function err_mom = objective(x) set(groot, 'DefaultAxesLineWidth', 1.5); set(groot, 'DefaultLineLineWidth', 4); set(groot, 'DefaultAxesTickLabelInterpreter','latex'); set(groot, 'DefaultLegendInterpreter','latex'); set(groot, 'DefaultAxesFontSize',22); intmeth = 'linear'; printr = 0; optset('bisect', 'tol', 1e-32); % Calibrated Parameters p.beta = x(1); % discount factor p.alpha = x(2); % weight on housing in preferences p.R = x(3); p.phi = x(4); % productivity non-market p.F0m = x(5); % fixed cost of refinancing p.B = x(6); p.rh = (1 + x(7))^(1/4) - 1; % interest rate p.r1 = x(8); p.r2 = x(9); % parameters governing shape of rl curve % Assigned Parameters p.rl = (1 - 0.028)^(1/4) - 1; % lower bound on liquid rate p.T = 61*4; % last period of life p.D = 30*4; % maturity of mortgages p.sigma = 2; % CRRA p.gamma = 1; % Frisch elasticity of non-market production p.rm = (1 + 0.025)^(1/4) - 1; % interest rate p.Fs = 0.06; % fixed cost of selling home p.F1m = 0.005; % proportional cost of refinancing p.wbar = 0.00001; % parameter governing luxuriousness of bequests p.thetam = 0.85; % maximum LTV p.thetay = 0.214; % maximum PTI rhoz = 0.9908; sz = 0.0761; se = (1 - 0.55)^(1/2)*0.4869; % Krueger Perri (2011) show 55% of the variance of trans compon is measurement error so subtract time = (1 : 1 : p.T)'; p.lambdat = exp(0.07982636 - 0.02322307 * (time/4 + 25) + 0.00105409 * (time/4 + 25).^2 - 0.00001028 * (time/4 + 25).^3); p.thetay = p.thetay*(1 - 0.3126./(1 + exp(18.629 - 0.3049*(time/4 + 25)))); p.mbar = p.rm/(1 - (1 + p.rm)^(-p.D)); % minimum payment required per 1 of initial debt % Quality of Approximation p.na = 50; % number of nodes for liquid assets p.nl = 75; % number of nodes for liquidity p.no = 9; % number of nodes for omega (fraction of loan outstanding) p.nt = 3; % number of possible initial LTV p.nh = 7; % number of nodes for housing p.nz = 9; % points for exogenous income z p.ne = 3; % Discretize Income Process [zgrid, Fzz] = rouwenhorst(rhoz, sz, p.nz); p.zgrid = exp(zgrid'); [Fz, d] = eigs(Fzz', 1, 'largestabs'); Fz = Fz/sum(Fz); Fz = full(Fz); % ergodic distribution of z [egrid, we] = qnwnorm(p.ne, 0, se^2); p.egrid = exp(egrid); % Discretize other state variables amin = 0; amax = 100; p.agrid = amin + (amax - amin)*nodeunif(p.na, 0, 1).^2; omin = 0; omax = 1; p.ogrid = omin + (omax - omin)*nodeunif(p.no, 0, 1); tmin = 0.5; tmax = p.thetam; p.tgrid = tmin + (tmax - tmin)*nodeunif(p.nt, 0, 1); hmin = 5; % minimum house size hmax = 40; % maximum house size p.hgrid = hmin + (hmax - hmin)*nodeunif(p.nh, 0, 1).^1.5; ymax = max(p.lambdat)*zgrid(end)*egrid(end); lmin = -0.1; lmax = 125; p.lgrid = lmin + (lmax - lmin)*nodeunif(p.nl, 0, 1).^1.5; % Construct grids: svbarh = gridmake(p.agrid, p.ogrid, p.tgrid, p.hgrid, p.zgrid); % grid for expected value of homeowners svbarr = gridmake(p.agrid, p.zgrid); % grid for expected value of renters svh = gridmake(p.agrid, p.ogrid, p.tgrid, p.hgrid, p.zgrid, p.egrid); % grid for value of homeowners prior to making h choice svr = gridmake(p.agrid, p.zgrid, p.egrid); % grid for value of renters prior to making h choice swh = gridmake(p.lgrid, p.ogrid, p.tgrid, p.hgrid, p.zgrid); % grid for W functions swr = gridmake(p.lgrid, p.zgrid); ind2h = kron((1:1:p.no*p.nt*p.nh*p.nz)', ones(p.nl, 1)); % index of all other state-variables to speed up evaluations (Bangladesh) ind2r = kron((1:1: p.nz)', ones(p.nl, 1)); vbarh = zeros(p.na*p.no*p.nt*p.nh*p.nz, p.T + 1); % expected values of homeowners vbarr = zeros(p.na*p.nz, p.T + 1); % expected values of renters vh = zeros(p.na*p.no*p.nt*p.nh*p.nz*p.ne, p.T); % value of homeowners prior to making h choice (envelope over 5 possible options) vr = zeros(p.na*p.nz*p.ne, p.T); % value of renters prior to making h choice (envelope over possible options) wh = zeros(p.nl*p.no*p.nt*p.nh*p.nz, p.T); % value of homeowners after making h choice wr = zeros(p.nl*p.nz, p.T); % value of renters after making h choice