I am trying to simulate a model and I get the following error message:
There are 34 eigenvalue(s) larger than 1 in modulus
for 34 forward-looking variable(s)
The rank condition is verified.
??? Error using ==> print_info
Blanchard Kahn conditions are not satisfied: indeterminacy due to rank failure
What does that mean? How does this case (case 5 in print_info) differ from case 4?
Thanks
print_info case 4: indeterminacy means that you have less eigenvalues larger than one in modulus as you have forward-looking variables.
case 5 means that you have the right number of eignevalues larger than one in modulus, but that the matrix that should be inverted to find the solution isn’t full rank.
Now, what is surprising is that check doesn’t seem to detect the rank failure.
Could you please post your example or mail it to me?
Thanks
Michel
Good Morning Mr Juillard .In fact I have seen many messages concerning the blanchard and khan conditions and I try to change the parametrzation and the timing convention but I could not resolve the problem .Also model_diagnostics(M_,options_,oo_) did not give me any result.Could you find what the problem is please? If you find suffisiant time for responding me.
Thank you
Selima
this is the mode file
//----------------------------------------------------------------
// 1. Endogenous variables
//----------------------------------------------------------------
var
EAUS_RER EA_B EA_BF EA_BY EA_C EA_CI EA_CJ EA_CY EA_D EA_FH EA_FI
EA_FJ EA_FX EA_G EA_GAMMAB EA_GAMMAI EA_GAMMAIDER EA_GAMMAIMC EA_GAMMAIMCDAG
EA_GAMMAIMI EA_GAMMAIMIDAG EA_GAMMAU EA_GAMMAUDER EA_GAMMAV EA_GAMMAVI
EA_GAMMAVIDER EA_GAMMAVJ EA_GAMMAVJDER EA_GH EA_GI EA_GJ EA_GX EA_GY EA_H
EA_HC EA_HI EA_I EA_II EA_IM EA_IMC EA_IMCY EA_IMI EA_IMIY EA_IMY EA_IY EA_K
EA_KD EA_KI EA_LAMBDAI EA_LAMBDAJ EA_M EA_MC EA_MI EA_MJ EA_ND EA_NDI EA_NDJ
EA_NI EA_NJ EA_PH EA_PHTILDE EA_PI EA_PIC EA_PIC4 EA_PIH EA_PIM
EA_PY EA_Q EA_QC EA_QI EA_R EA_RER EA_RK EA_RP EA_RR
EA_RRSTAR EA_SH EA_SI EA_SJ EA_SX EA_T EA_TAUC EA_TAUD EA_TAUK EA_TAUN
EA_TAUWF EA_TAUWH EA_TB EA_TI EA_TJ EA_TOT EA_TR EA_TRI EA_TRJ EA_TRY EA_TY
EA_U EA_UTILI EA_UTILJ EA_VI EA_VJ EA_W EA_WI EA_WITILDE EA_WJ EA_WJTILDE EA_Y
EA_YGAP EA_YGROWTH EA_YS EA_YSHARE EA_Z USEA_RER US_B US_BF US_BY US_C US_CI
US_CJ US_CY US_D US_FH US_FI US_FJ US_FX US_G US_GAMMAI
US_GAMMAIDER US_GAMMAIMC US_GAMMAIMCDAG US_GAMMAIMI US_GAMMAIMIDAG US_GAMMAU
US_GAMMAUDER US_GAMMAV US_GAMMAVI US_GAMMAVIDER US_GAMMAVJ US_GAMMAVJDER US_GH
US_GI US_GJ US_GX US_GY US_H US_HC US_HI US_I US_II US_IM US_IMC US_IMCY
US_IMI US_IMIY US_IMY US_IY US_K US_KD US_KI US_LAMBDAI US_LAMBDAJ US_M US_MC
US_MI US_MJ US_ND US_NDI US_NDJ US_NI US_NJ US_PH US_PHTILDE US_PI US_PIC
US_PIC4 US_PIH US_PIIM US_PIM US_PIMTILDE US_PY US_Q US_QC US_QI US_R
US_RK US_RR US_RRSTAR US_SH US_SI US_SJ US_SX US_T US_TAUC US_TAUD US_TAUK
US_TAUN US_TAUWF US_TAUWH US_TI US_TJ US_TR US_TRI US_TRJ US_TRY US_TY US_U
US_UTILI US_UTILJ US_VI US_VJ US_W US_WI US_WITILDE US_WJ US_WJTILDE US_Y
US_YGAP US_YGROWTH US_YS US_YSHARE US_Z EA_PIMTILDE EA_PIIM EA_RERDEP
;
//----------------------------------------------------------------
// 2. Exogenous variables
//----------------------------------------------------------------
varexo EA_EPSR EA_EPSRP EA_EPSTAUC EA_EPSTAUD EA_EPSTAUK EA_EPSTAUN
EA_EPSTAUWF EA_EPSTAUWH EA_EPSTR EA_EPSZ EA_EPSG
US_EPSG US_EPSR US_EPSTAUC US_EPSTAUD US_EPSTAUK
US_EPSTAUN US_EPSTAUWF US_EPSTAUWH US_EPSTR US_EPSZ EA_EPSILONM US_EPSILONM;
//----------------------------------------------------------------
// 3. Parameters
//----------------------------------------------------------------
parameters EA_ALPHA EA_BETA EA_BFYTARGET EA_BYTARGET EA_CHIH EA_CHII EA_CHIJ EA_CHIX
EA_DELTA EA_ETA EA_ETAI EA_ETAJ EA_GAMMAB1 EA_GAMMAI1 EA_GAMMAIMC1 EA_QBAR
EA_GAMMAIMI1 EA_GAMMAU1 EA_GAMMAU2 EA_GAMMAV1 EA_GAMMAV2 EA_KAPPA EA_MUC EA_MUI EA_NUC
EA_NUI EA_OMEGA EA_PHIRGY EA_PHIRPI EA_PHIRR EA_PHITB EA_PI4TARGET EA_PSI
EA_RHOG EA_RHOR EA_RHORP EA_RHOTAUC EA_RHOTAUD EA_RHOTAUK EA_RHOTAUN EA_RHOTAUWF
EA_RHOTAUWH EA_RHOTR EA_RHOZ EA_SIGMA EA_SIZE EA_THETA EA_XIH EA_XII EA_XIJ
EA_XIX EA_ZETA US_ALPHA US_BETA US_BYTARGET US_CHIH US_CHII US_CHIJ US_CHIX
US_DELTA US_ETA US_ETAI US_ETAJ US_GAMMAI1 US_GAMMAIMC1 US_GAMMAIMI1 US_GAMMAU1
US_GAMMAU2 US_GAMMAB1 US_GAMMAV1 US_GAMMAV2 US_KAPPA US_MUC US_MUI US_NUC US_NUI US_OMEGA
US_PHIRGY US_PHIRPI US_PHIRR US_PHITB US_PI4TARGET US_PSI US_RHOG US_RHOR US_RHOTAUC
US_RHOTAUD US_RHOTAUK US_RHOTAUN US_RHOTAUWF US_RHOTAUWH US_RHOTR US_RHOZ
US_SIGMA US_SIZE US_THETA US_XIH US_XII US_XIJ US_XIX US_ZETA EA_TAUKBAR EA_PIBAR EA_PSIBAR US_RER EA_PYBAR EA_YBAR
