clear all; clc; tic; s = 2; r = 0.045; rho = 0.05; w = .1; I=500; amin = -0.02; amax = 1; a = linspace(amin,amax,I)'; da = (amax-amin)/(I-1); maxit=20000; crit = 10^(-6); dVf = zeros(I,1); dVb = zeros(I,1); c = zeros(I,1); %INITIAL GUESS v0 = (w + r.*a).^(1-s)/(1-s)/rho; v = v0; for n=1:maxit V = v; % forward difference dVf(1:I-1) = (V(2:I)-V(1:I-1))/da; dVf(I) = 0; %will never be used % backward difference dVb(2:I) = (V(2:I)-V(1:I-1))/da; dVb(1) = (w + r.*amin).^(-s); %state constraint boundary condition I_concave = dVb > dVf; %indicator whether value function is concave (problems arise if this is not the case) %consumption and savings with forward difference cf = dVf.^(-1/s); muf = w + r.*a - cf; %consumption and savings with backward difference cb = dVb.^(-1/s); mub = w + r.*a - cb; %consumption and derivative of value function at steady state c0 = w + r.*a; dV0 = c0.^(-s); % dV_upwind makes a choice of forward or backward differences based on % the sign of the drift If = muf > 0; %positive drift --> forward difference Ib = mub < 0; %negative drift --> backward difference I0 = (1-If-Ib); %at steady state %make sure the right approximations are used at the boundaries %STATE CONSTRAINT: USE BOUNDARY CONDITION UNLESS muf > 0 Ib(I) = 1; If(I) = 0; dV_Upwind = dVf.*If + dVb.*Ib + dV0.*I0; %important to include third term c = dV_Upwind.^(-1/s); Vchange = c.^(1-s)/(1-s) + dV_Upwind.*(w + r.*a - c) - rho.*V; %% This is the update % the following CFL condition seems to work well in practice Delta = .9*da/max(w + r.*a); v = v + Delta*Vchange; dist(n) = max(abs(Vchange)); if dist(n)