%RAMSEY_FOC.M
function foc = ramsey_foc;
%foc = ramsey_foc: first-order conditions of the Ramsey problem in ``Optimal,  Simple, and Implementable Monetary and Fiscal Rules,'' by S. Schmitt-Grohe and Martin Uribe (2006)
%(c) Stephanie Schmitt-Grohe and Martin Uribe, May 2006

%Notes
% x_cu=x_t and x_cup=x_{t+1}
% x_fu1=x_{t+1} 
%x_fu1p=x_{t+2}
% x_ba1=x_{t-1} 
% x_ba1p=x_t
% x_ba2=x_{t-2} and x_ba2p=x_{t-1}

%define symbolic variables
allsyms


%Utility function
objective_cu =  ((c_cu * (1-h_cu)^GAMA )^(1-SIGMA) - 1) / (1-SIGMA);


[constraints_cu, constraints_ba1, constraints_fu1] = constraints;

%The Lagrange Multipliers
xi_cu  = [xi1_cu  xi2_cu  xi3_cu  xi4_cu  xi5_cu  xi6_cu  xi7_cu  xi8_cu  xi9_cu  xi10_cu  xi11_cu  xi12_cu  xi13_cu  xi14_cu xi15_cu xi16_cu];


xi_fu1  = [xi1_fu1  xi2_fu1  xi3_fu1  xi4_fu1  xi5_fu1  xi6_fu1  xi7_fu1  xi8_fu1  xi9_fu1  xi10_fu1  xi11_fu1  xi12_fu1  xi13_fu1  xi14_fu1 xi15_fu1 xi16_fu1];

xi_ba1  = [xi1_ba1  xi2_ba1  xi3_ba1  xi4_ba1  xi5_ba1  xi6_ba1  xi7_ba1  xi8_ba1  xi9_ba1  xi10_ba1  xi11_ba1  xi12_ba1  xi13_ba1  xi14_ba1 xi15_ba1 xi16_ba1];


%The Lagrangian
lagrangian = BETTA^(-1) * (xi_ba1 * constraints_ba1) + (objective_cu+xi_cu * constraints_cu) + BETTA * (xi_fu1 * constraints_fu1);


endogvar_cu = [k_cup iv_cu c_cu h_cu la_cu w_cu q_cu u_cu pai_cu mc_cu m_cu ptil_cu x1_cu x2_cu output_cu s_cup r_cu ]; 

endogvar_cu=endogvar_cu(:);

endogvar_ba1p = [k_fu1 iv_ba1p c_ba1p h_ba1p la_ba1p w_ba1p q_ba1p u_ba1p pai_ba1p mc_ba1p m_ba1p ptil_ba1p x1_ba1p x2_ba1p output_ba1p s_fu1 r_ba1p]; 

endogvar_ba1p=endogvar_ba1p(:);

foc = [jacobian(lagrangian,xi_cu) jacobian(lagrangian,endogvar_cu)+jacobian(lagrangian,endogvar_ba1p)];

foc = foc(:);