EA_UPSILONT EA_UPSILONTR EA_ZBAR EA_GYBAR EA_TRYBAR EA_TAUCBAR EA_TAUDBAR EA_TAUNBAR EA_TAUWHBAR EA_TAUWFBAR US_QBAR
US_TAUKBAR US_PIBAR US_PSIBAR US_PYBAR US_YBAR US_UPSILONT US_UPSILONTR US_ZBAR US_GYBAR US_TRYBAR US_TAUCBAR US_TAUDBAR US_TAUNBAR US_TAUWHBAR US_TAUWFBAR;
// --------------------------- //
//4. PARAMETER VALUES(calibration): EURO AREA //
// --------------------------- //
// COUNTRY SIZE (POPULATION)
EA_SIZE = 0.4194;
// SIZE OF HOUSEHOLD J
EA_OMEGA = 0.25;
// PREFERENCES
EA_BETA = 0.99;
EA_SIGMA = 2.00;
EA_KAPPA = 0.60;
EA_ZETA = 2.00;
// CAPITAL ACCUMULATION
EA_DELTA = 0.025;
// WAGE SETTING
EA_ETA = 6.00;
EA_ETAI = 6.00;
EA_ETAJ = 6.00;
EA_XII = 0.75;
EA_XIJ = 0.75;
EA_CHII = 0.75;
EA_CHIJ = 0.75;
// MONEY DEMAND
//EA_VI = 0.75;
//EA_EPSILONM =-0.75;
// INTERMEDIATE-GOOD TECHNOLOGY
EA_ALPHA = 0.30;
EA_PSI = 0.20;
// PRICE SETTING
EA_THETA = 6.00;
EA_XIH = 0.90;
EA_XIX = 0.30;
EA_CHIH = 0.50;
EA_CHIX = 0.50;
// FINAL-GOOD TECHNOLOGIES
EA_NUC = 0.919622;
EA_MUC = 1.5;
EA_NUI = 0.418629;
EA_MUI = 1.5;
// EXPENDITURE SHARES
//EA_IY = 0.22;
//EA_IMCY = 0.13;
//EA_IMIY = 0.05;
// ADJUSTMENT AND TRANSACTION COST
EA_GAMMAV1 = 0.289073;
EA_GAMMAV2 = 0.150339;
EA_GAMMAI1 = 3.00;
EA_GAMMAU1 = 0.032765;
EA_GAMMAU2 = 0.007;
EA_GAMMAIMC1 = 2.50;
EA_GAMMAIMI1 = 0.00;
EA_GAMMAB1 = 0.01;
// FISCAL POLICY
EA_BYTARGET = 2.40;
EA_PHITB = 0.10;
EA_GYBAR = 0.18;
EA_TRYBAR = 0.195161;
EA_TAUCBAR = 0.183;
EA_TAUKBAR = 0.184123;
EA_TAUNBAR = 0.122;
EA_TAUWHBAR = 0.118;
EA_TAUWFBAR = 0.219;
EA_UPSILONT = 1.20;
EA_UPSILONTR = 2/3;
// MONETARY POLICY
EA_PI4TARGET = 1.02;
EA_PHIRR = 0.95;
EA_PHIRPI = 2.00;
EA_PHIRGY = 0.10;
EA_interest_EXOG=EA_BETA^(-1)*EA_PI4TARGET^(1/4);
// NET FOREIGN ASSET POSITION
EA_BFYTARGET = 0.00;
// AUTOREGRESSIVE SHOCKS
EA_RHOZ = 0.90;
EA_RHOR = 0.90;
EA_RHOG = 0.90;
EA_RHOTR = 0.90;
EA_RHOTAUC = 0.90;
EA_RHOTAUK = 0.90;
EA_RHOTAUN = 0.90;
EA_RHOTAUD = 0.90;
EA_RHOTAUWH = 0.90;
EA_RHOTAUWF = 0.90;
EA_PYBAR =1.00645740523434;
EA_YBAR = 3.62698111871356;
EA_RHORP = 0.9;
EA_PIBAR = 0.961117319822928;
EA_PSIBAR = 0.725396223742712;
EA_QBAR = 0.961117319822928;
EA_TAUDBAR = 0;
EA_ZBAR = 1;
// --------------------------------- //
// PARAMETER VALUES(calibration): UNITED STATES //
// --------------------------------- //
// COUNTRY SIZE (POPULATION)
US_SIZE = 0.5806;
// SIZE OF HOUSEHOLD J
US_OMEGA = 0.25;
// PREFERENCES
US_BETA = 0.99;
US_SIGMA = 2.00;
US_KAPPA = 0.60;
US_ZETA = 2.00;
// CAPITAL ACCUMULATION
US_DELTA = 0.025;
// WAGE SETTING
US_ETA = 6.00;
US_ETAI = 6.00;
US_ETAJ = 6.00;
US_XII = 0.75;
US_XIJ = 0.75;
US_CHII = 0.75;
US_CHIJ = 0.75;
// MONEY DEMAND
//US_VI = 2.38;
//US_EPSILONM =-0.75;
// INTERMEDIATE-GOOD TECHNOLOGY
US_ALPHA = 0.30;
US_PSI = 0.20;
// PRICE SETTING
US_THETA = 6.00;
US_XIH = 0.90;
US_XIX = 0.30;
US_CHIH = 0.50;
US_CHIX = 0.50;
// FINAL-GOOD TECHNOLOGIES
US_NUC = 0.899734;
US_MUC = 1.5;
US_NUI = 0.673228;
US_MUI = 1.5;
// EXPENDITURE SHARES
//US_IY = 0.22;
//US_IMCY = 0.06;
//US_IMIY = 0.07;
// ADJUSTMENT AND TRANSACTION COST
US_GAMMAV1 = 0.028706;
US_GAMMAV2 = 0.150339;
US_GAMMAI1 = 3.00;
US_GAMMAU1 = 0.034697;
US_GAMMAU2 = 0.007;
US_GAMMAIMC1 = 2.50;
US_GAMMAIMI1 = 0.00;
US_GAMMAB1 = 0.01;
// FISCAL POLICY
US_BYTARGET = 2.40;
US_PHITB = 0.10;
US_GYBAR = 0.16;
US_TRYBAR = 0.079732;
US_TAUCBAR = 0.077;
US_TAUKBAR = 0.184123;
US_TAUNBAR = 0.154;
US_TAUWHBAR = 0.071;
US_TAUWFBAR = 0.071;
US_UPSILONT = 1.20;
US_UPSILONTR = 2/3;
// MONETARY POLICY
US_PI4TARGET = 1.02;
US_PHIRR = 0.95;
US_PHIRPI = 2.00;
US_PHIRGY = 0.10;
US_interest_EXOG=US_BETA^(-1)*US_PI4TARGET^(1/4);
// AUTOREGRESSIVE SHOCKS
US_RHOZ = 0.90;
US_RHOR = 0.90;
US_RHOG = 0.90;
US_RHOTR = 0.90;
US_RHOTAUC = 0.90;
US_RHOTAUK = 0.90;
US_RHOTAUN = 0.90;
US_RHOTAUD = 0.90;
US_RHOTAUWH = 0.90;
US_RHOTAUWF = 0.90;
US_PYBAR = 0.992282866960427;
US_TAUDBAR = 0;
US_YBAR = 3.92445610588497;
US_PIBAR = 1.01776829477927;
US_PSIBAR = 0.784891221176995;
US_QBAR = 1.01776829477927;
US_ZBAR = 1;
US_RER=1;
//----------------------------------------------------------------
// 5. Model
//----------------------------------------------------------------
model;
// Original Model Code:
// Original monetary policy rule for EA and US (Coenen, McAdams and Straub (2008))
//EA_R^4-1 = EA_PHIRR*(EA_R(-1)^4-1)+(1-EA_PHIRR)(EA_RRSTAR^4EA_PI4TARGET-1+EA_PHIRPI*(EA_PIC4-EA_PI4TARGET))+EA_PHIRGY*(EA_Y/EA_Y(-1)-1)+EA_EPSR ;
//US_R^4-1 = US_PHIRR*(US_R(-1)^4-1)+(1-US_PHIRR)(US_RRSTAR^4US_PI4TARGET-1+US_PHIRPI*(US_PIC4-US_PI4TARGET))+US_PHIRGY*(US_Y/US_Y(-1)-1)+US_EPSR ; //Taylor rule