ch = zeros(p.nl*p.no*p.nt*p.nh*p.nz, p.T); % consumption homeowners after making h choice cr = zeros(p.nl*p.nz, p.T); % consumption of renters after making h choice aprimeh = zeros(p.nl*p.no*p.nt*p.nh*p.nz, p.T); % liquid savings of homeowners after making h choice aprimer = zeros(p.nl*p.nz, p.T); % liquid savings of renters after making h choice cmaxh = bisect('savings', 1e-13, 1e5, p.lgrid, p, 'h', amin); % c that implies a' = amin cmaxr = bisect('savings', 1e-13, 1e5, p.lgrid, p, 'r', amin); cminh = bisect('savings', 1e-13, 1e5, p.lgrid, p, 'h', amax); % c that implies a' = amax cminr = bisect('savings', 1e-13, 1e5, p.lgrid, p, 'r', amax); cmaxh = repmat(cmaxh, p.no*p.nt*p.nh*p.nz, 1); cmaxr = repmat(cmaxr, p.nz, 1); cminh = repmat(cminh, p.no*p.nt*p.nh*p.nz, 1); cminr = repmat(cminr, p.nz, 1); lh = zeros(p.na*p.no*p.nt*p.nh*p.nz*p.ne, p.T); % liquidity of homeowners after making h choice (envelope over 5 possible options) lr = zeros(p.na*p.nz*p.ne, p.T); % liquidity of renters after making h choice (envelope over possible options) omegaprimeh = zeros(p.na*p.no*p.nt*p.nh*p.nz*p.ne, p.T); omegaprimer = zeros(p.na*p.nz*p.ne, p.T); thetaprimeh = zeros(p.na*p.no*p.nt*p.nh*p.nz*p.ne, p.T); thetaprimer = zeros(p.na*p.nz*p.ne, p.T); hprimeh = zeros(p.na*p.no*p.nt*p.nh*p.nz*p.ne, p.T); hprimer = zeros(p.na*p.nz*p.ne, p.T); dh = zeros(p.na*p.no*p.nt*p.nh*p.nz*p.ne, p.T); dr = zeros(p.na*p.nz*p.ne, p.T); % Terminal value of bequests vbarh(:, p.T + 1) = p.B*(p.wbar + (1 + p.rl)*svbarh(:,1) + (1 - p.Fs - svbarh(:,2).*svbarh(:,3)*(1 + p.rl)).*svbarh(:,4)).^(1 - p.sigma)/(1 - p.sigma); vbarr(:, p.T + 1) = p.B*(p.wbar + (1 + p.rl)*svbarr(:,1)).^(1 - p.sigma)/(1 - p.sigma); for t = p.T : -1 : 1 % tic EVh = griddedInterpolant({p.agrid, (1: 1:p.no*p.nt*p.nh*p.nz)'}, reshape(vbarh(:, t + 1), p.na, p.no*p.nt*p.nh*p.nz), intmeth, 'linear'); EVr = griddedInterpolant({p.agrid, (1: 1: p.nz)'}, reshape(vbarr(:, t + 1), p.na, p.nz), intmeth, 'linear'); % solve consumption-savings choice ch(:, t) = solve_golden('wfunc', cminh, cmaxh, swh, ind2h, EVh, p, 'h'); cr(:, t) = solve_golden('wfunc', cminr, cmaxr, swr, ind2r, EVr, p, 'r'); [~, aprimeh(:,t)] = savings(ch(:, t), swh, p, 'h'); [~, aprimer(:,t)] = savings(cr(:, t), swr, p, 'r'); wh(:, t) = wfunc(ch(:, t), swh, ind2h, EVh, p, 'h'); wr(:, t) = wfunc(cr(:, t), swr, ind2r, EVr, p, 'r'); Whinterp = griddedInterpolant({p.lgrid, (1: 1: p.no*p.nt*p.nh*p.nz)'}, reshape(wh(:, t), p.nl, p.no*p.nt*p.nh*p.nz), intmeth, 'linear'); Wrinterp = griddedInterpolant({p.lgrid, (1: 1: p.nz)'}, reshape(wr(:, t), p.nl, p.nz), intmeth, 'linear'); % Solve discrete choice problem of renters A = svr(:,1); Y = p.lambdat(t)*svr(:,2).*svr(:,3); znow = repmat(kron((1: 1 : p.nz)', ones(p.na, 1)), p.ne, 1); % index of z in (a, z, e) space for renters [lr(:,t), omegaprimer(:,t), thetaprimer(:,t), hprimer(:,t), vr(:,t), dr(:,t)] = solveh(svr, Whinterp, Wrinterp, p, p.thetay(t), 'r', A, Y, znow); % Solve discrete choice problem of housing A = svh(:,1); Y = p.lambdat(t)*svh(:,5).*svh(:,6); znow = repmat(kron((1: 1 : p.nz)', ones(p.na*p.no*p.nt*p.nh, 1)), p.ne, 1); % index of z in (a, omega, theta, h, z, e) space for owners hnow = repmat(kron((1: 1 : p.nh)', ones(p.na*p.no*p.nt, 1)), p.nz*p.ne, 1); % index of h in (a, omega, theta, h, z, e) space for owners tnow = repmat(kron((1: 1 : p.nt)', ones(p.na*p.no, 1)), p.nh*p.nz*p.ne, 1); % index of theta in (a, omega, theta, h, z, e) space for owners [lh(:,t), omegaprimeh(:,t), thetaprimeh(:,t), hprimeh(:,t), vh(:,t), dh(:,t)] = solveh(svh, Whinterp, Wrinterp, p, p.thetay(t), 'h', A, Y, znow, hnow, tnow); % Compute expected value and update vbar for i = 1 : p.ne vbarh(:,t) = vbarh(:,t) + we(i)*kronm({p.na*p.no*p.nt*p.nh, Fzz}, vh((i-1)*p.na*p.no*p.nt*p.nh*p.nz + 1 : i*p.na*p.no*p.nt*p.nh*p.nz, t)); vbarr(:,t) = vbarr(:,t) + we(i)*kronm({p.na, Fzz}, vr((i-1)*p.na*p.nz + 1 : i*p.na*p.nz, t)); end %fprintf('%6i %6.3f\n',[t toc]); end simulate