File size: 12,284 Bytes
2a8276b |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 |
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.r1 = x(7);
p.r2 = x(8); % parameters governing shape of rl curve
p.rh = (1 + x(9))^(1/4) - 1; % interest rate
p.rr = x(10);
p.hr = x(11);
% Assigned Parameters
p.nu = 3; % parameter in exponential distribution
p.pidelta = [0.975; 0.025]; % probability of expenditure shocks
p.delta = [0; 0.0625]; % expenditure shock, fraction of home (quarterly so divide by 4)
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; % mortgage interest rate
p.Fs = 0.06; % fixed cost of selling home
p.F1m = 0.005; % proportional cost of refinancing
p.wbar = 1; % 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 = 75; % number of nodes for liquid assets
p.nat = 75; % number of nodes for atilde = (1 + rl)*a - delta*h
p.nl = 75; % number of nodes for liquidity
p.no = 11; % number of nodes for omega (fraction of loan outstanding)
p.nt = 5; % 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);
zgrid = exp(zgrid');
p.zgrid = 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);
egrid = exp(egrid);
p.egrid = egrid;
% Discretize other state variables
amin = -0.4;
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 = 0.25;
tmax = p.thetam; % allow to cover fixed cost
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;
ymin = min(p.lambdat)*zgrid(1)*egrid(1);
ymax = max(p.lambdat)*zgrid(end)*egrid(end);
lmin = -1; % keep it reasonably negative so they know that's a bad state to find yourself in
lmax = 125;
p.lgrid = lmin + (lmax - lmin)*nodeunif(p.nl, 0, 1).^1.5;
atmin = -p.delta(2)*hmax + (1 + p.rl)*amin;
atmax = (1 + p.rh)*amax;
p.atgrid = atmin + (atmax - atmin)*nodeunif(p.nat, 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.atgrid, p.ogrid, p.tgrid, p.hgrid, p.zgrid, p.egrid); % grid for value of homeowners prior to making h choice
svr = gridmake(p.atgrid, 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);
svht = gridmake(p.agrid, p.ogrid, p.tgrid, p.hgrid, p.zgrid, p.egrid); % grid for computing intermediate value function (creier prajit)
svrt = gridmake(p.agrid, p.zgrid, p.egrid);
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));
ind3h = kron((1:1:p.no*p.nt*p.nh*p.nz*p.ne)', ones(p.na, 1)); % index of all other state-variables to speed up evaluations (Bangladesh)
ind3r = kron((1:1: p.nz*p.ne)', ones(p.na, 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.nat*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.nat*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
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);
% Terminal value of bequests
rlh = 1./(1 + exp(-p.r1*(svbarh(:,1) - p.r2)))*(p.rh - p.rl) + p.rl;
rlr = 1./(1 + exp(-p.r1*(svbarr(:,1) - p.r2)))*(p.rh - p.rl) + p.rl;
vbarh(:, p.T + 1) = p.pidelta(1)*p.B*(p.wbar + (1 + rlh).*svbarh(:,1) + (1 - p.Fs - svbarh(:,2).*svbarh(:,3)*(1 + p.rm) - p.delta(1)).*svbarh(:,4)).^(1 - p.sigma)/(1 - p.sigma) + ...
p.pidelta(2)*p.B*(p.wbar + (1 + rlh).*svbarh(:,1) + (1 - p.Fs - svbarh(:,2).*svbarh(:,3)*(1 + p.rm) - p.delta(2)).*svbarh(:,4)).^(1 - p.sigma)/(1 - p.sigma);
vbarr(:, p.T + 1) = p.B*(p.wbar + (1 + rlr).*svbarr(:,1)).^(1 - p.sigma)/(1 - p.sigma);
for t = p.T : -1 : 1
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');
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
At = svr(:,1);
Y = p.lambdat(t)*svr(:,2).*svr(:,3);
znow = repmat(kron((1: 1 : p.nz)', ones(p.nat, 1)), p.ne, 1); % index of z in (a, z, e) space for renters
[~, ~, ~, ~, vr(:,t)] = solveh(svr, Whinterp, Wrinterp, p, p.thetay(t), 'r', At, Y, znow);
% Solve discrete choice problem of housing
At = svh(:,1);
Y = p.lambdat(t)*svh(:,5).*svh(:,6);
znow = repmat(kron((1: 1 : p.nz)', ones(p.nat*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.nat*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.nat*p.no, 1)), p.nh*p.nz*p.ne, 1); % index of theta in (a, omega, theta, h, z, e) space for owners
[~, ~, ~, ~, vh(:,t)] = solveh(svh, Whinterp, Wrinterp, p, p.thetay(t), 'h', At, Y, znow, hnow, tnow);
% We need to interpolate to calculate the expected value before the delta shocks are realized, but after the z,e shocks are realized
Vhinterp = griddedInterpolant({p.atgrid, (1: 1:p.no*p.nt*p.nh*p.nz*p.ne)'}, reshape(vh(:, t), p.nat, p.no*p.nt*p.nh*p.nz*p.ne), intmeth, 'linear');
Vrinterp = griddedInterpolant({p.atgrid, (1: 1: p.nz*p.ne)'}, reshape(vr(:, t), p.nat, p.nz*p.ne), intmeth, 'linear');
% Compute expected value and update vbar
% 1. Step 1: integrate delta shocks by interpolate value of home and rent (which are functions of atilde)
vhtemp = p.pidelta(1)*Vhinterp((1 + interest(svht(:,1), p)).*svht(:,1) - p.delta(1)*svht(:, 4), ind3h) + p.pidelta(2)*Vhinterp((1 + interest(svht(:,1), p)).*svht(:,1) - p.delta(2)*svht(:, 4), ind3h);
vrtemp = Vrinterp((1 + interest(svrt(:,1), p)).*svrt(:,1), ind3r);
for i = 1 : p.ne
vbarh(:,t) = vbarh(:,t) + we(i)*kronm({p.na*p.no*p.nt*p.nh, Fzz}, vhtemp((i - 1)*p.na*p.no*p.nt*p.nh*p.nz + 1 : i*p.na*p.no*p.nt*p.nh*p.nz));
vbarr(:,t) = vbarr(:,t) + we(i)*kronm({p.na, Fzz}, vrtemp((i - 1)*p.na*p.nz + 1 : i*p.na*p.nz));
end
end
simulate
|