// Model equations Euro area
//Households
//Household I
EA_UTILI = (1/(1-EA_SIGMA)(EA_CI-EA_KAPPAEA_CI(-1))^(1-EA_SIGMA)-1/(1+EA_ZETA)EA_NI^(1+EA_ZETA)+EA_BETAEA_UTILI(+1)) ; // Value function of HH I (Bellman)
EA_LAMBDAI*(1+EA_TAUC+EA_GAMMAVI+EA_VIEA_GAMMAVIDER) = (EA_CI-EA_KAPPAEA_CI(-1))^(-EA_SIGMA) ; // equation (4), FOC with respect to consumption
EA_R = EA_BETA^(-1)EA_LAMBDAI/EA_LAMBDAI(+1)EA_PIC(+1) ; // equation (8), FOC with respect to bonds
EA_VI^2EA_GAMMAVIDER = 1-EA_BETAEA_LAMBDAI(+1)/(EA_LAMBDAIEA_PIC(+1)) ; // equation (10), FOC with respect to money
EA_VI = (1+EA_TAUC)EA_CI/EA_MI ; // consumption based velocity
EA_GAMMAVI = EA_GAMMAV1EA_VI+EA_GAMMAV2/EA_VI-2(EA_GAMMAV1EA_GAMMAV2)^(1/2) ; // transaction cost
EA_GAMMAVIDER = EA_GAMMAV1-EA_GAMMAV2EA_VI^(-2) ; // derivative of transaction cost
EA_KI = (1-EA_DELTA)EA_KI(-1)+(1-EA_GAMMAI(-1))EA_II(-1) ; // capital accumulation
EA_GAMMAI(-1) = EA_GAMMAI1/2(EA_II/EA_II(-1)-1)^2 ; // investment adjustment cost
EA_GAMMAIDER = EA_GAMMAI1(EA_II/EA_II(-1)-1)/EA_II(-1) ; // derivative of investment adjustment cost
EA_GAMMAU = ((EA_BETA^(-1)-1+EA_DELTA)EA_QBAR-EA_DELTAEA_TAUKBAREA_PIBAR)/((1-EA_TAUKBAR)EA_PIBAR)(EA_U-1)+EA_GAMMAU2/2(EA_U-1)^2 ; //capital utilisation cost
EA_GAMMAUDER = ((EA_BETA^(-1)-1+EA_DELTA)EA_QBAR-EA_DELTAEA_TAUKBAREA_PIBAR)/((1-EA_TAUKBAR)EA_PIBAR)+EA_GAMMAU2(EA_U-1) ; //derivative capital utilisation cost
EA_RK = EA_GAMMAUDEREA_PI ; //equation (7), FOC with respect to utilization rate
EA_PI = EA_Q*(1-EA_GAMMAI-EA_GAMMAIDEREA_II)+EA_BETAEA_LAMBDAI(+1)/EA_LAMBDAIEA_Q(+1)EA_GAMMAIDER(+1)EA_II(+1)^2/EA_II ; //equation (5), FOC with respect to investment
EA_Q = EA_BETAEA_LAMBDAI(+1)/EA_LAMBDAI((1-EA_TAUK(+1))(EA_RK(+1)EA_U(+1)-EA_GAMMAU(+1)EA_PI(+1))+EA_TAUK(+1)EA_DELTAEA_PI(+1)+(1-EA_DELTA)EA_Q(+1)) ; //equation (6), FOC with respect to capital
EA_WITILDE^(1+EA_ETAIEA_ZETA) = EA_ETAI/(EA_ETAI-1)EA_FI/EA_GI ;
EA_FI = EA_WI^(EA_ETAI(1+EA_ZETA))EA_NDI^(1+EA_ZETA)+EA_XIIEA_BETA(EA_PIC(+1)/(EA_PIC^EA_CHIIEA_PI4TARGET^(1/4*(1-EA_CHII))))^(EA_ETAI*(1+EA_ZETA))EA_FI(+1) ;
EA_GI = EA_LAMBDAI(1-EA_TAUN-EA_TAUWH)EA_WI^EA_ETAIEA_NDI+EA_XIIEA_BETA(EA_PIC(+1)/(EA_PIC^EA_CHIIEA_PI4TARGET^(1/4(1-EA_CHII))))^(EA_ETAI-1)EA_GI(+1) ;
EA_WI^(1-EA_ETAI) = (1-EA_XII)EA_WITILDE^(1-EA_ETAI)+EA_XIIEA_WI(-1)^(1-EA_ETAI)(EA_PIC(-1)^EA_CHIIEA_PI4TARGET^(1/4(1-EA_CHII))/EA_PIC)^(1-EA_ETAI) ;
//Household J
EA_UTILJ = (1/(1-EA_SIGMA)(EA_CJ-EA_KAPPAEA_CJ(-1))^(1-EA_SIGMA)-1/(1+EA_ZETA)EA_NJ^(1+EA_ZETA)+EA_BETAEA_UTILJ(+1)) ; // Value function of HH J (Bellman)
(1+EA_TAUC+EA_GAMMAVJ)EA_CJ+EA_MJ = (1-EA_TAUN-EA_TAUWH)EA_WJEA_NJ+EA_TRJ-EA_TJ+EA_MJ(-1)EA_PIC^(-1) ; // budget constraint for household J, eq. 14
EA_LAMBDAJ(1+EA_TAUC+EA_GAMMAVJ+EA_VJEA_GAMMAVJDER) = (EA_CJ-EA_KAPPAEA_CJ(-1))^(-EA_SIGMA) ; // FOC with respect to consumption
EA_VJ^2EA_GAMMAVJDER = 1-EA_BETAEA_LAMBDAJ(+1)/(EA_LAMBDAJEA_PIC(+1)) ; // FOC with respect to money
EA_VJ = (1+EA_TAUC)EA_CJ/EA_MJ ; // consumption based velocity
EA_GAMMAVJ = EA_GAMMAV1EA_VJ+EA_GAMMAV2/EA_VJ-2*(EA_GAMMAV1EA_GAMMAV2)^(1/2) ; // transaction cost
EA_GAMMAVJDER = EA_GAMMAV1-EA_GAMMAV2EA_VJ^(-2) ; // derivation of transaction cost
EA_WJTILDE^(1+EA_ETAJEA_ZETA) = EA_ETAJ/(EA_ETAJ-1)EA_FJ/EA_GJ ;
EA_FJ = EA_WJ^(EA_ETAJ(1+EA_ZETA))EA_NDJ^(1+EA_ZETA)+EA_XIJEA_BETA(EA_PIC(+1)/(EA_PIC^EA_CHIJEA_PI4TARGET^(1/4(1-EA_CHIJ))))^(EA_ETAJ*(1+EA_ZETA))EA_FJ(+1) ;
EA_GJ = EA_LAMBDAJ(1-EA_TAUN-EA_TAUWH)EA_WJ^EA_ETAJEA_NDJ+EA_XIJEA_BETA(EA_PIC(+1)/(EA_PIC^EA_CHIJEA_PI4TARGET^(1/4(1-EA_CHIJ))))^(EA_ETAJ-1)EA_GJ(+1) ;
EA_WJ^(1-EA_ETAJ) = (1-EA_XIJ)EA_WJTILDE^(1-EA_ETAJ)+EA_XIJEA_WJ(-1)^(1-EA_ETAJ)(EA_PIC(-1)^EA_CHIJEA_PI4TARGET^(1/4(1-EA_CHIJ))/EA_PIC)^(1-EA_ETAJ) ;
//Firms
//Intermediate-good firms
EA_YS = EA_ZEA_KD^EA_ALPHAEA_ND^(1-EA_ALPHA)-EA_PSIBAR ; // production function, eq.17
EA_RK = EA_ALPHA*(EA_YS+EA_PSIBAR)/EA_KDEA_MC ; // FOC with respect to capital
EA_MC = 1/(EA_ZEA_ALPHA^EA_ALPHA*(1-EA_ALPHA)^(1-EA_ALPHA))EA_RK^EA_ALPHA((1+EA_TAUWF)EA_W)^(1-EA_ALPHA) ; // eq.18, marginal cost
EA_NDI = ((1-EA_OMEGA)(EA_WI/EA_W)^(-EA_ETA)EA_ND) ; // demand function for labor I, eq. 22
EA_NDJ = (EA_OMEGA(EA_WJ/EA_W)^(-EA_ETA)EA_ND) ; // demand function for labor J, eq. 22
EA_ND^(1-1/EA_ETA) =((1-EA_OMEGA)^(1/EA_ETA)EA_NDI^(1-1/EA_ETA)+EA_OMEGA^(1/EA_ETA)EA_NDJ^(1-1/EA_ETA)) ; // combination of differentiated labour, eq.19
EA_D = EA_PYEA_Y-EA_RKEA_KD-(1+EA_TAUWF)EA_WEA_ND ; // definition of dividends, used later in the budget constraint of the government to compute tax income from dividends
EA_PHTILDE/EA_PH = EA_THETA/(EA_THETA-1)EA_FH/EA_GH ;
