//*************************************************************************************************************************************************************************************************** // A TWO COUNTRY MONETARY UNION MODEL //*************************************************************************************************************************************************************************************************** // // // Observational variables // // gy : domestic gdp // gyea : union gdp // gcpi : domestic inflation // gcpiea : union inflation // gc : consumption growth // gi : investment growth // gl : employment growth // gw : wage inflation // // Model variables // // LY_A : log gdp over technology // LK_A : log capital over technology // TB_Z : log trade balance over domestic absorption // LI : log investment // GUC : consumtpion utility growth // LC : log consumtion // Q : Tobin's Q // FApf : foreign recursive price variable A // FBpf : foreign recursive price variable B // MC_star : union marginal cost // LPfcop : optimal foreign real price // RK : return on capital // LY : log gdp // FAph : domestic recursive price variable A // FBph : domestic recursive price variable BA // LPhcop : optimal foreign real price // LA : technological level // PIwop : inflation of optimal wage // Ld : labor demand // LWop : optimal wage // FAw : domestic recursive wage variable A // FBw : domestic recursive wage variable B // Sw // lamda : lagrange multiplier // PI : doemstic inflation // PIh : domestic producer inflation // PIf : foreign producer inflation and foreign inflation // LPhc : domestic real price // LPfc : foreign real price // gyea : foreign output growth // G_C : consumption growth // G_I : investment growth // G_Z : absorption growth // G_K : capital growth // G_L : employment growth // G_A : technological growth // G_Y : output growth // PIw : wage inflation // LY_Z : log gdp to absorption ratio // LC_Z : log consumption to absorption ratio // LZ_Z_star: log domestic over foreign absorption // LI_K : log investment to capital ratio // LI_Z : log investment to absorption ratio // LZ : log absorption // LK : log capital // LY_star : log foreign gdp // LW : log real wage // LL : log employment // MC : log marginal cost // MUL ; log marginal utility of labor // MUC_Z ; log MUc over absorptio n // R : interest rate // //*************************************************************************************************************************************************************************************************** // VARIABLES DEFINITION //*************************************************************************************************************************************************************************************************** var G_Y_star LY_A LK_A TB_Z LI GUC LC Q FApf FBpf MC_star LPfcop RK LY FAph FBph LPhcop LA PIwop Ld LWop FAw FBw Sw lamda lamda_star PI PIh PIf LPhc LPfc G_C G_I G_Z G_K G_L G_A G_Y PIw LY_Z LC_Z LZ_Z_star LI_K LI_Z LZ LK LY_star LW LL MC MUL R E_K E_PI E_P E_M E_L E_PIF E_Y_star E_TB_Z LD_Z G_D; // r gy gyea gcpi gcpiea gc gi gl gw varexo EPS_A EPS_PI EPS_R EPS_K EPS_P EPS_MC_star EPS_PIF EPS_TB_Z EPS_Y_star EPS_L; // parameters betae alphae h sigmae phi eta rhoA rhoL rhoPIF rhoPI rhoR rhoM rhoK rhoW rhoP rhoY_star rhoTB_Z rhoMC_star phi1 phi2 phi3 phi4 gamai deltae gamap G G_star Infl alphae_star xi viw etaw viph etaph etapf vipf MC_ss LW_ss LL_ss MUC_Z_ss R_ss RK_ss I_Y C_Y LY_star_ss LWop_ss lamda_ss MUL_ss LPhcop_ss LPhc_ss LPfc_ss Ld_ss Sw_ss LC_Z_ss LZ_Z_star_ss MC_star_ss LY_ss LPfcop_ss; gamai = 2; //InveStment adjuStment coSt parameter deltae = 0.025 ; //DEprECIATION RATE viw = 0.6 ; //Calvo parameters for wage etaw = 3.5 ; // elaSticity of subStitution of labor service / elasticity of substitution between labor varieties gamap = 0.55 ; //wage share in output viph = 0.5 ; // fraction of firms