARs systems, and discussed how long- and short-run implications from economic theory can inform the identification of these systems using associated long- and short-run restrictions that give rise to Structural VARs. Slides: [Macro Data Sources and Tools]**Tue Jan 31st:**We finished our discussion on how theory informs the identification of SVARs; some applications have been studied that include the identification of productivity shocks, investment shocks, redistributive shocks, and spillover effects. The purpose of this course is to master macroeconomic models with heterogeneous agents. Therefore, we will need both aggregate and household/individual data to discipline the quantitative implications of our models. Today we have introduced several sources of household- and individual-level data. This includes sources of cross-sectional data, panels of cross-sectional data, and panel data. Then we have discussed in detail the facts regarding household level inequality (using SCF and PSID data) and over the business cycle (using the CPS-MORG) and exposed some of the puzzles that arise in benchmark theories and that we will attack once we learn how to solve models with heterogenous agents. "Fishing for facts" we have also discussed the empirical relationship between education, HIV status, and risky sexual behavior over stages of the HIV epidemic (using DHS data). Finally we have reviewed the literature on skilled biased technical change to explain the rise of wage inequality (using CPS and Census data). In this context, we have discussed the hypothesis regarding the complementarity between capital-equipment and skilled labor to explain skill biased technical change; and explored the possibility of changes in the substitutability of factors that represent increases in human capital specificity over time.**Tue Feb 7th:**Ting-Wei, Jaevin, and Raul presented their results for HWK1 on business cycle measurement in much detail. We looked more carefully at consumption, income, wealth, and labor supply inequality over time and over the life cycle. To do so we have extensively used the facts documented in a recent special issue of the Review of Economic Dynamics, Krueger, Perri, Pistaferri, and Violante (2010). We started our discussion on how to estimate earnings, income, or wage processes. Slides: [Micro Data Sources and Tools]**Tue Feb 14th:**We have discussed in detail the estimation of the permanent-transitory model. We have discussed the practical of identifying age, cohort, and time effects; we have used Heathcolte, Storesletten, and Violante (2005), Ameriks and Zeldes (2004), and Krueger and Fernandez-Villaverde (2007) as working examples. We ended our session with a discussion on selection bias. In particular we discussed median wage regressions using imputed wages as in Olivetti and Petrongolo (2009) and model structure as in French (2005). Finally, Daniel and Duksang presented HWK2 on cross-country differences in income per capita.**Tue Feb 21st:**We discussed methods to approximate functions. We first introduced local methods, and discussed its pros and cons. We then moved to global methods: discretization; spectral methods (polynomial interpolation, Chebyshev); and finite element methods (splines) in detail. We have discussed shape-preserving Schumaker splines. One useful specification of splines is B-splines, we discussed them in detail. Finally, we analyzed weighted residual methods such as least squares, collocation, and Bubnov-Garlekin. Slides: [Function Approximation]**Tue Feb 28th:**We finished our discusstion on numerical methods. We learned how to compute numerical derivatives and integrals using Gaussian-quadratures and Quasi- and Monte Carlo methods; Slides: [Numerical Integration and Differentiation]. We then discussed how to solve for multivariate nonlinear systems of equations (bisection, newton, broyden). We went in detail on Gauss-Jacobi and Gauss-Seidel methods. We discussed advantages and disadvantages; Slides: [Nonlinear Systems]. We ended up our discussion with derivative-free methods (direct search methods). In particular, we have discussed the Nelder-Mead algorithm, a simplex method; Slides: [Numerical Optimization].**Tue March 6th:**We went over value function iteration to solve for deterministic and stochastic neoclassical economies. Slides: [Value Function Methods]. Ting-Wei and Jaevin proposed their projects, and Fen presented her research.**Tue March 20th:**We discussed the (log)linearization of standard models for the business cycle. We used Blanchard-Khan eigenvalue matrix decompostion and Uhlig's method of undetermined coeffiecients. We extended the analysis to ISTC shocks, capital utilization, habit persistence, and the Cho and Cooley version of explicitely modelling both intensive and extensive margins of the labor input. Slides: [(Log-)Linearized Euler Equations Methods]. Grant discussed his results for the transition hwk.**Tue March 26th:**We discussed aggregation of heterogeneity in initial wealth endowments with quasi-homothetic preferences and complete markets, i.e., Gorman aggregation. We have analyzed under what conditions we can formulate an economy with heterogenous agents as a representative agent economy because aggregaet equilibrium quantities and prices do not depend on the distribution of the individual quantities across agents. We have also analyzed the equilibrium dynamics of the wealth distribution. We have also used the Negishi method to compute the competitive equilibrium prices and allocations of complete markets economies with heterogenous agents for which the welfare theorems hold. Finally, we have pushed aggregation theorems beyond Gorman with Maliar and Maliar (2001,2003) that extend the RA representation to non-homothetic preferences and insurable shocks to labor productivity. Slides: [Heterogeneity in Complete Markets.pdf].**Tue April 3rd:**We discussed consumption insurance in incomplete markets. First, we studied the case of certainty equivalence, and analyzed the consumption response to income shocks in this scenario. Second, we introduced precautionary savings via (1) prudence and (2) liquidity constraints. Third, we discussed numerical solutions of models with precautionary savings. We discussed the case of infinite and finite horizons, as well as solving for the value function versus using the Euler equation to solve for the policy function. Finally, we also discussed the Endogenous grid method. Slides: [Heterogeneity in Incomplete Markets I.pdf]**Tue April 17th:**We posed a macroeconomic model with heterogenous agents. We defined the sequential markets equilibrium, the recursive competitive equilibrium, and the stationary recursive equilibrium. We discussed theoretical results of existence and uniqueness. Finally, we discussed the computation of the general equilibrium by guessing the interest rate and iterating on it according to the clearing condition of the capital market. Slides: [Heterogeneity in Incomplete Markets II.pdf]. Finally we discussed a search and matching model to understand risky sexual behavior and the diffusion of the HIV epidemic.**Tue April 24th:**Projects and other presentations.**Tue May 1st:**We discussed macroeconomic models with heterogeneous agents that are subject to unexpected aggregate shocks (probability zero events) and discussed an algorithm to compute the transition from old to new steady states in the presence of heterogeneity. We have then discussed measures of welfare and why including the transition (rather than looking at steady states) is important to conduct welfare assesments. Then we have discussed the Krusell-Smith economy in which heterogenous (in income) agents are subject to expected aggregate shocks in TFP. We have discussed the quasi-aggregation theorem implied from the Krusell-Smith economy, and assessed their quantitative results. Slides: [Heterogeneity in Incomplete Markets with Aggregate Risk.pdf]

Students should expect one homework per foreseeable Tuesday:

- [Homework 1] Business cycle measurement, and Structural VARs. Due Feb 7.
- [Homework 2] TFP, population, health. Accounting for international income per capita differences. Due Feb 14.
- [Homework 3] Identification and estimation of income processes using panel data: The permanent-transitory model. Due Feb 28.
- [Homework 4] Function approximation. Due March 6.
- [Homework 5] Solving a transition in a neoclassical economy. Due March 20.
- [Homework 6] Value function iteration: The stochastic optimal growth economy. Due March 20.
- [Homework 7] Solve a business cycle model. Due April 3.
- [Homework 8] Solve a macroeconomic model with heterogenous agents and endogenous labor choice. Due May 25.