EA_FH = EA_MCEA_H+EA_XIHEA_BETAEA_LAMBDAI(+1)/EA_LAMBDAI(EA_PIH(+1)/(EA_PIH^EA_CHIHEA_PI4TARGET^(1/4(1-EA_CHIH))))^EA_THETAEA_FH(+1) ;
EA_GH = EA_PHEA_H+EA_XIHEA_BETAEA_LAMBDAI(+1)/EA_LAMBDAI*(EA_PIH(+1)/(EA_PIH^EA_CHIHEA_PI4TARGET^(1/4(1-EA_CHIH))))^(EA_THETA-1)EA_GH(+1) ;
EA_PH^(1-EA_THETA) = (1-EA_XIH)EA_PHTILDE^(1-EA_THETA)+EA_XIH(EA_PH(-1)/EA_PIC)^(1-EA_THETA)(EA_PIH(-1)^EA_CHIHEA_PI4TARGET^(1/4(1-EA_CHIH)))^(1-EA_THETA) ;
EA_PIH = EA_PH/EA_PH(-1)EA_PIC ;
US_PIMTILDE/US_PIM = EA_THETA/(EA_THETA-1)EA_FX/EA_GX ;
EA_FX = US_SIZE/EA_SIZEUS_IMEA_MC+EA_XIXEA_BETAEA_LAMBDAI(+1)/EA_LAMBDAI*(US_PIIM(+1)/(US_PIIM^EA_CHIXEA_PI4TARGET^(1/4(1-EA_CHIX))))^EA_THETAEA_FX(+1) ;
EA_GX = EAUS_RERUS_PIMUS_SIZE/EA_SIZEUS_IM+EA_XIXEA_BETAEA_LAMBDAI(+1)/EA_LAMBDAI*(US_PIIM(+1)/(US_PIIM^EA_CHIXEA_PI4TARGET^(1/4(1-EA_CHIX))))^(EA_THETA-1)EA_GX(+1) ;
US_PIM^(1-EA_THETA) = (1-EA_XIX)US_PIMTILDE^(1-EA_THETA)+EA_XIX(US_PIM(-1)/US_PIC)^(1-EA_THETA)(US_PIIM(-1)^EA_CHIXUS_PI4TARGET^(1/4(1-EA_CHIH)))^(1-EA_THETA) ;
US_PIIM = US_PIM/US_PIM(-1)*US_PIC ;
EAUS_RER = EA_RER/US_RER ;
//Final-good firms
EA_QC^((EA_MUC-1)/EA_MUC) = EA_NUC^(1/EA_MUC)EA_HC^(1-1/EA_MUC)+(1-EA_NUC)^(1/EA_MUC)((1-EA_GAMMAIMC)EA_IMC)^(1-1/EA_MUC) ; // eq. 27
1^(1-EA_MUC) = EA_NUCEA_PH^(1-EA_MUC)+(1-EA_NUC)(EA_PIM/EA_GAMMAIMCDAG)^(1-EA_MUC) ; // equation for the price of a unit of private consumption good
EA_HC = EA_NUCEA_PH^(-EA_MUC)EA_QC ; // eq. 30, demand function for domestic intermediate good
EA_GAMMAIMC = EA_GAMMAIMC1/2(EA_IMC/EA_QC/(EA_IMC(-1)/EA_QC(-1))-1)^2 ; // adjumstment cost of imported intermediate good used for final consumption good
EA_GAMMAIMCDAG = 1-EA_GAMMAIMC-EA_GAMMAIMC1*(EA_IMC/EA_QC/(EA_IMC(-1)/EA_QC(-1))-1)EA_IMC/EA_QC/(EA_IMC(-1)/EA_QC(-1)) ; // second equation, page 2560
EA_QI^((EA_MUI-1)/EA_MUI) = EA_NUI^(1/EA_MUI)EA_HI^(1-1/EA_MUI)+(1-EA_NUI)^(1/EA_MUI)((1-EA_GAMMAIMI)EA_IMI)^(1-1/EA_MUI) ; // eq. 31, non-tradable final private investment good
EA_PI^(1-EA_MUI) = EA_NUIEA_PH^(1-EA_MUI)+(1-EA_NUI)(EA_PIM/EA_GAMMAIMIDAG)^(1-EA_MUI) ; // equation for the price of a unit of private investment good
EA_HI = EA_NUI*(EA_PH/EA_PI)^(-EA_MUI)EA_QI ; // demand function for domestic intermediate investment good bundle
EA_GAMMAIMI = EA_GAMMAIMI1/2(EA_IMI/EA_QI/(EA_IMI(-1)/EA_QI(-1))-1)^2 ; // adjumstment cost of imported intermediate good used for final investment good
EA_GAMMAIMIDAG = 1-EA_GAMMAIMI-EA_GAMMAIMI1*(EA_IMI/EA_QI/(EA_IMI(-1)/EA_QI)-1)*EA_IMI/EA_QI/(EA_IMI(-1)/EA_QI(-1)) ; // second equation, page 2560
//Fiscal authority
EA_PH(-1)*EA_G(-1)+EA_TR(-1)+EA_B(-1)EA_PIC(-1)^(-1)+EA_M(-2)EA_PIC(-1)^(-1) = EA_TAUC(-1)EA_C(-1)+(EA_TAUN(-1)+EA_TAUWH(-1))(EA_WI(-1)EA_NDI(-1)+EA_WJ(-1)EA_NDJ(-1))+EA_TAUWF(-1)EA_W(-1)EA_ND(-1)+EA_TAUK(-1)(EA_RK(-1)EA_U(-1)-(EA_GAMMAU(-1)+EA_DELTA)EA_PI(-1))EA_K(-1)+EA_TAUD(-1)EA_D(-1)+EA_T(-1)+EA_R(-1)^(-1)EA_B+EA_M(-1) ; // budget constraint, eq. 33
EA_PHEA_G = EA_GYEA_PYBAREA_YBAR ; //government purchases of the final public consumption good as a fraction of steady state nominal output
EA_TR = EA_TRYEA_PYBAREA_YBAR ; //government transfers as a fraction of steady state nominal output
EA_T/(EA_PYBAREA_YBAR) = EA_PHITB(EA_B/(EA_PYBAREA_YBAR)-EA_BYTARGET) ; // eq. 34, lump-sum taxes as fraction of steady state nominal output
EA_TI = EA_UPSILONTEA_T ;
EA_TRI = EA_UPSILONTREA_TR ;
//Monetary authority
// original monetary policy rule at the beginning of the original model code
EA_PIC4 = EA_PIC*EA_PIC(-1)*EA_PIC(-2)*EA_PIC(-3) ; // annual inflation
EA_RR-1 = EA_R/EA_PIC(+1)-1 ; // real interest rate
EA_RRSTAR-1 = 1/EA_BETA-1 ; // equilibrium nominal interest rate
//Aggregation
EA_C = (1-EA_OMEGA)EA_CI+EA_OMEGAEA_CJ ; // consumption
EA_M = (1-EA_OMEGA)EA_MI+EA_OMEGAEA_MJ ; // money
EA_K = (1-EA_OMEGA)EA_KI ; // capital
EA_I = (1-EA_OMEGA)EA_II ; // investment
EA_TRJ = (1/EA_OMEGAEA_TR-(1-EA_OMEGA)/EA_OMEGAEA_TRI) ; // transfers to HH J
EA_TJ = (1/EA_OMEGAEA_T-(1-EA_OMEGA)/EA_OMEGAEA_TI) ; // taxes HH J
EA_GAMMAV = (1-EA_OMEGA)EA_CIEA_GAMMAVI+EA_OMEGAEA_CJEA_GAMMAVJ ; // total transaction costs
EA_NI = EA_SIEA_NDI ;
EA_SI = (1-EA_XII)(EA_WITILDE/EA_WI)^(-EA_ETAI)+EA_XII*(EA_WI(-1)/EA_WI)^(-EA_ETAI)(EA_PIC/(EA_PIC(-1)^EA_CHIIEA_PI4TARGET^(1/4*(1-EA_CHII))))^EA_ETAIEA_SI(-1) ;
EA_NJ = EA_SJEA_NDJ ;
EA_SJ = (1-EA_XIJ)(EA_WJTILDE/EA_WJ)^(-EA_ETAJ)+EA_XIJ(EA_WJ(-1)/EA_WJ)^(-EA_ETAJ)(EA_PIC/(EA_PIC(-1)^EA_CHIJEA_PI4TARGET^(1/4*(1-EA_CHIJ))))^EA_ETAJEA_SJ(-1) ;
EA_UEA_K = EA_KD ;
EA_YS = EA_SHEA_H+EA_SXUS_SIZE/EA_SIZEUS_IM ;
EA_H = EA_HC+EA_HI+EA_G ; // eq. 32
EA_IM = EA_IMI+EA_IMC ; // eq. 32
