%primal_form.m
%This program computes symbolic math expressions for the first-order conditions of the Ramsey problem in the model of 
%Schmitt-Grohe, Stephanie and Martin Uribe, 
%`Foreign Demand for Domestic Currency and The Optimal Rate of Inflation'' %Journal of Money, Credit and Banking, forthcoming. 
%
%(c) Stephanie Schmitt-Grohe and Martin Uribe
% Date: January 5,  2011


clear all

syms THETA BETTA A B 
syms c h  v g yf
syms c_b h_b  v_b g_b yf_b
syms c_f h_f  v_f g_f yf_f 
syms  la 

%Satiation level of velocity
vbar = sqrt(B/A);

%Current Period (t)
U = log(c) + THETA * log(1-h);
Uc = diff(U,'c');
Uh = diff(U,'h');
S = A*v + B/v - 2*sqrt(A*B);
Sv = diff(S,'v');
Svv = diff(Sv,'v');
GAMA = 1 + S + v*Sv;
vf = v;


%One period back (t-1) 
U_b  = log(c_b) + THETA * log(1-h_b);
Uc_b = diff(U_b,'c_b');
S_b = A*v_b + B/v_b - 2*sqrt(A*B);
Sv_b = diff(S_b,'v_b');
GAMA_b = 1 + S_b + v_b*Sv_b;
vf_b = v_b;

%One period forward (t+1)
U_f  = log(c_f) + THETA * log(1-h_f);
Uc_f = diff(U_f,'c_f');
Uh_f = diff(U_f,'h_f');
S_f = A*v_f + B/v_f - 2*sqrt(A*B);
Sv_f = diff(S_f,'v_f');
GAMA_f = 1 + S_f + v_f*Sv_f;
vf_f = v_f;


%Feasibility constraint in the current period (t)
h = - (yf / vf -yf_b / vf_b * (1-v_b^2 * Sv_b) * Uc_b / GAMA_b * GAMA  / BETTA / Uc  - (1+S) * c - g);

%Feasibility constraint in t+1
h_f = -( yf_f / vf_f -yf / vf * (1-v^2 * Sv) * Uc / GAMA * GAMA_f / BETTA / Uc_f - (1+S_f) * c_f - g_f);

U = eval(U);
U_f  = eval(U_f);
Uh = eval(Uh);
Uh_f = eval(Uh_f);

%Period-t term of Implementability constraint
implementability = Uc * c + Uh *h;
implementability_f = Uc_f * c_f + Uh_f *h_f;


%Lagrangean
L = U + BETTA * U_f + la * implementability + la * BETTA * implementability_f;

Lc = diff(L,'c');
Lv = diff(L,'v');

% In steady state,  c_b=c=c_f, and so forth
xx = [c_b v_b yf_b g_b c_f v_f yf_f g_f];
yy = [c v yf g c v yf g];

Lcs = subs(Lc,xx,yy,0);
Lvs = subs(Lv,xx,yy,0);