%%%     THIS FILE SOLVES THE RICCATI EQUATION PRESENTED
%%%  IN LECTURE THREE OF RECURSIVE MACROECONOMICS OF MARTIN ELLISON 
%--------------------------------------------------------------------------
%The central bank asigns equal weights on inflation and output deviations from
%the target and a smaller weight on deviations in the interest rate from the
%target. Objective is to minimise the present discounted value of expected
%losses.

%--------------------------------------------------------------------------
%                           DEFINE PARAMETERS
%--------------------------------------------------------------------------

clear;
beta=0.99;
Q=zeros(1,1);
R=zeros(2,2);
A=zeros(2,2);
B=zeros(2,1);
Q(1,1)=0.1;
R(1,1)=1;
R(1,2)=0;
R(2,1)=0;
R(2,2)=1;
A(1,1)=0.75;
A(1,2)=0;
A(2,1)=0;
A(2,2)=0.25;
B(1,1)=-0.5;
B(2,1)=-0.5;

d=1;
D=0;
I=0;
i=0;
P0=-0.000001*eye(2);
format long

%--------------------------------------------------------------------------
%                   ITERATE THE RICCATI EQUATION
%--------------------------------------------------------------------------

while d>0.0000000001
    P1=R+beta*A'*P0*A-(beta*A'*P0*B)*(inv(Q+beta*B'*P0*B))*(beta*B'*P0*A);
    Pd=P1-P0;
    d=max(abs(Pd));
    d=max(d');
    D=[D d];
    P0=P1;
    i=i+1;
    I=[I i];
end
P=P0;

%--------------------------------------------------------------------------
%      CALCULATE THE POLICY FUNCTION COEFFICIENTS AND DISPLAY RESULTS
%--------------------------------------------------------------------------

F=-inv(Q+beta*B'*P*B)*(beta*B'*P*A);
ID=[I(2:length(I))' D(2:length(I))'];
disp('          i                d');
disp(ID);
disp('          SOLUTIONS');
disp('F');
disp(F);
disp('P');
disp(P);