EA_SH = (1-EA_XIH)(EA_PHTILDE/EA_PH)^(-EA_THETA)+EA_XIH*(EA_PIH/(EA_PIH(-1)^EA_CHIHEA_PI4TARGET^(1/4(1-EA_CHIH))))^EA_THETAEA_SH(-1) ;
EA_SX = (1-EA_XIX)(US_PIMTILDE/US_PIM)^(-EA_THETA)+EA_XIX*(US_PIIM/(US_PIIM(-1)^EA_CHIXEA_PI4TARGET^(1/4(1-EA_CHIH))))^EA_THETAEA_SX(-1) ;
EA_QC = EA_C+EA_GAMMAV ;
EA_QI = EA_I+EA_KEA_GAMMAU ;
EA_PYEA_Y = EA_QC+EA_PIEA_QI+EA_PHEA_G+EAUS_RERUS_PIMUS_SIZE/EA_SIZEUS_IM-EA_PIM*(EA_IMC*(1-EA_GAMMAIMC)/EA_GAMMAIMCDAG+EA_IMI*(1-EA_GAMMAIMI)/EA_GAMMAIMIDAG) ;
EA_Y = EA_YS ;
log(EA_Z) = (1-EA_RHOZ)log(EA_ZBAR)+EA_RHOZlog(EA_Z(-1))+EA_EPSZ ;
EA_GY = (1-EA_RHOG)EA_GYBAR+EA_RHOGEA_GY(-1)+EA_EPSG ;
EA_TRY = (1-EA_RHOTR)EA_TRYBAR+EA_RHOTREA_TRY(-1)+EA_EPSTR ;
EA_TAUC = (1-EA_RHOTAUC)EA_TAUCBAR+EA_RHOTAUCEA_TAUC(-1)+EA_EPSTAUC ;
EA_TAUD = (1-EA_RHOTAUD)EA_TAUDBAR+EA_RHOTAUDEA_TAUD(-1)+EA_EPSTAUD ;
EA_TAUK = (1-EA_RHOTAUK)EA_TAUKBAR+EA_RHOTAUKEA_TAUK(-1)+EA_EPSTAUK ;
EA_TAUN = (1-EA_RHOTAUN)EA_TAUNBAR+EA_RHOTAUNEA_TAUN(-1)+EA_EPSTAUN ;
EA_TAUWH = (1-EA_RHOTAUWH)EA_TAUWHBAR+EA_RHOTAUWHEA_TAUWH(-1)+EA_EPSTAUWH ;
EA_TAUWF = (1-EA_RHOTAUWF)EA_TAUWFBAR+EA_RHOTAUWFEA_TAUWF(-1)+EA_EPSTAUWF ;
EA_CY = EA_C/(EA_PYEA_Y) ;
EA_IY = EA_PIEA_I/(EA_PYEA_Y) ;
EA_IMY = EA_PIMEA_IM/(EA_PYEA_Y) ;
EA_IMCY = EA_PIMEA_IMC/(EA_PYEA_Y) ;
EA_IMIY = EA_PIMEA_IMI/(EA_PYEA_Y) ;
EA_BY = EA_B/(EA_PYBAREA_YBAR) ;
EA_TY = EA_T/(EA_PYBAREA_YBAR) ;
EA_YGAP = EA_Y/EA_YBAR-1 ;
EA_YGROWTH = EA_Y/EA_Y(-1) ;
EA_YSHARE = EA_SIZEEA_PYEA_Y/EA_RER/(EA_SIZEEA_PYEA_Y/EA_RER+US_SIZEUS_PYUS_Y/US_RER) ;
EA_EPSILONM = -1/81/(EA_R*(EA_GAMMAV2*EA_R+EA_R-1)) ;
// Model equations US
//Households
//Household I
US_UTILI = (1/(1-US_SIGMA)(US_CI-US_KAPPAUS_CI(-1))^(1-US_SIGMA)-1/(1+US_ZETA)US_NI^(1+US_ZETA)+US_BETAUS_UTILI(+1)) ; //Value function of HH I (Bellman)
US_LAMBDAI*(1+US_TAUC+US_GAMMAVI+US_VIUS_GAMMAVIDER) = (US_CI-US_KAPPAUS_CI(-1))^(-US_SIGMA) ; //equation (4), FOC with respect to consumption
US_R = US_BETA^(-1)US_LAMBDAI/US_LAMBDAI(+1)US_PIC(+1) ; //equation (8), FOC with respect to bonds
US_VI^2US_GAMMAVIDER = 1-US_BETAUS_LAMBDAI(+1)/(US_LAMBDAIUS_PIC(+1)) ; //equation (10), FOC with respect to money
US_VI = (1+US_TAUC)US_CI/US_MI ; // consumption based velocity
US_GAMMAVI = US_GAMMAV1US_VI+US_GAMMAV2/US_VI-2(US_GAMMAV1US_GAMMAV2)^(1/2) ; //transaction cost
US_GAMMAVIDER = US_GAMMAV1-US_GAMMAV2US_VI^(-2) ; // derivative of transaction cost
US_KI = (1-US_DELTA)US_KI(-1)+(1-US_GAMMAI(-1))US_II(-1) ; // capital accumulation
US_GAMMAI(-1) = US_GAMMAI1/2(US_II/US_II(-1)-1)^2 ; //investment adjustment cost
US_GAMMAIDER = US_GAMMAI1(US_II/US_II(-1)-1)/US_II(-1) ; // derivative of investment adjustment cost
US_GAMMAU = ((US_BETA^(-1)-1+US_DELTA)US_QBAR-US_DELTAUS_TAUKBARUS_PIBAR)/((1-US_TAUKBAR)US_PIBAR)(US_U-1)+US_GAMMAU2/2(US_U-1)^2 ; //capital utilisation cost
US_GAMMAUDER = ((US_BETA^(-1)-1+US_DELTA)US_QBAR-US_DELTAUS_TAUKBARUS_PIBAR)/((1-US_TAUKBAR)US_PIBAR)+US_GAMMAU2(US_U-1) ; //derivative capital utilisation cost
US_RK = US_GAMMAUDERUS_PI ; //equation (7), FOC with respect to utilization rate
US_PI = US_Q*(1-US_GAMMAI-US_GAMMAIDERUS_II)+US_BETAUS_LAMBDAI(+1)/US_LAMBDAIUS_Q(+1)US_GAMMAIDER(+1)US_II(+1)^2/US_II ; //equation (5), FOC with respect to investment
US_Q = US_BETAUS_LAMBDAI(+1)/US_LAMBDAI((1-US_TAUK(+1))(US_RK(+1)US_U(+1)-US_GAMMAU(+1)US_PI(+1))+US_TAUK(+1)US_DELTAUS_PI(+1)+(1-US_DELTA)US_Q(+1)) ; //equation (6), FOC with respect to capital
US_WITILDE^(1+US_ETAIUS_ZETA) = US_ETAI/(US_ETAI-1)US_FI/US_GI ;
US_FI = US_WI^(US_ETAI(1+US_ZETA))US_NDI^(1+US_ZETA)+US_XIIUS_BETA(US_PIC(+1)/(US_PIC^US_CHIIUS_PI4TARGET^(1/4*(1-US_CHII))))^(US_ETAI*(1+US_ZETA))US_FI(+1) ;
US_GI = US_LAMBDAI(1-US_TAUN-US_TAUWH)US_WI^US_ETAIUS_NDI+US_XIIUS_BETA(US_PIC(+1)/(US_PIC^US_CHIIUS_PI4TARGET^(1/4(1-US_CHII))))^(US_ETAI-1)US_GI(+1) ;
US_WI^(1-US_ETAI) = (1-US_XII)US_WITILDE^(1-US_ETAI)+US_XIIUS_WI(-1)^(1-US_ETAI)(US_PIC(-1)^US_CHIIUS_PI4TARGET^(1/4(1-US_CHII))/US_PIC)^(1-US_ETAI) ;
//Household J
US_UTILJ = (1/(1-US_SIGMA)(US_CJ-US_KAPPAUS_CJ(-1))^(1-US_SIGMA)-1/(1+US_ZETA)US_NJ^(1+US_ZETA)+US_BETAUS_UTILJ(+1)) ; //Value function of HH J (Bellman)
(1+US_TAUC+US_GAMMAVJ)US_CJ+US_MJ = (1-US_TAUN-US_TAUWH)US_WJUS_NJ+US_TRJ-US_TJ+US_MJ(-1)US_PIC^(-1) ; // budget constraint for household J, eq. 14
US_LAMBDAJ(1+US_TAUC+US_GAMMAVJ+US_VJUS_GAMMAVJDER) = (US_CJ-US_KAPPAUS_CJ(-1))^(-US_SIGMA) ; //FOC with respect to consumption
US_VJ^2US_GAMMAVJDER = 1-US_BETAUS_LAMBDAJ(+1)/(US_LAMBDAJUS_PIC(+1)) ; //FOC with respect to money
US_VJ = (1+US_TAUC)US_CJ/US_MJ ; // consumption based velocity
US_GAMMAVJ = US_GAMMAV1US_VJ+US_GAMMAV2/US_VJ-2*(US_GAMMAV1US_GAMMAV2)^(1/2) ; //transaction cost
