%FIGURE3.M
function figure3 
%This program produces figure 3 of the paper ``Deep Habits,'' by Morten Ravn, Stephanie Schmitt-Grohe, and Martin Uribe (Review of Economic Studies)
%
%Calls: gx_hx.m,  and ir.m
% 
%(c) Stephanie Schmitt-Grohe and Martin Uribe
%
%Date December 23, 2003


%SHOCKS
%shock=1 preference
%shock=2 government
%shock=3 technology

for shock=1:3

%Compute analytical derivatives of equilibrium conditions
[fx,fxp,fy,fyp,f] = dh;

%Numerical evaluation

%Assign values to parameters and steady-state variables

[RHO, THETA_S,THETA_D,THETA_G,GBAR,SIGMA,CHI,GAMA,BETTA,DELTA,ALFA,ETA,RHOZ,RHOG,RHOV,nu,nug,w,h,c,s,u,k,ivv,markup,a,g,v,stockg,output,xx,nup,nugp,wp,hp,cp,sp,up,kp,ivvp,markupp,ap,gp,vp,stockgp,outputp,xxp,NU,NUG,W,H,C,S,U,K,IVV,MARKUP,A,G,V,STOCKG,OUTPUT,XX,PHI,EHW]=dh_ss(-0.1,-0.1,-0.1);

approx = 1; %Order of approximation desired

%Compute numerical derivatives of f
num_eval

%Compute a first-order accurate approximation to the policy functions
[gx,hx] = gx_hx(nfy,nfx,nfyp,nfxp); 

T=21;  %Length of impulse response function

%Compute impulse responses
IR = ir(gx,hx,[0 0 0 (shock==1)*XX (shock==2)*1 (shock==3)*1],T);
Ir(:,:,shock) = IR;

t=(0:size(IR,1)-1)';

u= squeeze(Ir(:,7,:));

c = squeeze(Ir(:,5,:));
h = squeeze(Ir(:,4,:));
w = squeeze(Ir(:,3,:));
ivv = squeeze(Ir(:,6,:)); 
output = squeeze(Ir(:,2,:));
markup = squeeze(Ir(:,1,:));
nu = squeeze(Ir(:,8,:));
nug = squeeze(Ir(:,9,:));
s = squeeze(Ir(:,12,:));
g = squeeze(Ir(:,15,:));
a = squeeze(Ir(:,16,:));

subplot(3,4,1+(shock-1)*4)
plot(t,markup(:,shock),'-')
title('Markup')

subplot(3,4,2+(shock-1)*4)
plot(t,output(:,shock))
title('Output')

subplot(3,4,3+(shock-1)*4)
plot(t,w(:,shock))
title('Wage')

subplot(3,4,4+(shock-1)*4)
plot(t,c(:,shock))
title('Consumption')

end
shg

%DH.M
function [fx,fxp,fy,fyp,f] = dh
%This program computes a log-linear approximation to the  function f for an economy with catching up with the Joneses good by good. The model is described in ``Deep Habits''
%The function f defines  the DSGE model (a p denotes next-period variables) : 
%  E_t f(yp,y,xp,x) =0. 
%
%Output: Analytical first derivative of f
%
%Calls: anal_deriv.m 
%
%(c) Stephanie Schmitt-Grohe and Martin Uribe
%
%Date December 23, 2003

%Define parameters
syms RHO THETA_S THETA_D THETA_G SIGMA CHI GAMA BETTA DELTA ALFA ETA RHOZ RHOG RHOV  GBAR PHI  

%Define variables 

syms xx xxp s sp c cp h hp w wp u up k kp ivv ivvp nu nup nug nugp markup markupp a ap g gp stockg stockgp output outputp v vp  

%Give functional form for  production, and utility functions

%Utility function
util = ((xx-v)^(1-SIGMA) -1) / (1-SIGMA) + GAMA * ((1-h)^(1-CHI)-1) / (1-CHI);
utilp = ((xxp-vp)^(1-SIGMA) -1) / (1-SIGMA) + GAMA * ((1-hp)^(1-CHI)-1) / (1-CHI);

pf = a*k^(ALFA)*h^(1-ALFA);
pfp = ap * kp^(ALFA)*hp^(1-ALFA);

