@ DIMENSION OF CONTROL SPACE (NC), STATE AND COSTATE VECTORS (NS),
EXOGENEOUS STATE VECTOR (NN) @
@ ORDERING OF VARIABLES:
C: C(T) N(T)
S: K(T) LAMBDA(T)
E: A(T) @
NAMEC="C"|"N";
NAMEK="K";
NAMEL="LAMBDA";
NAMEE="A";
NC=2; NS=1; NN=1;
@ ===================================================================
@
@
ECONOMIC PARAMETER VALUES
@
@ ===================================================================
@
@ FACTOR SHARES (SK,SN) AND THE ELASTICITY OF SUBSTITUTION (ZETAKN).
WE USE COBB-DOUGLAS SPECIFICATION @
SN=0.58; SK=1-SN;
ZETAKN=1.00;
XIKK=-SN*ZETAKN; XIKN= SN*ZETAKN;
XINN=-SK*ZETAKN; XINK= SK*ZETAKN;
@ GROWTH RATE OF TECHNICAL CHANGE AND DEPRECIATION @
GAMMAX=1.004; DELTAK=0.025;
@ DISCOUNT FACTOR AND REAL INTEREST RATE @
RR=0.065/4;
BSTAR=GAMMAX/(1+RR);
@ STEADY STATE SHARE OF TIME TO MARKET ACTIVITIES @
NBAR=0.200;
@ ====================================================================
@
@
STEADY STATE CALCULATIONS
@
@ ====================================================================
@
@ - THE TRANSFORMED ECONOMY @
@ THE RATIO PHI AND CAP-OUTPUT RATIO ARE @
PHI=GAMMAX/(GAMMAX-(1-DELTAK));
KYRATIO=(BSTAR*SK)/(GAMMAX-BSTAR*(1-DELTAK));
@ STEADY STATE CONSUMPTION AND INVESTMENT SHARES @
SI=KYRATIO*GAMMAX/PHI; SC=1-SI;
ABAR=10;
YSS=(ABAR^(1/SN))*(KYRATIO^(SK/SN))*NBAR;
CSS=SC*YSS;
@ PARAMETERS OF PREFERENCES
SIGMA IS THE ABSOULTE VALUE OF THE ELASTICITY OF MARGINAL UTILITY
OF CONSUMPTION @
SIGMA=1;
@ THETA WE DETERMINE SO THAT IT IS CONSISTENT WITH THE STEADY
STATE NUMBER OF HOURS WORKED @
THETA=((SC*NBAR)/(SN*(1-NBAR)))/(1+(SC*NBAR)/(SN*(1-NBAR)));
@ FROM WHICH WE CAN GET THE ELASTICITIES OF PREFERENCES @
@ WE HAVE TWO MAIN POSSIBILITIES:
SIGMA=1 - ADDITIVELY
SEP.
SIGMA><1 - MULTIPLICATIVELY
SEP. @
XICC=THETA*(1-SIGMA)-1; XILC=THETA*(1-SIGMA);
@ THE OTHER TWO ELASTICITIES ARE FREE SO LONG AS XICL=0 IF SIGMA=1 @
XICL=(1-THETA)*(1-SIGMA); XILL=(1-THETA)*(1-SIGMA)-1;
@ STEADY STATE ELASTICITIES OF MARGINAL PRODUCT OF CAPITAL @
ETAA=(GAMMAX-BSTAR*(1-DELTAK))/GAMMAX;
ETAK=-SN*ETAA;
ETAN=SN*ETAA;
ETAI=BSTAR*DELTAK*SI*YSS;
@ ============================================================= @
@
BASIC SYSTEM MATRICES
@
@ ============================================================= @
@ MATRICES IN CONTROL SYSTEM @
@ THE CONTROLS ARE: C(t) N(t) @
MCC=ZEROS(NC,NC);
MCC[1,1]=XICC; MCC[1,2]=-XICL*NBAR/(1-NBAR);
MCC[2,1]=XILC; MCC[2,2]=-XILL*NBAR/(1-NBAR)-XINN;
MCS=ZEROS(NC,NS*2);
MCS[1,1]=0; MCS[1,2]=1;
MCS[2,1]=XINK; MCS[2,2]=1;
MCE=ZEROS(NC,NN);
MCE[1,1]=0;
MCE[2,1]=1;
@ MATRICES IN STATE EQUATIONS @
MSS0=ZEROS(2*NS,2*NS);
MSS0[1,1]=ETAK; MSS0[1,2]=1;
MSS0[2,1]=SI*PHI; MSS0[2,2]=0;
MSS1=ZEROS(2*NS,2*NS);
MSS1[1,1]=0;
MSS1[1,2]=-1;
MSS1[2,1]=-SK-SI*(PHI-1); MSS1[2,2]=0;
MSC0=ZEROS(2*NS,NC);
MSC0[1,1]=0; MSC0[1,2]=-ETAN;
MSC0[2,1]=0; MSC0[2,2]=0;
MSC1=ZEROS(2*NS,NC);
MSC1[1,1]=0; MSC1[1,2]=0;
MSC1[2,1]=-SC; MSC1[2,2]=SN;
MSE0=ZEROS(2*NS,NN);
MSE0[1,1]=-ETAA;
MSE0[2,1]=0;
MSE1=ZEROS(2*NS,NN);
MSE1[1,1]=0;
MSE1[2,1]=1;
@ =============================================================== @
@ RELATIONS LINKING EXTRA FLOW VARIABLES (OUTPUT, PRODUCTIVITY
AND INVESTMENT) TO
FUNDAMENTAL CONTROLS AND STATES @
@ =============================================================== @