US_GAMMAVJDER = US_GAMMAV1-US_GAMMAV2US_VJ^(-2) ; //derivation of transaction cost
US_WJTILDE^(1+US_ETAJUS_ZETA) = US_ETAJ/(US_ETAJ-1)US_FJ/US_GJ ;
US_FJ = US_WJ^(US_ETAJ(1+US_ZETA))US_NDJ^(1+US_ZETA)+US_XIJUS_BETA(US_PIC(+1)/(US_PIC^US_CHIJUS_PI4TARGET^(1/4(1-US_CHIJ))))^(US_ETAJ*(1+US_ZETA))US_FJ(+1) ;
US_GJ = US_LAMBDAJ(1-US_TAUN-US_TAUWH)US_WJ^US_ETAJUS_NDJ+US_XIJUS_BETA(US_PIC(+1)/(US_PIC^US_CHIJUS_PI4TARGET^(1/4(1-US_CHIJ))))^(US_ETAJ-1)US_GJ(+1) ;
US_WJ^(1-US_ETAJ) = (1-US_XIJ)US_WJTILDE^(1-US_ETAJ)+US_XIJUS_WJ(-1)^(1-US_ETAJ)(US_PIC(-1)^US_CHIJUS_PI4TARGET^(1/4(1-US_CHIJ))/US_PIC)^(1-US_ETAJ) ;
//Firms
//Intermediate-good firms
US_YS = US_ZUS_KD^US_ALPHAUS_ND^(1-US_ALPHA)-US_PSIBAR ; // production function, eq.17
US_RK = US_ALPHA*(US_YS+US_PSIBAR)/US_KDUS_MC ; // FOC with respect to capital
US_MC = 1/(US_ZUS_ALPHA^US_ALPHA*(1-US_ALPHA)^(1-US_ALPHA))US_RK^US_ALPHA((1+US_TAUWF)US_W)^(1-US_ALPHA) ; // eq.18, marginal cost
US_NDI = ((1-US_OMEGA)(US_WI/US_W)^(-US_ETA)US_ND) ; // demand function for labor I, eq. 22
US_NDJ = (US_OMEGA(US_WJ/US_W)^(-US_ETA)US_ND) ; // demand function for labor J, eq. 22
US_ND^(1-1/US_ETA) = ((1-US_OMEGA)^(1/US_ETA)US_NDI^(1-1/US_ETA)+US_OMEGA^(1/US_ETA)US_NDJ^(1-1/US_ETA)) ; // combination of differentiated labour, eq.19
US_D = US_PYUS_Y-US_RKUS_KD-(1+US_TAUWF)US_WUS_ND ; // definition of dividends, used later in the budget constraint of the government to compute tax income from dividends
US_PHTILDE/US_PH = US_THETA/(US_THETA-1)US_FH/US_GH ;
US_FH = US_MCUS_H+US_XIHUS_BETAUS_LAMBDAI(+1)/US_LAMBDAI(US_PIH(+1)/(US_PIH^US_CHIHUS_PI4TARGET^(1/4(1-US_CHIH))))^US_THETAUS_FH(+1) ;
US_GH = US_PHUS_H+US_XIHUS_BETAUS_LAMBDAI(+1)/US_LAMBDAI*(US_PIH(+1)/(US_PIH^US_CHIHUS_PI4TARGET^(1/4(1-US_CHIH))))^(US_THETA-1)US_GH(+1) ;
US_PH^(1-US_THETA) = (1-US_XIH)US_PHTILDE^(1-US_THETA)+US_XIH(US_PH(-1)/US_PIC)^(1-US_THETA)(US_PIH(-1)^US_CHIHUS_PI4TARGET^(1/4(1-US_CHIH)))^(1-US_THETA) ;
US_PIH = US_PH/US_PH(-1)US_PIC ;
EA_PIMTILDE/EA_PIM = US_THETA/(US_THETA-1)US_FX/US_GX ;
US_FX = EA_SIZE/US_SIZEEA_IMUS_MC+US_XIXUS_BETAUS_LAMBDAI(+1)/US_LAMBDAI*(EA_PIIM(+1)/(EA_PIIM^US_CHIXUS_PI4TARGET^(1/4(1-US_CHIX))))^US_THETAUS_FX(+1) ;
US_GX = USEA_REREA_PIMEA_SIZE/US_SIZEEA_IM+US_XIXUS_BETAUS_LAMBDAI(+1)/US_LAMBDAI*(EA_PIIM(+1)/(EA_PIIM^US_CHIXUS_PI4TARGET^(1/4(1-US_CHIX))))^(US_THETA-1)US_GX(+1) ;
EA_PIM^(1-US_THETA) = (1-US_XIX)EA_PIMTILDE^(1-US_THETA)+US_XIX(EA_PIM(-1)/EA_PIC)^(1-US_THETA)(EA_PIIM(-1)^US_CHIXEA_PI4TARGET^(1/4(1-US_CHIH)))^(1-US_THETA) ;
EA_PIIM = EA_PIM/EA_PIM(-1)*EA_PIC ;
USEA_RER = US_RER/EA_RER ;
//Final-good firms
US_QC^((US_MUC-1)/US_MUC) = US_NUC^(1/US_MUC)US_HC^(1-1/US_MUC)+(1-US_NUC)^(1/US_MUC)((1-US_GAMMAIMC)US_IMC)^(1-1/US_MUC) ; // eq. 27
1^(1-US_MUC) = US_NUCUS_PH^(1-US_MUC)+(1-US_NUC)(US_PIM/US_GAMMAIMCDAG)^(1-US_MUC) ; // equation for the price of a unit of private consumption good
US_HC = US_NUCUS_PH^(-US_MUC)US_QC ; // eq. 30, demand function for domestic intermediate good
US_GAMMAIMC = US_GAMMAIMC1/2(US_IMC/US_QC/(US_IMC(-1)/US_QC(-1))-1)^2 ; // adjumstment cost of imported intermediate good used for final consumption good
US_GAMMAIMCDAG = 1-US_GAMMAIMC-US_GAMMAIMC1*(US_IMC/US_QC/(US_IMC(-1)/US_QC(-1))-1)US_IMC/US_QC/(US_IMC(-1)/US_QC(-1)) ; // second equation, page 2560
US_QI^((US_MUI-1)/US_MUI) = US_NUI^(1/US_MUI)US_HI^(1-1/US_MUI)+(1-US_NUI)^(1/US_MUI)((1-US_GAMMAIMI)US_IMI)^(1-1/US_MUI) ; // eq. 31, non-tradable final private investment good
US_PI^(1-US_MUI) = US_NUIUS_PH^(1-US_MUI)+(1-US_NUI)(US_PIM/US_GAMMAIMIDAG)^(1-US_MUI) ; // equation for the price of a unit of private investment good
US_HI = US_NUI*(US_PH/US_PI)^(-US_MUI)US_QI ; // demand function for domestic intermediate investment good bundle
US_GAMMAIMI = US_GAMMAIMI1/2(US_IMI/US_QI/(US_IMI(-1)/US_QI(-1))-1)^2 ; // adjumstment cost of imported intermediate good used for final investment good
US_GAMMAIMIDAG = 1-US_GAMMAIMI-US_GAMMAIMI1*(US_IMI/US_QI/(US_IMI(-1)/US_QI)-1)*US_IMI/US_QI/(US_IMI(-1)/US_QI(-1)) ; // second equation, page 2560
//Fiscal authority
US_PH(-1)*US_G(-1)+US_TR(-1)+US_B(-1)US_PIC(-1)^(-1)+US_M(-2)US_PIC(-1)^(-1) = US_TAUC(-1)US_C(-1)+(US_TAUN(-1)+US_TAUWH(-1))(US_WI(-1)US_NDI(-1)+US_WJ(-1)US_NDJ(-1))+US_TAUWF(-1)US_W(-1)US_ND(-1)+US_TAUK(-1)(US_RK(-1)US_U(-1)-(US_GAMMAU(-1)+US_DELTA)US_PI(-1))US_K(-1)+US_TAUD(-1)US_D(-1)+US_T(-1)+US_R(-1)^(-1)US_B+US_M(-1) ; // budget constraint, eq. 33
US_PHUS_G = US_GYUS_PYBARUS_YBAR ; //government purchases of the final public consumption good as a fraction of steady state nominal output