%Write equations (ei, i=1:n)

e1 = -xx + c/s^THETA_S; %Habit-adjusted consumption

%Labor supply
e2 = w/s^THETA_S + diff(util,'h') / diff(util,'xx');

%Euler equation
e3 = - diff(util,'xx') / s^THETA_S + BETTA * diff(utilp,'xxp') / sp^THETA_S * (1-DELTA+up);

%Resource constraint
e4 = -(pf - PHI) + c + ivv + g;

%Evoluiton of capital
e5 = -kp + (1-DELTA) * k + ivv;

%Markup
e6 = -markup + diff(pf,'h') / w;

%nu
e7 = - (c+g+ivv) / ETA + nu * c + nug * g + ivv * (1-1/markup);

%Firm's FOC w.r.t. s
e8 = - (1-nu - 1/markup ) / (RHO-1) + BETTA * diff(utilp,'xxp') / sp^THETA_S / diff(util,'xx') * s^THETA_S * (THETA_D * (1-ETA) * nup * cp / sp + RHO * (1-nup-1/markupp) / (RHO-1));


%Firm's FOC w.r.t. stockg
e9 =  - (1-nug - 1/markup ) / (RHO-1) + BETTA * diff(utilp,'xxp') / sp^THETA_S / diff(util,'xx') * s^THETA_S * (THETA_G * (1-ETA) * nugp * gp / stockgp + RHO * (1-nugp-1/markupp) / (RHO-1));

%MRT = w/u
e10 = -w / u + diff(pf,'h') / diff(pf,'k');

%Evolution of habit stock
e11 = - sp + RHO * s + (1-RHO) * c;
%Evolution of gov't habit stock
e12 = -stockgp + RHO * stockg + (1-RHO) * g;

%Technology shock
e13 = log(ap) - RHOZ * log(a);

%Government Purchases shock
e14 = log(gp/GBAR) - RHOG * log(g/GBAR);

%Evolution of preference shock
e15 = - vp + RHOV * v;

%Definition of output
e16 = -output + pf - PHI;


%Create function f
f = [e1;e2;e3;e4;e5;e6;e7;e8;e9;e10;e11;e12;e13;e14;e15;e16];


% Define the vector of controls in period t, y, and t+1, yp, and the vector of states in period t, x, and t+1, xp

x = [stockg s k g a];
xp = [stockgp sp kp gp ap];

y = [markup output w h c ivv u nu nug xx];

yp = [markupp outputp wp hp cp ivvp up nup nugp xxp];


%Make f a function of the logarithm of the state and control vector
f = subs(f, [x,y,xp,yp], exp([x,y,xp,yp]));


x = [stockg s k v g a];
xp = [stockgp sp kp vp gp ap];

approx = 1; %Order of approximation desired
%Compute analytical derivatives of f
[fx,fxp,fy,fyp]=anal_deriv(f,x,y,xp,yp,approx);

%DH_SS.M
function [RHO, THETA_S,THETA_D,THETA_G,GBAR,SIGMA,CHI,GAMA,BETTA,DELTA,ALFA,ETA,RHOZ,RHOG,RHOV,nu,nug,w,h,c,s,u,k,ivv,markup,a,g,v,stockg,output,xx,nup,nugp,wp,hp,cp,sp,up,kp,ivvp,markupp,ap,gp,vp,stockgp,outputp,xxp,NU,NUG,W,H,C,S,U,K,IVV,MARKUP,A,G,V,STOCKG,OUTPUT,XX,PHI,EHW,m]=dh_ss(theta_s,theta_d,theta_g);
%This program produces the deep structural parameters and computes the steady state of the relative deep habit model described in  ``Deep Habits.'' by Morten Ravn, Stephanie Schmitt-Grohe, and Martin Uribe 
%(c) Stephanie Schmitt-Grohe and Martin Uribe, December 23, 2003