with non-reoptimizing price etaph = 2 ; //price elasticity of demand for a specific varieties of good vipf = 0.7 ; etapf = 2.5 ; alphae_star = 0.005; // Home bias of the foreign economy alphae = 0.85; // Home bias betae = 0.9925; //9926 to match observations Rate of time preferences h = 0.51; // Habit persiStence sigmae = 2.5; // coefficient of relative risk aversion phi = 2; // Inverse labor elaSticity eta = 1.5; // ElaSticity of subStitution between domeStic and foreign goods //Growth rate Steady Swate G_star = 0.0025; // Union steady_state growth G = 0.0025; // Home country steady state growth Infl = 0.004; // Steadystate inflation //AR parameters rhoA = 0.9; rhoPI = 0.9; rhoPIF = 0.7659; rhoL = 0.3371; rhoR = 0.9043; rhoM = 0; rhoK = 0.99; rhoW = 0.85; rhoP = 0.99; rhoY_star = 0.8963; rhoTB_Z = 0; rhoMC_star = 0.9; //Monetary policy parameters phi1 = 0;//1.7906 ; phi2 = 0;//.3259; phi3 = 0;//.1059; phi4 = 0;//0.0471; etaws = 2:0.5:5; viws = 0.6:0.05:0.8; etaphs = 2:0.5:5; viphs = 0.5:0.05:0.8; etapfs = 2:0.5:5; vipfs = 0.6:0.05:0.8; etas = 1.5:0.5:5; hs=0.5:0.075:0.8; //Results=[]; //for i=1:length(etaws) // etaw = etaws(i); // for j=1:length(viws) // viw = viws(j); // for k=1:size(etaphs) // etaph=etaphs(k); // for l=1:size(viphs) // viph=viphs(l); // for m=1:size(etapfs) // etapf=etapfs(m); // for n=1:size(vipfs) // viph=viphs(n); // for o=1:size(etas) // eta=etas(o); // for p=1:size(hs) // h=hs(p); //Steady-State value R_ss = 1/betae-1+Infl; //(1-gamap)*(deltae+G)*(1/I_Y)+(1-deltae); RK_ss = 1/betae-1; LPhcop_ss = log( ( (1-viph*(1+Infl)^(etaph-1)) / (1-viph))^(1/(1-etaph)) ); LPhc_ss = 0 ; LPfc_ss = 0 ; LY_ss = 1 ; MC_ss = (etaph-1)/etaph * exp(LPhcop_ss)*(1-viph*betae*(1+G)^(1-sigmae) * (1+Infl)^etaph) / (1-viph*betae * (1+G)^(1-sigmae) * (1+Infl)^(etaph-1)); I_Y = ((1-gamap) * MC_ss * (deltae+G)) / ((1/betae - 1+deltae+G)) ; // Investment share C_Y = 1 - I_Y ; // Consumption share LC_Z_ss = log(C_Y) ; LZ_Z_star_ss= log(alphae_star/alphae); Sw_ss = 1 ; LK_ss = LY_ss + log(I_Y) - log(deltae+G); Ld_ss = LY_ss/gamap - (1-gamap)/gamap*LK_ss; //1/gamap - (1-gamap)/gamap*( log(1-gamap)+1-log(1/betae-(1-deltae-G)) ) ; LL_ss = log(Sw_ss) + Ld_ss ; LW_ss = log(gamap)+log(MC_ss) + LY_ss - Ld_ss ; MUC_Z_ss = -sigmae * ( LC_Z_ss+log(1-h) ) ; lamda_ss = exp( LC_Z_ss+LY_ss+log(1-h) )^-sigmae ; LWop_ss = LW_ss ; xi = (1-viw*betae)/(1-viw*betae*(1+G)^-sigmae*(1+Infl))*(etaw-1)/etaw*exp(LWop_ss)*lamda_ss/exp(LL_ss)^phi ; //Value implied by the model to ensure a steady state LY_star_ss = LY_ss + log(alphae/alphae_star) + (LPfc_ss-LPhc_ss)*(1-eta) ; MUL_ss = -xi*exp(LL_ss)^phi ; LPfcop_ss = log( ( ( 1 - vipf*(1+Infl)^(etapf-1) ) / (1-vipf) )^(1/(1-etapf)) ) ; MC_star_ss = (etapf-1)/etapf * exp(LPfcop_ss) * ( 1 - vipf*betae* (1+G_star)^(1-sigmae) * (1+Infl)^etapf ) / ( 1 - vipf*betae* (1+G_star)^(1-sigmae) * (1+Infl)^(etapf-1) ); model; //*************************************************************************************************************************************************************************************************** // UTILITY BLOCK (Equation 1 to 5) //*************************************************************************************************************************************************************************************************** // Marginal utility of labor MUL = -xi*(exp(LL))^phi * exp(E_L); //Marginal utility of consumption lamda = (exp(LC)*(1-h/(1+G_C-G)))^-sigmae * exp(E_P); // Marginal utility growth, not usaed in the model GUC = -sigmae*G_C + log(lamda) - log(lamda(-1)); // Union Marginal utility of consumption lamda_star= log(((C_Y * (exp(LY_star)-h*exp(LY_star(-1)))))^-sigmae);//*exp(E_P) //*************************************************************************************************************************************************************************************************** // prODUCTION BLOCK (Equation 6 to 