US_TR = US_TRYUS_PYBARUS_YBAR ; //government transfers as a fraction of steady state nominal output
US_T/(US_PYBARUS_YBAR) = US_PHITB(US_B/(US_PYBARUS_YBAR)-US_BYTARGET) ; // eq. 34, lump-sum taxes as fraction of steady state nominal output
US_TI = US_UPSILONTUS_T ;
US_TRI = US_UPSILONTRUS_TR ;
//Monetary authority
// original monetary policy rule at the beginning of the original model code
US_PIC4 = US_PIC*US_PIC(-1)*US_PIC(-2)*US_PIC(-3) ; // annual inflation
US_RR-1 = US_R/US_PIC(+1)-1 ; // real interest rate
US_RRSTAR-1 = 1/US_BETA-1 ; // equilibrium nominal interest rate
//Aggregation
US_C = (1-US_OMEGA)US_CI+US_OMEGAUS_CJ ; // consumption
US_M = (1-US_OMEGA)US_MI+US_OMEGAUS_MJ ; // money
US_K = (1-US_OMEGA)US_KI ; // capital
US_I = (1-US_OMEGA)US_II ; // investment
US_TRJ = (1/US_OMEGAUS_TR-(1-US_OMEGA)/US_OMEGAUS_TRI) ; // transfers to HH J
US_TJ = (1/US_OMEGAUS_T-(1-US_OMEGA)/US_OMEGAUS_TI) ; // taxes HH J
US_GAMMAV = (1-US_OMEGA)US_CIUS_GAMMAVI+US_OMEGAUS_CJUS_GAMMAVJ ; // total transaction costs
US_NI = US_SIUS_NDI ;
US_SI = (1-US_XII)(US_WITILDE/US_WI)^(-US_ETAI)+US_XII*(US_WI(-1)/US_WI)^(-US_ETAI)(US_PIC/(US_PIC(-1)^US_CHIIUS_PI4TARGET^(1/4*(1-US_CHII))))^US_ETAIUS_SI(-1) ;
US_NJ = US_SJUS_NDJ ;
US_SJ = (1-US_XIJ)(US_WJTILDE/US_WJ)^(-US_ETAJ)+US_XIJ(US_WJ(-1)/US_WJ)^(-US_ETAJ)(US_PIC/(US_PIC(-1)^US_CHIJUS_PI4TARGET^(1/4*(1-US_CHIJ))))^US_ETAJUS_SJ(-1) ;
US_UUS_K = US_KD ;
US_YS = US_SHUS_H+US_SXEA_SIZE/US_SIZEEA_IM ;
US_H = US_HC+US_HI+US_G ; // eq. 32
US_IM = US_IMI+US_IMC ; // eq. 32
US_SH = (1-US_XIH)(US_PHTILDE/US_PH)^(-US_THETA)+US_XIH*(US_PIH/(US_PIH(-1)^US_CHIHUS_PI4TARGET^(1/4(1-US_CHIH))))^US_THETAUS_SH(-1) ;
US_SX = (1-US_XIX)(EA_PIMTILDE/EA_PIM)^(-US_THETA)+US_XIX*(EA_PIIM/(EA_PIIM(-1)^US_CHIXUS_PI4TARGET^(1/4(1-US_CHIH))))^US_THETAUS_SX(-1) ;
US_QC = US_C+US_GAMMAV ;
US_QI = US_I+US_KUS_GAMMAU ;
US_PYUS_Y = US_QC+US_PIUS_QI+US_PHUS_G+USEA_REREA_PIMEA_SIZE/US_SIZEEA_IM-US_PIM*(US_IMC*(1-US_GAMMAIMC)/US_GAMMAIMCDAG+US_IMI*(1-US_GAMMAIMI)/US_GAMMAIMIDAG) ;
US_Y = US_YS ;
log(US_Z) = (1-US_RHOZ)log(US_ZBAR)+US_RHOZlog(US_Z(-1))+US_EPSZ ;
US_GY = (1-US_RHOG)US_GYBAR+US_RHOGUS_GY(-1)+US_EPSG ;
US_TRY = (1-US_RHOTR)US_TRYBAR+US_RHOTRUS_TRY(-1)+US_EPSTR ;
US_TAUC = (1-US_RHOTAUC)US_TAUCBAR+US_RHOTAUCUS_TAUC(-1)+US_EPSTAUC ;
US_TAUD = (1-US_RHOTAUD)US_TAUDBAR+US_RHOTAUDUS_TAUD(-1)+US_EPSTAUD ;
US_TAUK = (1-US_RHOTAUK)US_TAUKBAR+US_RHOTAUKUS_TAUK(-1)+US_EPSTAUK ;
US_TAUN = (1-US_RHOTAUN)US_TAUNBAR+US_RHOTAUNUS_TAUN(-1)+US_EPSTAUN ;
US_TAUWH = (1-US_RHOTAUWH)US_TAUWHBAR+US_RHOTAUWHUS_TAUWH(-1)+US_EPSTAUWH ;
US_TAUWF = (1-US_RHOTAUWF)US_TAUWFBAR+US_RHOTAUWFUS_TAUWF(-1)+US_EPSTAUWF ;
US_CY = US_C/(US_PYUS_Y) ;
US_IY = US_PIUS_I/(US_PYUS_Y) ;
US_IMY = US_PIMUS_IM/(US_PYUS_Y) ;
US_IMCY = US_PIMUS_IMC/(US_PYUS_Y) ;
US_IMIY = US_PIMUS_IMI/(US_PYUS_Y) ;
US_BY = US_B/(US_PYBARUS_YBAR) ;
US_TY = US_T/(US_PYBARUS_YBAR) ;
US_YGAP = US_Y/US_YBAR-1 ;
US_YGROWTH = US_Y/US_Y(-1) ;
US_YSHARE = US_SIZEUS_PYUS_Y/US_RER/(US_SIZEUS_PYUS_Y/US_RER+EA_SIZEEA_PYEA_Y/EA_RER) ;
US_EPSILONM = -1/81/(US_R*(US_GAMMAV2US_R+US_R-1)) ;
1 = US_R(1-EA_GAMMAB)EA_BETAEA_LAMBDAI(+1)/EA_LAMBDAIEA_RERDEP(+1)/US_PIC(+1) ;
EA_GAMMAB = EA_GAMMAB1(exp(EA_REREA_BF/US_PIC/(EA_PYEA_Y)-EA_BFYTARGET)-1)-EA_RP ;
EA_RP = EA_RHORPEA_RP(-1)+EA_EPSRP ;
EA_RERDEP = EA_RER/EA_RER(-1) ;
EA_TOT = EA_PIM/(EA_RERUS_PIM) ;
EA_TB = EA_RERUS_PIMUS_SIZE/EA_SIZEUS_IM-EA_PIMEA_IM ;
EA_BF/US_R(-1) = EA_BF(-1)+EA_TB(-1)/EA_RER(-1) ;
EA_SIZEEA_BF+US_SIZEUS_BF = 0 ;
end;
initval;
EAUS_RER=0.937575549426388;
EA_B=8.76096481337854;
EA_BF=0;
EA_BY=2.4;
EA_C=2.18953931985619;
EA_CI=2.30653728902394;
EA_CJ=1.83854541235293;
EA_CY=0.599807724330806;
EA_D=8.130685761198e-016;
EA_FH =23.3942599750924;
EA_FI =130.027056955912;
EA_FJ =4.81581692429301;
EA_FX =0.779768566246692;
EA_G =0.652856601368441;
EA_GAMMAB =0;
EA_GAMMAI =0;
EA_GAMMAIDER =0;
EA_GAMMAIMC =0;
EA_GAMMAIMCDAG =1;
EA_GAMMAIMI =0;
EA_GAMMAIMIDAG =1;
EA_GAMMAU =0;
EA_GAMMAUDER =0.0340909090909092;
EA_GAMMAV =0.00070188348844825;
EA_GAMMAVI =0.000320562175834483;
EA_GAMMAVIDER =0.0218035974789136;
EA_GAMMAVJ =0.000320562175834478;
EA_GAMMAVJDER =0.0218035974789136;
EA_GH =28.0731119701109;
EA_GI =9.36344272645511;
EA_GJ =4.91231876574661;
EA_GX =0.935722279496031;
EA_GY =0.18;
EA_H =2.97412451734512;
EA_HC =1.99483961529246;
EA_HI =0.326428300684219;
EA_I =0.835577951476644;
EA_II =1.11410393530219;
EA_IM =0.706271096709553;
EA_IMC =0.196186415752654;
EA_IMCY =0.0500000000000002;
EA_IMI =0.510084680956899;
EA_IMIY =0.13;
EA_IMY =0.18;
EA_IY =0.22;
EA_K =33.4231180590658;
EA_KD =33.4231180590658;
EA_KI =44.5641574120877;
EA_LAMBDAI =0.97925563259573;