@ THE ORDER OF VARIABLES IS:
Y(T) AP(T) I(T) @
NAMEXC="Y"|"AP"|"I";
NXF=3;
@ 1. EXTRA FLOWS ON CONTROLS [C N]' @
FVC=ZEROS(3,NC);
FVC[1,1]=0; FVC[1,2]=SN;
FVC[2,1]=0; FVC[2,2]=SN-1;
FVC[3,1]=-SC/SI; FVC[3,2]=SN/SI;
@ 2. EXTRA FLOWS ON STATES AND SHOCKS [K A]' @
FVKE=ZEROS(3,NS+NN);
FVKE[1,1]=SK; FVKE[1,2]=1;
FVKE[2,1]=SK; FVKE[2,2]=1;
FVKE[3,1]=SK/SI; FVKE[3,2]=1/SI;
@ EXTRA FLOWS ON CO-STATES [LAMBDA] @
FVL=ZEROS(3,NS);
@ =============================================================== @
@ FUNDAMENTAL STATE-COSTATE
DIFFERENCE EQUATION
@
@ =============================================================== @
MSss0 = MSS0 - MSC0*(INV(MCC))*MCS;
MSss1 = MSS1 - MSC1*(INV(MCC))*MCS;
MSse0 = MSE0 + MSC0*(INV(MCC))*MCE;
MSse1 = MSE1 + MSC1*(INV(MCC))*MCE;
@ THE FUNDAMENTAL DIFFERENCE EQUATION IS PUT IN NORMAL FORM @
W = -(INV(MSss0))*MSss1;
R = (INV(MSss0))*MSse0;
Q = (INV(MSss0))*MSse1;
@ =============================================================== @
@ EIGENVECTOR-EIGENVALUE DECOMPOSITION OF STATE TRANSITION MATRIX @
@ =============================================================== @
@ FIRST WE FIND THE REAL PARTS OF THE EIGENVALUES (X1)
AND EIGENVECTORS (X3) @
{X1,X2,X3,X4}=EIGRG2(W);
@ SECONDLY WE FIND THE INDICATOR (IND1) OF THE ORDER OF THE MAXIMUM
ABSOLUTE EIGENVALUES (AMU) @
AMU=ABS(X1);
IND1=SORTIND(AMU);
@ THIRDLY WE ORDER THE COLUMNS OF THE EIGENVECTORS (X3) BY THIS
INDICATOR RESULTING IN P @
P=ZEROS(2*NS,2*NS);
I=1;
DO UNTIL I>2*NS;
P[1:2*NS,I]=X3[1:2*NS,IND1[I,1]];
I=I+1;
ENDO;
@ FINALLY WE FORM A DIAGONAL MATRIX (MU) IN WHICH THE DIAGONAL HAVE
THE EIGENVALUES IN ASCENDING ABSOLUTE VALUE, I.E. WE USE THE
INDICATOR FUNCTION IND1 AGAIN @
MU=ZEROS(2*NS,2*NS);
I=1;
DO UNTIL I>2*NS;
MU[I,I]=X1[IND1[I,1],1];
I=I+1;
ENDO;
@ WE NOW HAVE P AND MU FOR WHICH WE KNOW THAT P*MU*P^-1=W (ALSO
X3*DIAG(X1)*X3=W) @
@ =============================================================== @
@
PARTITIONING THE MATRICES
@
@ =============================================================== @
MU1=MU[1:NS,1:NS];
MU2=MU[NS+1:2*NS,NS+1:2*NS];
P11=P[1:NS,1:NS];
P12=P[1:NS,NS+1:2*NS];
P21=P[NS+1:2*NS,1:NS];
P22=P[NS+1:2*NS,NS+1:2*NS];
PS=INV(P);
PS11=PS[1:NS,1:NS];
PS12=PS[1:NS,NS+1:2*NS];
PS21=PS[NS+1:2*NS,1:NS];
PS22=PS[NS+1:2*NS,NS+1:2*NS];
RKE=R[1:NS,1:NN];
RLE=R[NS+1:2*NS,1:NN];
QKE=Q[1:NS,1:NN];
QLE=Q[NS+1:2*NS,1:NN];
@ =============================================================== @
@
COMPOSITE EXPRESSIONS
@
@ =============================================================== @
SP1=-(INV(MU2))*(PS21*RKE+PS22*RLE);
SP2=-(INV(MU2))*(PS21*QKE+PS22*QLE);
KLK=P11*MU1*(INV(P11));
KTL=(P11*MU1*PS12+P12*MU2*PS22)*(INV(PS22));
@ =============================================================== @
@
COMPUTATION OF STEADY STATE
@
@ =============================================================== @
@ THE ORDERING OF VARIABLES (DESCRIBED IN MDR.X1) IS:
K A L C N Y AP C @
YSS=(ABAR^(1/SN))*(KYRATIO^(SK/SN))*NBAR;
CSS=SC*YSS;
ISS=SI*YSS;
KSS=ISS/(GAMMAX-(1-DELTAK));
APSS=YSS/NBAR;
ASS=ABAR;
NSS=NBAR;
LSS=1; @ THIS VALUE IS WRONG BUT WE ARE NOT INTERESTED
IN
THE EVOLUTION
OF SHADOW PRICES @
@ STEADY STATE VALUES IN SSVAL @
SSVAL=KSS|ASS|LSS|CSS|NSS|YSS|APSS|ISS;
NAME=NAMEK|NAMEE|NAMEL|NAMEC|NAMEXC;