Wouter J. den Haan - CFM-LSE Numerical Summer School

ADVANCED


Instructors:

• Wouter J. den Haan, two days
• Pontus Rendahl, three days

Prerequisites!!!: This course is not for everybody. It is important that you have a solid knowledge of macroeconomic models taught at the graduate level before you start a course on numerically solving these models. So make sure to read the course prerequisites carefully. If you don't have the right background yet, then it is better to wait a year.

In addition to the prerequisites of the essentials course you also need the following.

• Knowledge on solving dynamic representative-agent models using Dynare and projection methods (You will need to be able to use Dynare to solve some of the computer assignments).
• Understanding of the economics of classic models with heterogenous agents such as the Aiyagari and the Krusell-Smith model.
• Some understanding of the economics of occasionally binding constraints.

Course outline:

Two days - Solving and simulating models with heterogeneous agents

• Overview: We will look at popular algorithms used to solve models with heterogeneous agents and with aggregate risk. We will go through their implementation and discuss their strengths and weaknesses. The pioneering algorithm of Krusell and Smith (1998) is often reliable, but it is also quite slow and we will discuss improvements. In particular, we will discuss how to efficiently compute a stochastic simulation which avoids sampling uncertainty, and we will discuss alternative techniques which avoids simulation all together. We will discuss ways to impose market clearing, which in some applications is a non-trivial and important issue. we will teach you certain "tricks" to deal with this. We will also discuss how to deal with portfolio problems, asset pricing, and the introduction of money in these types of models. Lastly, we will discuss how to exploit linearization techniques when issues like the zero lower bound are present.
• Topics:
  • Simulation and distributions
  • Krusell & Smith algorithm to solve models with heterogeneous agents and aggregate uncertainty
  • Avoiding sampling uncertainty
  • Xpa algorithm to solve models with heterogeneous agents aggregate uncertainty
  • Obtaining the ergodic distribution quickly (without simulating) as the Eigenvector of the matrix in the transition equation.
• Applications & exercises - Monday and Tuesday: Solving models with heterogeneous agents.

Two days - Continuous-time models

• Overview: Most macroeconomic analysis takes place in discrete time. But some problems are better dealt with in continuous time. These two days we focus on continuous-time models and explore numerical algorithms to solve them. During these days, we will first illustrate how one may transition between continuous and discrete time by letting the interval-length between time-periods become arbitrarily small. Second, we will focus on how to solve the Hamilton-Jacobi-Bellman equation, which is the continuous time analogue to the regular Bellman equation. When doing this, we will consider two common approached: the explicit and implicit finite difference method. Lastly, we will analyze a workhorse model with incomplete markets set in continuous time.
• Topics:
  • Continuous-time models
  • Moving from discrete to continuous time
  • Algorithms to solve the Hamilton-Jacobi-Bellman equation
  • Heterogeneous agents in continuous time
• Applications & exercises - Wednesday and Thursday:Solve two continuous-time models: The Diamond-Mortensen-Pissarides model and the Aiyagari model.

One day - Occasionally binding constraints and bounded rationality

• Overview: Many interesting macroeconomic models have occasionally binding constraints. These can take the form of borrowing constraints, irreversibility constraints, or collateral constraints, to name a few. Such models are quite challenging to solve, and this part of the course will go through some methods to deal with these issues in both linear and nonlinear frameworks. The part concerning nonlinear models will focus on both liquidity constraints as well as irreversible investment. While the part concerning linear models will put an emphasis on regime switching models, in which the economy can (stochastically) transition between different regimes. A case in point is the zero lower bound.
• Applications & exercises -Friday: Solve a linear model at the zero lower bound and calculate the fiscal multiplier (c.f. Eggertsson (2011)).

LSE Economics