EA_LAMBDAJ =1.54123305743258;
EA_M =3.45363335385316;
EA_MC =0.838714504361949;
EA_MI =3.63817815055376;
EA_MJ =2.89999896375135;
EA_ND =1.81678486262395;
EA_NDI =0.877517383251523;
EA_NDJ =0.994178476830804;
EA_NI =0.877517383251523;
EA_NJ =0.994178476830804;
EA_PH =1.00645740523434;
EA_PHTILDE =1.00645740523434;
EA_PI =0.961117319822928;
EA_PIC =1.0049629315732;
EA_PIC4 =1.02;
EA_PIH =1.0049629315732;
EA_PIIM =1.0049629315732;
EA_PIM =0.930340154176811;
EA_PIMTILDE =0.930340154176811;
EA_PY =1.00645740523434;
EA_Q =0.961117319822928;
EA_QC =2.19024120334464;
EA_QI =0.835577951476644;
EA_R =1.01241680983897;
EA_RER =0.937575549426388;
EA_RERDEP =1;
EA_RK =0.0327653631757817;
EA_RP =0;
EA_RR =1.00741706786548;
EA_SH =1;
EA_SI =1;
EA_SJ =1;
EA_SX =1;
EA_T =0;
EA_TAUC =0.183;
EA_TAUD =0;
EA_TAUK =0.184122534797561;
EA_TAUN =0.122;
EA_TAUWF =0.219;
EA_TAUWH =0.118;
EA_TB =0;
EA_TI =0;
EA_TJ =0;
EA_TOT =0.924371115298412;
EA_TR =0.712415745652017;
EA_TRI =0.47494382568524;
EA_TRJ =1.42483150555235;
EA_TRY =0.195160901337473;
EA_TY =0;
EA_U =1;
EA_UTILI =-177.80959564505;
EA_UTILJ =-229.178322918907;
EA_VJ =0.75;
EA_W =1.153802633989;
EA_WI =1.2416037978402;
EA_WITILDE =1.2416037978402;
EA_WJ =1.01257698454711;
EA_WJTILDE =1.01257698454711;
EA_Y =3.62698111871356;
EA_YGAP =0;
EA_YGROWTH =1;
EA_YS =3.62698111871356;
EA_YSHARE =0.419354838709677;
EA_Z =1;
EA_EPSILONM=-0.75;
EA_RRSTAR =1.00741706786548;
EA_VI=0.75;
USEA_RER = 1.06658071513469;
US_B = 9.34600933441895;
US_BF = 0;
US_BY = 2.4;
US_C = 2.41414187416651;
US_CI = 2.69734517849423;
US_CJ = 1.56453196118335;
US_CY = 0.619937375480895;
US_D = 1.01192186569047e-014;
US_FH = 26.4782322019398;
US_FI = 828.361762914362;
US_FJ = 30.6800652931245;
US_FX = 0.600773607720124;
US_G = 0.627912976941598;
US_GAMMAI = 0;
US_GAMMAIDER = 0;
US_GAMMAIMC = 0;
US_GAMMAIMCDAG = 1;
US_GAMMAIMI = 0;
US_GAMMAIMIDAG = 1;
US_GAMMAU = 0;
US_GAMMAUDER = 0.0340909090909093;
US_GAMMAV = 0.000243870558389395;
US_GAMMAVI = 0.000101017492384781;
US_GAMMAVIDER = 0.00216519376842896;
US_GAMMAVJ = 0.000101017492384779;
US_GAMMAVJDER = 0.00216519376842894;
US_GH = 31.7738786423278;
US_GI = 14.3472077461836;
US_GJ = 14.2151126631918;
US_GX = 0.720928329264149;
US_GY = 0.16;
US_H = 3.41427681211994;
US_HC = 2.1976954192956;
US_HI = 0.588668415882739;
US_I = 0.841760867101418;
US_II = 1.12234782280189;
US_IM = 0.471595002779754;
US_IMC = 0.217659232052194;
US_IMCY = 0.0599999999999999;
US_IMI = 0.25393577072756;
US_IMIY = 0.0700000000000001;
US_IMY = 0.13;
US_IY = 0.219999999999998;
US_K = 33.6704346840567;
US_KD = 33.6704346840567;
US_KI = 44.8939129120756;
US_LAMBDAI = 0.793738951061822;
US_LAMBDAJ = 2.35929293297691;
US_M = 1.09244991532661;
US_MC = 0.826902389133687;
US_MI = 1.22060536018415;
US_MJ = 0.707983580753977;
US_ND = 2.02692319692714;
US_NDI = 0.842738211796422;
US_NDJ = 1.28040341539794;
US_NI = 0.842738211796422;
US_NJ = 1.28040341539794;
US_PH = 0.992282866960424;
US_PHTILDE = 0.992282866960425;
US_PI = 1.01776829477927;
US_PIC = 1.0049629315732;
US_PIC4 = 1.02;
US_PIH = 1.0049629315732;
US_PIIM = 1.0049629315732;
US_PIM = 1.07346805902745;
US_PIMTILDE = 1.07346805902745;
US_PY = 0.992282866960427;
US_Q = 1.01776829477927;
US_QC = 2.4143857447249;
US_QI = 0.841760867101418;
US_R = 1.01241680983897;
US_RK = 0.0346966464129299;
US_RR = 1.00741706786548;
US_SH = 0.999999999999997;
US_SI = 1;
US_SJ = 1;
US_SX = 1;
US_T = 0;
US_TAUC = 0.077;
US_TAUD = 0;
US_TAUK = 0.184122534797571;
US_TAUN = 0.154;
US_TAUWF = 0.071;
US_TAUWH = 0.071;
US_TI = 0;
US_TJ = 0;
US_TR = 0.310489186066476;
US_TRI = 0.206992788641056;
US_TRJ = 0.620978378342736;
US_TRY = 0.0797317892477657;
US_TY = 0;
US_U = 1;
US_UTILI = -152.9846874364;
US_UTILJ = -312.074198693985;
US_VJ = 2.38;
US_W = 1.25570101402423;
US_WI =1.38543840813496;
US_WITILDE = 1.38543840813496;
US_WJ = 1.0759481039628;
US_WJTILDE = 1.0759481039628;
US_Y = 3.92445610588497;
US_YGAP = 0;
US_YGROWTH = 1;
US_YS = 3.92445610588497;
US_YSHARE = 0.580645161290323;
US_Z = 1;
US_VI = 2.38;
US_EPSILONM=-0.75;
US_RRSTAR =1.00741706786548;
end;
steady(solve_algo=3, maxit=100);
resid(1);
check; //=> to check the Blanchard and Khan conditions
shocks;
var EA_EPSR=0;
var EA_EPSRP=0;
var EA_EPSTAUC=0;
var EA_EPSTAUD=0;
var EA_EPSTAUK=0;
var EA_EPSTAUN=0;
var EA_EPSTAUWF=0;
var EA_EPSTAUWH=0;
var EA_EPSTR=0;
var EA_EPSZ=0;
var US_EPSG=0;
var US_EPSR=0;
var US_EPSTAUC=0;
var US_EPSTAUD=0;
var US_EPSTAUK=0;
var US_EPSTAUN=0;
var US_EPSTAUWF=0;
var US_EPSTAUWH=0;
var US_EPSTR=0;
var US_EPSZ=0;
end;
stoch_simul (order=1, irf = 20, ar=100);