# Quantitative Macroeconomics [F2018]

"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).

## Course Description:

Quantitative Macroeconomics (Unit I) follows the first year PhD macro sequence. The goal of this course is to equip you with a wide set of tools to (i) solve macroeconomic models with heterogenous agents aka Aiyagari-Bewley-Hugget-Imrohoroglu (ABHI) economies 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.

The ABHI economies are the industry standard in macro. These economies can take the form of infinite horizon, lifecycle environments, and overlapping generations. Importantly, 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.

This course is demanding and I expect you to be engaged continuously from day one. The grade will be some weighted average of regular homeworks.

We meet Mondays and Wednesdays 15:00-16:30 in the UAB Seminar room.

## Class Diary:

• Wed Sep 12: We went over the syllabus [.pdf] and discussed the rules of the game.
• Mon Sep 17: We discussed some examples of quantitative experiments that we do in macro (e.g., how much of X explains Y? how much does welfare increase/decrease after a redistributive policy?). As an example, we focused on the Lucas paradox (or puzzle) emerging from the global allocation of capital (MPKs against aggregate K or TFP). We then discussed the typical steps in a quantitative experiment. Slides: [What is Quantitative Macro?].
• Wed Sep 19: We posed the projection methods algorithm. Slides: [Projection Methods: An Algorithm]. This requires the knowledge of several numerical techniques that we will cover in the following days. We started with numerical differentiation and integration. Slides: [Numerical Differentiation and Integration] . We then moved into function approximation covering local methods, and highlighting the pros and cons. [Function Approximation]
• Tue Sep 25: We continued our discussion on function approximation entering the territory of global methods. These are essential for our industry standard quant macro models where agents are heterogenous in some dimensions of interest. We started with spectral methods that use the entire domain in the approximation. We discussed the choice of interpolation nodes and basis functions. We discussed the problems arising from monomials as basis functions. We then discussed some nice propoerties of Chebyshev nodes (as opposed to equally spaced nodes) and orthogonal polynomials. We focused on Chebyshev polynomials.
• Wed Sep 26: We continued our discussion on global methods. We now discussed finite-element methods/splines (a good reference is the chapter by Ellen McGrattan (1998). Finite-element methods are particularly useful when we want local support for example to problems with functions that have kinks (e.g., inequality constraints that occasionally bind). We discussed linear splines and cubic splines. We briefly discussed Schumacker splines that preserve monotonicity and concavity. Finally, we open a new subject and discussed how to solve systems of nonlinear equations. We started with bisection and then went over Newton and quasi-newton methods in univariate and multivariate cases. We also discussed in some detail Gauss-Jacobi and Gauss-Seidel as alternative methods.
• Mon Oct 1: We discussed numerical optimization, in particular, derivative-free methods such as the Nelder-Mead algorithm. Slides: [Numerical Optimization]. We started to implement the VFI on the neoclassical growth model with global methods. We started with descretization. Slides: [Value Function Methods: Discrete and Continuous Methods].
• Please, refresh your knowledge on dynamic programming (Value Function Iteration, VFI) if you need to. This is a must. Slides: [Dynamic Programming].

• Wed Oct 4: Adam discussed your HWK 1 on different approximations including a 2-dimensional approximation of a CES production function using Cheby nodes and polynomials. Maria discussed the construction and the secular properties of the labor share in HWK 2.
• Mon Oct 8: We discussed in some detail HWK3, a neoclassical transition and an assesment of tax progressivity in a 2-period GE model with permanent and transitory heterogeneity in income. We continued our discussion on VFI adding stochastic components to the model.
• Wed Oct 10: We continued our discussion on VFI focusing on continuous methos to approximate the value function. We focused on B-splines. We also discussed the simulation of the model and replication of some business cycle statistics including some impulse response functions.
• Mon Oct 15: We discussed the use of tensor products for high-order dimensionality. Slides: [Tensors and the Curse of Dimensionality]. We started our discussion of ABHI economies. Slides: [Aiyagari-Bewley-Hugget-Imrohoroglu Economies].
• Thur Oct 25: We continued our discussion of ABHI economies. We also discussed how to check for the accuracy of our approximated solutions. Slides: [Accuracy Tests]. Finally, we discussed how to conduct policy evaluation. Slides: [Welfare Analysis and Policy Evaluation].

## Homeworks

Students should expect one homework per foreseeable week of class:

• [Homework 1] [Due Fri Sep 28] Numerical differentiation and function approximation, univariate and multivariate cases. Example: A CES production function.
• [Homework 2] [Due Wed Oct 3] Secular behavior of the labor share of income.
• [Homework 3] [Due Wed Oct 10] Two models: (1) A Neoclassical transitions (or miracle) and (2) a 2-period GE consumption model with stochastic income and progressive income taxation.
• [Homework 4] [Due Wed Oct 17] Value function iteration of the neoclassical growth model, and discussion of business cycle properties of model and data.
• [Homework 5] [Due Thu Nov 1] ABHI Economy.