7) //*************************************************************************************************************************************************************************************************** //Home Aggregate price index 1 = ((1-alphae)*(exp(LPhc))^(1-eta) + alphae*(exp(LPfc))^(1-eta))^(1/(1-eta)) ; // Production per unit of technology LY_A = (LK_A)*(1-gamap) + (Ld)*gamap; //*************************************************************************************************************************************************************************************************** // CONSUMPTION BLOCK (Equation 8 to 10) //*************************************************************************************************************************************************************************************************** //Euler eq (lamda) =(lamda(+1))/((betae)*(1+R-PI(+1))); // Perfect risk sharing equation combined with Euler consumption of the foreign economy, domestic househoLds are taking into expectations how the foreign economy is evolving to adjust their consumption //(lamda) = (alphae_star/alphae)^-sigmae*exp(lamda_star(+1))/(betae*(1+R-PIf(+1)))*exp(LPfc); // Euler equation for the foreign economy //exp(lamda_star)= exp(lamda_star(+1))/(betae*(1+R-PIf(+1))); //*************************************************************************************************************************************************************************************************** // INVESTMENT BLOCK (Equation 11 to 13) //*************************************************************************************************************************************************************************************************** // Tobin's Q equation Q = 1/((1+R-PI(+1)))*( 1+RK(+1)-(1-deltae-G)+(1-deltae-G)*Q(+1)); 1 = Q*exp(E_K)*(1-gamai/2*(G_I - G)^2-gamai*(G_I - G)*G_I) + betae*Q(+1)*exp(E_K(+1))*gamai*(G_I(+1) - G)*G_I(+1)^2; G_K - G = exp(LI_K)*(1-gamai/2*(G_I - G)^2)*exp(E_K) - deltae-G;// //*************************************************************************************************************************************************************************************************** // DOMESTIC CALVO prICE BLOCK (Equation 14 to 17) //*************************************************************************************************************************************************************************************************** FAph = ((1-alphae)*exp(LZ)+alphae_star*exp(LY_star)*exp(LPhc-LPfc)^-eta)* MC * exp(LPhcop)^(-etaph-1) + viph*betae*lamda(+1)*(1+G)^-sigmae/(lamda)*exp(LPhcop-LPhcop(+1))^(-etaph-1)*(1/(1+PI(+1)))^-etaph*(1+G)*FAph(+1) ; FBph = ((1-alphae)*exp(LZ)+alphae_star*exp(LY_star)*exp(LPhc-LPfc)^-eta) * exp(LPhcop)^(-etaph) + viph*betae*lamda(+1)*(1+G)^-sigmae/(lamda)*exp(LPhcop-LPhcop(+1))^(-etaph)*(1/(1+PI(+1)))^(1-etaph)*(1+G)*FBph(+1) ; etaph * FAph= (etaph-1)* FBph *exp(E_PI); 1 = viph*(1+PIh)^(etaph-1) + (1-viph)*(exp(LPhcop)*exp(E_TB_Z))^(1-etaph) ; //*************************************************************************************************************************************************************************************************** // UNION CALVO prICE BLOCK (Equation 18 to 21) //*************************************************************************************************************************************************************************************************** FApf = exp(LY_star)*MC_star*exp(LPfcop)^(-etapf-1) + vipf*betae*lamda_star(+1)/(lamda_star)*exp(LPfcop-LPfcop(+1))^(-etapf-1)*(1/(1+PIf(+1)))^-etapf*(1+G_star)^(1-sigmae)*FApf(+1); FBpf = exp(LY_star)*exp(LPfcop)^(-etapf) + vipf*betae*lamda_star(+1)/(lamda_star)*exp(LPfcop-LPfcop(+1))^(-etapf)*(1/(1+PIf(+1)))^(1-etapf)*(1+G_star)^(1-sigmae)*FBpf(+1) ; //FApf = exp(LY_star)*MC_star*exp(LPfcop)^(-etapf-1) + vipf*betae*(betae*(1+R-PIf(+1)))*exp(LPfcop-LPfcop(+1))^(-etapf-1)*(1/(1+PIf(+1)))^-etapf*(1+G_star)^(1-sigmae)*FApf(+1) ; //FBpf = exp(LY_star)*exp(LPfcop)^(-etapf) + vipf*betae*(betae*(1+R-PIf(+1)))*exp(LPfcop-LPfcop(+1))^(-etapf)*(1/(1+PIf(+1)))^(1-etapf)*(1+G_star)^(1-sigmae)*FBpf(+1) ; etapf * FApf = (etapf-1) * FBpf*exp(E_PIF); 1 = vipf*(1+PIf)^(etapf-1) + (1-vipf)*(exp(LPfcop))^(1-etapf) ; //*************************************************************************************************************************************************************************************************** // MC BLOCK (Equation 20 to 22) //*************************************************************************************************************************************************************************************************** MC * (1-gamap) * exp(LK(-1))^ -gamap * exp(Ld+LA)^ gamap = 1+RK - (1-deltae-G) ; MC * gamap*exp(LA) * exp(LK(-1))^(1-gamap) * exp(Ld+LA)^(gamap-1) = exp(LW) ; MC_star = (1-rhoMC_star)*MC_star_ss + rhoMC_star*MC_star(-1) +EPS_MC_star;//+E_TB_Z; //*************************************************************************************************************************************************************************************************** // CALVO WAGE SETTING BLOCK (Equation 23 to 28) //*************************************************************************************************************************************************************************************************** FAw = (etaw-1)/etaw * exp(LWop)*(lamda)*exp(LW-LWop)^etaw*exp(Ld) + viw*betae*((1+PI(+1))/((1+G)*(1+PI)))^(etaw-1)*(1+PIwop)^(etaw-1)*FAw(+1)*((1+Infl)*(1+G)^-sigmae); FBw = - MUL*exp(LWop-LW)^-etaw*exp(Ld) + viw*betae*((PIwop(+1)+1)*(1+PI(+1))/((1+G)*(1+PI)))^etaw*FBw(+1) ; FAw = FBw ; exp(LL) = Sw*exp(Ld) ; Sw = (1-viw)*(exp(LWop-LW))^(-etaw) + viw*((1+PIw-G) * (1+PI(-1))/(1+PI))^(-etaw) * Sw(-1) ; 1 = (1-viw)*exp(LWop-LW)^(1-etaw) + viw*( (1+G)/(1+PIw) * (1+PI(-1))/(1+PI) ) ^(1-etaw) ; //*************************************************************************************************************************************************************************************************** // RESSOURCE CONSTRAINT BLOCK (Equation 29 to 31) //*************************************************************************************************************************************************************************************************** //gdp/absorption = (home production + export) exp(LY_Z) = ( (1-alphae)*exp(LPhc)^(1-eta) + alphae_star/exp(LZ_Z_star)*exp((LPhc-LPfc))^-eta); //gdp/absorption= 1 + TradeBalance/absorption exp(LY_Z)=TB_Z + 1 ; // 1 = invest/absorption + consump/absorption 1 = exp(LI_Z) + exp(LC_Z) ; //Trqde bqlqnce definition TB_Z = -alphae*exp(LPfc)^(1-eta) + alphae_star*exp(LPhc)*exp(LPhc- LPfc)^(-eta)/exp(LZ_Z_star); //Current account condition requiered to close the model following Schmitt-Grohe 2001, equation divided by debt/absorption TB_Z/exp(LD_Z) =(1+G_D)/(1+R-0.000742*((exp(LD_Z)-0.6))) -1; //real debt growth relative to absorption growth G_D -G_Z = R_ss - G + LD_Z - LD_Z(-1); //*************************************************************************************************************************************************************************************************** // DEFINITION BLOCK (Equation 32 to 51) //*************************************************************************************************************************************************************************************************** // Euro area gdp G_Y_star - G_star= + (LY_star - LY_star(-1)); LY_star = (1-rhoY_star)*LY_star_ss + rhoY_star* LY_star(-1) +(E_Y_star); //Log Technology level LA = rhoA *LA(-1)+ EPS_A;// LA(-1) ;//+ pr * E_Y_star //G_A = G + (rhoA -1)* LA(-1)+EPS_A;//+ EPS_A; //Dom GDP growth G_Y = G + LY - LY(-1) ; G_C = G + LC - LC(-1) ; //Domest consumption growth G_K = G + LK - LK(-1) ; //D capital g G_I = G + LI - LI(-1) ; //D investment