% Program mdrendowment.m

% Chapter 1.  The basic model

% One good, endownment model with perfect capital mobility

% This program is part of the book "Open Economy Macroeconomics in
% Developing Countries" (forthcoming MIT Press) by Carlos A. Vegh. Use of
% this program or modifications thereof should be acknowledged by citing
% this manuscript.  This program is a modification of the original
% King, Plosser, and Rebelo's (1988) programs. I am grateful to Guillermo
% Vuletin for his help in writing this program.  

% Revised: June 2007

% This program requires as the following outputs of program dynendowment.m

% nc,ns,ne,nf
% Mcc,Mcs,Mce
% FVc,FVke,FVl
% MU,PS21,PS22
% Rke,Qke,
% KTL,KEC
% SP1,SP2

% Housekeeping (matrices needed only for pfe.m)

clear W;
clear R;
clear Q;


% COMPUTATION OF DECISION RULES


% This program computes the Markovian Decision Rules (mdr) for the 
% specified linear dynamic model.  The structure of forcing variables 
% can be altered without rerunning program #1 (dyn.m).

% Let the exogenous variables be generated by a (ne x ne) vector
% AR(1) process. Let the transition matrix be RHO.


% Transition Matrix of Forcing Variables

%     Z    G    PI
rr=input('Serial Correlation of shocks   ');

RHO= eye(ne,ne)*rr;




% The shadow price solution requires the evaluation of a (matrix)
% discounted forward sum.  The strategy employed in the following
% set of computations is to exploit the standard forecasting formulas
% for scalar discounted sums (generalizations along Hansen-Sargent
% lines for ARMA processes (if desirable) should be relatively direct.)

% The composite expression that summarizes the exogenous influences
% in (A48) is

FL = SP1*RHO + SP2;

% This is an ns x ne matrix.  Partion this matrix into ns row vectors.
% Then, there is a individual difference equation for each of the
% transformed shadow prices equation, which can be solved in
% the forward direction, with a simple application of the forecasting
% formula for the relevant linear combination of the discounted sum
% of exogenous variables. This is accomplished in the following do-loop.

IRHO = eye(RHO);

for i=1:ncs;
    q = FL(i,1:ne);
    mu2i = 1/MU2(i,i);
    dsum =inv((IRHO-mu2i*RHO));
    LE(i,1:ne) = q*dsum;
end
LE;

clear mu2i; clear q; clear dsum; clear i;

% The matrix LE links the transformed shadow price to the exogenous
% (forcing) state variables. 

% THE CAPITAL DECISION RULE

% To derive the decision rule for capital, we need to combine the
% influences of the transformed shadow price with the "direct"
% influence of exogenous variables, as in (A56).  Call this KEC,
% where the memnonic is that this is the combined (deicison rule) 
% impact of the exogenous variables on capital

KEC = Rke*RHO + Qke + KTL*LE;

% Thus, the decision rule for capital is given by the coefficients
% KLK and KEC.  The decision rule for the untransformed shadow price
% is given by the coefficients ULE and ULK, where

ULE = PS22\LE;
ULK = -PS22\PS21;

% according to the formula (A55b).


% SYSTEM DYNAMICS

% The system dynamics are then generated by the system comprised
% of states (ns elements) and exogenous variables (ne
% elements).  In matrix form, these are governed by the matrix M
% built up as follows:

% State Transition Equation (Mke)

Mke =    [KLK             KEC
	  zeros(ne,ns)    RHO ];


% Note that this matrix incorporates the restriction that some dynamics are
% exogenous, by means of zero restrictions in M.


% INCORPORATION OF SHADOW PRICE, CONTROLS AND OTHER FLOWS


% First we incorporate in the flow matrices defined in dyn1.m the
% investment, trade balance and current account variables.  To that end we
% first update the number of flow variables


% The shadow price is linked to the state by the ns x (ns+ne) matrix

Lke =   [ULK   ULE];


%March 2007 comment: in this case, Lke is a 1x2 matrix, where the 1 is for
%the only co-state (lambda) and the 2 is for the state (a) + the exogenous variable (y). 


% The solutions for the controls can be obtained by a simple application
% of the optimal control equations to the optimal state equations. The
% (nc x ns+ne) matrices linking the evolution of the controls to the state and
% exogenous forcing variables (uncontrolled states) are calculated as follows.

% The matrix linking optimal controls to states and co-states is

Z=Mcc\Mcs;

% This can be paritioned to separate responses to states and co-states

MOck = Z(1:nc,1:ns);
MOcl = Z(1:nc,ns+1:ns+ncs);


% The matrix linking optimal controls to forcing variables is 

MOce = Mcc\Mce;

clear Z; clear Mcc; clear Mcs; clear Mce;

% These expressions can be combined with the structure for the optimal shadow
% price vector, given above, to obtain a link between optimal controls and the 
% optimal state vector.

MOcke = [MOck+MOcl*ULK MOce+MOcl*ULE];

%March 2007: in this case, this is a 1x2 (1 for the only control and 2 for
%state + exogenous. 

% The additional matrices derived in program #1 link (nf) flow variables
% to controls, states, costates and exogneous forcing variables.
% The flow variables are consequently linked to the k,e vector by:

Fke =[FVc*MOcke+FVke+FVl*Lke];

clear MOce; clear MOcl;

% RATES OF RETURN

% It is sometimes valuable to study commodity rates of return on the capital 
% goods of the economy.  The direct method of doing this is to analyze the
% rate of change of shadow prices.  From above, we know that 
% the shadow price is linked to the state by the ns x (ns+ne) matrix
% Lke =   [ULK   ULE] and that the capital stock and exogenous forcing 
% variables evolve according to
%
%    Mke =    [KLK             KEC
%              zeros(ne,ns)    RHO ];
%
% Thus, the conditional expectation of next period's shadow price is 
% given by multiplying [ULK*KLK (ULK*KEC+ULE*RHO)] and the state vector.
% Thus, using the fact that the rate of return is equal to minus the rate
% of change of price, it follows that the following matrix determines
% movements in expected rates of return.

RRke = [ULK-ULK*KLK ULE-(ULK*KEC+ULE*RHO)];


% Thus, the stacked vector of shadow prices, controls, other flows and
% returns is linked to the state vector by the (ns+nc+nf+ns x ns+ne) matrix

% ' H Matrix Linking Costate,Control,'
% ' Other Flows and Returns to State Vector'


H = [Lke
     MOcke
     Fke
     RRke];
 
     
% Eliminate spurious complex numbers in the output

if max(imag(H))<10e-10
	H=real(H);
end
if max(imag(Mke))<10e-10
	Mke=real(Mke);
end

% disp('Variables in H are ordered as follows')
% disp('lambda, c, TB')


clear Lke; clear MOcke; clear Fke; clear RRke;
clear Rke; clear Qke; 
clear SP1; clear SP2;
clear PS21; clear PS22; clear MU2
clear FL; clear KTL; clear ULK; clear ULE;
clear FVc; clear FVke; clear FVl;
clear MOck; clear LE; clear FL;
clear RHO; clear IRHO;
clear KLK; clear KEC;