%Calibrated parameters

THETA_S=theta_s;%superficial habit parameter 

THETA_D = theta_d;%deep habit parameter

THETA_G = theta_g; %govt deep habit parameter

SIGMA = 2; %intertemporal elasticity of substitution

EHW = 1.3; %Frisch elasticity of labor suppy

H = 0.2; %hours (Prescott, 1986)

ETA = 5.3; %Long-run Price elasticity of demand for a particular variety. Taken from the econometric estimates shown in the paper ``Deep Habits''


R = 1.04^(1/4); %Gross quarterly real interest rate (Rotemberg & Woodford, JPE, 1991)

SH = 0.75;%Labor share (Rotemberg & Woodford, JPE, 1991)

SG = 0.12; % Steady State Share of Government Purchases (Rotemberg & Woodford, JPE, 1991)

RHOZ = 0.9; %Persistence of technology shock z_t

RHOG = 0.9; %Persistence of Government purchases shock

RHOV = 0.9; %Persistence of preference shock


RHO = 0.85; %1-RHO is the weight on current consumption in the evolution of the stock of habit. S_t = RHO s_{t-1} + (1-RHO) C_{t}. Taken from the econometric estimates presented in ``Deep Habits.''

SC = 0.7; %Consumption share (Rotemberg & Woodford, JPE, 1991)
 
%Implied parameters

BETTA = 1/R; %Subjective discount factor

SI = 1-SC-SG; %INVESTMENT SHARE

m = SC * (1-BETTA * RHO) / (RHO-1) * (BETTA*(1-ETA) * THETA_D + (1-BETTA*RHO) / (RHO-1))^(-1) + SG * (1-BETTA*RHO) / (RHO-1) * (BETTA*THETA_G*(1-ETA)+(1-BETTA*RHO)/(RHO-1))^(-1) + SI;

MARKUP = 1/ (1-1/ETA/ m); %Markup


ALFA = 1 - SH; %capital elasticity of output

SK = SH * ALFA/(1-ALFA); %share of capital

DELTA = SI  * (1/BETTA-1)/ (SK-SI);


NU = (1-1/MARKUP) * (1-BETTA*RHO) / (RHO-1) / (BETTA*THETA_D*(1-ETA) + (1-BETTA*RHO) / (RHO-1));

NUG = (1-1/MARKUP) * (1-BETTA*RHO) / (RHO-1) / (BETTA*THETA_G*(1-ETA) + (1-BETTA*RHO) / (RHO-1)); %Lagrange multiplier on aggregate demand faced by monopolist i


U = 1/BETTA-1+DELTA; %rental rate of capital

K = (SI/DELTA/MARKUP)^(1/(1-ALFA))*H; %Steady state capital stock

OUTPUT = K^ALFA * H^(1-ALFA) / MARKUP; %output


W = U * (1-ALFA)/ALFA *K/H; %Wage rate

IVV = DELTA * K; %Investment

PHI = K^ALFA * H^(1-ALFA) - OUTPUT; %Fixed cost

C = SC * OUTPUT; %consumption

S = C; %Stock of habit

XX = C^(1-THETA_S); %Habit-adjusted consumption


CHI = (1-H)/H / EHW;
GAMA = W * (1-H)^CHI / C^(SIGMA*(1-THETA_S)+THETA_S) ;

V =0; %Steady-State value of preference shock


A = 1; %steady-state value of technology shock

G = SG * OUTPUT; %Government consumption

GBAR=G; %Unconditional mean of gov't consumption

STOCKG = G;

%Log values
nu=log(NU);
nug=log(NUG);
w=log(W);
h=log(H);
c=log(C);
s=log(S);
u=log(U);
k=log(K);
ivv=log(IVV);
markup=log(MARKUP);
a=log(A);
g = log(G);
stockg=g;
output = log(OUTPUT);
xx = log(XX);
v=0;
%Future values of logged variables
nup=log(NU);
nugp=log(NUG);
wp=log(W);
hp=log(H);
cp=log(C);
sp=log(S);
up=log(U);
kp=log(K);
ivvp=log(IVV);
markupp=log(MARKUP);
ap=log(A);
gp = log(G);
stockgp=g;
outputp = log(OUTPUT);
xxp = log(XX);
vp=v;