g G_Z = G + LZ - LZ(-1) ; //D absorption g G_L = Ld - Ld(-1) ; //D labor demand g PIwop = G + LWop - LWop(-1) ;//D optimal real wage inflation PIw = G + LW - LW(-1) ; //D real wage inflation PIh - PI = LPhc - LPhc(-1) ; //D home price i PIf - PI = LPfc - LPfc(-1) ; //D foreign price i LI_K = LI - LK(-1) ; //D log investment to capital ratio LZ - LY_star = LZ_Z_star ; //Relative absorption (foreign absorption is assumed to equal foreign GDP) LC_Z = LC - LZ ; //Log consumption to absorption rqtio LI_Z = LI - LZ ; //L investment t LY_Z = LY - LZ ; //L gdp t LY_A = LY-LA; //L g to technology ratio LK_A = LK(-1)-LA; //L capital t //*************************************************************************************************************************************************************************************************** // SHOCKS (Equation 52 to 59) //*************************************************************************************************************************************************************************************************** E_TB_Z = rhoTB_Z*E_TB_Z(-1) + EPS_TB_Z; E_K = rhoK*E_K(-1) + EPS_K; E_PI = rhoPI*E_PI(-1) + EPS_PI; E_PIF = rhoPIF*E_PIF(-1) + EPS_PIF; E_M = rhoM*E_M(-1) + EPS_R; E_P = rhoP*E_P(-1) + EPS_P; E_L = rhoL*E_L(-1) + EPS_L; E_Y_star= rhoY_star*E_Y_star(-1) + EPS_Y_star; //*************************************************************************************************************************************************************************************************** // MONETARY POLICY BLOCK (Equation 60) //*************************************************************************************************************************************************************************************************** R = rhoR*R(-1) + (1-rhoR)*(R_ss +phi1*(PIf-Infl) + phi2*(LY_star-LY_star_ss))+ phi3*(PIf-PIf(-1)) + phi4*(LY_star-LY_star(-1)) + E_M; //*************************************************************************************************************************************************************************************************** end; // CALCULATION OF THE StEADY StTE steady; check; shocks; var EPS_A; stderr 0; var EPS_PI; stderr 0.0011; var EPS_PIF; stderr 0.0151; var EPS_R; stderr 0; var EPS_K; stderr 0; var EPS_P; stderr 0; var EPS_L; stderr 0; var EPS_MC_star ; stderr 0; var EPS_TB_Z; stderr 0; var EPS_Y_star; stderr 0.0021; end; //OBSERVED VARIABLES //options_.ftol=1.e-6; estimated_params; phi1, normal_pdf, 1.7, 0.1; //phi2, normal_pdf, 0.125, 0.1; //phi3, normal_pdf, 0.3, 0.06; //phi4, normal_pdf, 0.0625, 0.05; stderr EPS_MC_star, GAMMA_PDF, 0.01, 0.006, 0, inf; stderr EPS_A, GAMMA_PDF, 0.01, 0.006, 0, inf; stderr EPS_PIF, GAMMA_PDF, 0.01, 0.006, 0, inf; stderr EPS_R, GAMMA_PDF, 0.01, 0.006, 0, inf; stderr EPS_K, GAMMA_PDF, 0.01, 0.006, 0, inf; stderr EPS_P, GAMMA_PDF, 0.01, 0.006, 0, inf; stderr EPS_L, GAMMA_PDF, 0.01, 0.006, 0, inf; stderr EPS_PI, GAMMA_PDF, 0.01, 0.006, 0, inf; stderr EPS_TB_Z, GAMMA_PDF, 0.005, 0.003, 0, inf; stderr EPS_Y_star, GAMMA_PDF,0.01, 0.006, 0, inf; //set to the observed relative variance of gdp growth rhoA, BETA_PDF, 0.85, 0.075,0,0.99; rhoR , BETA_PDF, 0.85, 0.075,0,0.99; //rhoM , BETA_PDF, 0.85, 0.1,0,0.99; rhoPI, BETA_PDF, 0.85, 0.1,0,0.99; rhoPIF, BETA_PDF, 0.85, 0.1,0,0.99; rhoL, BETA_PDF, 0.85, 0.1,0,0.99; rhoY_star, BETA_PDF, 0.85, 0.1,0,0.99; rhoK, BETA_PDF, 0.85, 0.1,0,0.99; rhoP, BETA_PDF, 0.85, 0.1,0,0.99; rhoTB_Z, BETA_PDF, 0.5, 0.2,0,0.99; //gamai, UNIFORM_PDF, 5, 4, 0, inf; end; varobs G_Y G_Y_star PI PIf G_I G_C PIw R G_L; close all; estimation(datafile=datae , //mode_file = MME7_mode, mode_compute=5, mode_check, mh_replic=0//nograph, ); // Results=[Results; oo_.MarginalDensity]; //stoch_simul(order=1,nograph); // end; // end; // end; // end; // end; // end; // end; //end; //shock_decomposition G_Y G_Y_star PI PIf G_I G_C PIw R G_L; Stoch_simul(order=1,nograph);