Our course is separated into two parts.

Part A is from basic numerical methods to solution of Representative Agent Models (permanent income, life-cycle) with VFI, PFI; and eventually going to Heterogenous Agents with Incomplete Markets. In the end, we need to solve Recursive Stationary Equilibrium of the Aiyagari-Bewley-Hugget-Imrohoroglu (ABHI) Model.

Part B accesses to Krusell and Smith (1998) model (Heterogenous agents, incomplete markets and aggregate uncertainty), and defaultable sovereign debt.

• Jonathan Heathcote, Kjetil Storesletten, and Gianluca Violante (2009). Quantitative Macroeconomics with Heterogeneous Households, Annual Review of Economics
• Guvenen, Fatih (2012). Macroeconomics with Heterogeneity: A Practical Guide, Richmond Fed QR
• Per Krusell, Anthony Smith (2006). quantitative macroeconomic models with heterogeneous agents, Advances in Economics and Econometrics: Theory and Applications, Ninth World Congress

• Dynamic General Equilibrium Modeling, Computational Methods and Applications: with relative codes for Fortran

• Applied Computational Economics and Finance: with CompEcon Toolbox for Matlab

Quantitative Economics, NYU by Gianluca Violante

VFI and PFI is the well-known basic algorithms of dynamic programming.

For Discretization, we practiced evenly-spaced grids, chebyshev polynomials (finer grids yield better approximations but are costly in terms of computer time, for algorithm containing numerical optimization and root-finding, Matlab performs worse than Python and Julia)

In our course, VFI and PFI are implemented by brute-force iteration of value function given discrete state space. Similar way could calculate decision variable by fminbnd (i.e. kp = fminbnd(@valfun,kmin,kmax); ), where since “fminbnd” assumes a continuous choice set, basically it will search over all values of kp between “kmin” and “kmax”, which will include points not in the original capital grid. For this reason, we need to interpolate values of the value function off of the grid. (i.e. g = interp1(kmat,v0,k,'linear'); val= utility(c) + beta*g;):

while dif>tol & its<maxits
   for i = 1:N
      k0 = kmat(i,1);
      k1 = fminbnd(@valfun,kmin,kmax);
      v1(i,1) =−valfun(k1);
      k11(i,1) = k1;
   end
   dif = norm(v1−v0);
   v0 = v1;
   its = its+1
end

for i = 1:N
   con(i,1) = kmat(i,1)ˆ(alpha)−k11(i,1) + (1−∆)*kmat(i,1);
   polfun(i,1) = kmat(i,1)ˆ(alpha)−k11(i,1) + (1−∆)*kmat(i,1);
end

Where the value function for a neoclassical growth model with no uncertainty and CRRA utility are calculated by interpolation.

function val=valfun(k)

global v0 beta ∆ alpha kmat k0 s

g = interp1(kmat,v0,k,'linear'); % smooths out previous value function

c = k0ˆalpha−k + (1−∆)*k0;       % consumption
if c≤0
   val =−9999−999*abs(c); % keeps it from going negative
else
   val=(1/(1−s))*(cˆ(1−s)−1) + beta*g;
end

val = −val; % make it negative since we're maximizing and code is to minimize.

eg.1: See Part_A/PS4-solution where we practiced VFI and PFI using a couple of improvements to speed up (e.g., concavity of value function, previous solution, monotonicity of decision rules).

eg.2: See Part_B/PS1-solution/main.m, which I use VFI associated with KS algorithm for solving Krusell and Smith model (~300 minutes MATLAB).

Costs and benefits of VFI and PFI: simple and have a solution for sure, but slow and inaccurate.

• Euler Equation iteration

EEI is faster than VFI (still pretty slow), but much more accurate. For EEI, we update policy function with Euler Equation.

• Endogenous Gridpoints Methods

similar to EEI, but avoid solving non-linear equations.

eg.1: Christopher Carroll’s endogenous gridpoint method

Perturbation method (local) approximate the policy function around the no uncertainty case, and this can deal with large problems but require model equations be differentiable, hence, cannot deal perfectly with zero lower bound and occasionally binding constraints.

To approximate the policy function, we could apply 1st-order taylor expansion, or higher-order, and some more recent resources from Matlab toolbox, the "Automatic Differentiation" method.

eg.1: Perturbation methods of solving Krusell&Smith(1998): Solving the Incomplete Markets Model withAggregate Uncertainty Using a Perturbation Method. Kim, Kim, and Kollmann.

eg.2 (can be <10 seconds MATLAB): with the borrowing constraint replaced by a penalty function and the finite-state Markov process replaced by a stochastic process with continuous support; or the most recent work done by Christian Bayer, "Solving heterogeneous agent models in discrete time with many idiosyncratic states by perturbation methods", (with Ralph Luetticke), July 2018, CEPR DP No. 13071, MALTAB code

eg.1: parameterization of the cross-sectional distribution: do not obtain next period's cross-sectional moments by simulation techniques,but by explicitly integrating the individual choices;

eg.2: no parameterization of the cross-sectional distribution: derives the aggregate laws of motion directly from the individual policy rules by simply aggregating them;

We learned this method during our Macro II by using Dynare, billions of resources could be found via Internet.

Perturbation + Projection methods:

I will replicate this method in my Final Project of Unit 2.

A guide to understand different methods on solving K&S model and comparisons of different solution methods to solve K&S model: Comparison of solutions to the incomplete markets model with aggregate uncertainty. Wouter den Haan

eg.1: Reiter Michael, 2009. "Solving heterogeneous-agent models by projection and perturbation," Journal of Economic Dynamics and Control, Elsevier, vol. 33(3), pages 649-665, March.

eg.2: Thomas Winberry, 2018 forthcoming. "A Method for Solving and Estimating Heterogeneous Agent Macro Models," Quantitative Economics

Parameterized Expectations Algorithm (PEA)

Explicit Aggregation (Xpa)

Euler Equation iteration with projection methods

...