clear all; clc; tic; s = 2; %CRRA utility with parameter s r = 0.03; %interest rate % s = 0.5; % r = 0.045; rho = 0.05; %discount rate z1 = .1; z2 = .2; z = [z1,z2]; la1 = 0.02; la2 = 0.03; la = [la1,la2]; I=500; amin = -0.02; %borrowing constraint amax = 2; a = linspace(amin,amax,I)'; da = (amax-amin)/(I-1); aa = [a,a]; zz = ones(I,1)*z; maxit= 100; crit = 10^(-6); Delta = 10; dVf = zeros(I,2); dVb = zeros(I,2); c = zeros(I,2); Aswitch = [-eye(I)*la(1),eye(I)*la(1);eye(I)*la(2),-eye(I)*la(2)]; %INITIAL GUESS v0(:,1) = (z(1) + r.*a).^(1-s)/(1-s)/rho; v0(:,2) = (z(2) + r.*a).^(1-s)/(1-s)/rho; % z_ave = la2/(la1+la2)*z(1) + la1/(la1+la2)*z(2); % v0(:,1) = (z_ave + r.*a).^(1-s)/(1-s)/rho; % v0(:,2) = (z_ave + r.*a).^(1-s)/(1-s)/rho; v = v0; for n=1:maxit V = v; V_n(:,:,n)=V; % forward difference dVf(1:I-1,:) = (V(2:I,:)-V(1:I-1,:))/da; dVf(I,:) = (z + r.*amax).^(-s); %will never be used, but impose state constraint a<=amax just in case % backward difference dVb(2:I,:) = (V(2:I,:)-V(1:I-1,:))/da; dVb(1,:) = (z + 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); ssf = zz + r.*aa - cf; %consumption and savings with backward difference cb = dVb.^(-1/s); ssb = zz + r.*aa - cb; %consumption and derivative of value function at steady state c0 = zz + r.*aa; dV0 = c0.^(-s); % dV_upwind makes a choice of forward or backward differences based on % the sign of the drift If = ssf > 0; %positive drift --> forward difference Ib = ssb < 0; %negative drift --> backward difference I0 = (1-If-Ib); %at steady state %make sure backward difference is used at amax %Ib(I,:) = 1; If(I,:) = 0; %STATE CONSTRAINT at amin: USE BOUNDARY CONDITION UNLESS sf > 0: %already taken care of automatically dV_Upwind = dVf.*If + dVb.*Ib + dV0.*I0; %important to include third term c = dV_Upwind.^(-1/s); u = c.^(1-s)/(1-s); %CONSTRUCT MATRIX X = - min(ssb,0)/da; Y = - max(ssf,0)/da + min(ssb,0)/da; Z = max(ssf,0)/da; for i=2:I-1 A1(i,i-1) = X(i,1); A1(i,i) = Y(i,1); A1(i,i+1) = Z(i,1); end A1(1,1)=Y(1,1); A1(1,2) = Z(1,1); A1(I,I)=Y(I,1); A1(I,I-1) = X(I,1); for i=2:I-1 A2(i,i-1) = X(i,2); A2(i,i) = Y(i,2); A2(i,i+1) = Z(i,2); end A2(1,1)=Y(1,2); A2(1,2) = Z(1,2); A2(I,I)=Y(I,2); A2(I,I-1) = X(I,2); AA = [A1,zeros(I,I);zeros(I,I),A2]; A = AA + Aswitch; B = (rho + 1/Delta)*eye(2*I,2*I) - A; u_stacked = [u(:,1);u(:,2)]; V_stacked = [V(:,1);V(:,2)]; b = u_stacked + V_stacked/Delta; V_stacked = B\b; %SOLVE SYSTEM OF EQUATIONS V = [V_stacked(1:I),V_stacked(I+1:2*I)]; Vchange = V - v; v = V; dist(n) = max(max(abs(Vchange))); if dist(n)