% ----- -------------------------------------------------------------- % Endogenous Variables % ----- -------------------------------------------------------------- var nubeta % discount factor shock (domestic) nun % labor supply shock (domestic) nuc % taste shock (domestic) nui % investment shock (domestic) nup % productivity shock (domestic) tauyd % output tax rate (domestic) taun % labor tax rate (domestic) tauk % capital tax rate (doemstic) taud % tax rate on doemstic consumption (domestic) taum % tax rate on imported consumption (domestic) gd % consumption on domestic goods by government (domestic) gm % consumption on imported goods by government (domestic) g % total governemnt spending (domestic) pid % domestic inflation rate (domestic) pi % cpi inflation (domestic) pidopt % reset price inflation (domestic) psigma % price dispersion (domestic) f % numerator of reset price inflation (domestic) gpr % denominator of reset price inflation (domestic) x % total consumption (domestic) c % total habit adjusted consumption in steady state (domestic) cd % domestic consumption (domestic) cm % imported consumption (domestic) n % labor supply (domestic) w % real wage rate (domestic) r % real rate of return on capital (domestic) k % stock of future capital (domestic) i % investment (domestic) yd % domestic output supplied/demanded (domestic) lambda2 % multiplier of investment/ marginal utility of investment (domestic) mc % marginal cost in steady state (domestic) lambda1 % real marginal utility of money (domestic) ydnat % natural rate of domestic output in steady state (domestic) % foreign endo variables nubetas % discount factor shock (foreign) nuns % labor supply shock (foreign) nucs % taste shock (foreign) nuis % investment shock (foreign) nups % productivity shock (foreign) tauyds % output tax rate (foreign) tauns % labor tax rate (foreign) tauks % capital tax rate (foreign) tauds % tax rate on doemstic consumption (foreign) taums % tax rate on imported consumption (foreign) gds % consumption on domestic goods by government (foreign) gms % consumption on imported goods by government (foreign) gs % total governemnt spending (foreign) pids % domestic inflation rate (foreign) pis % cpi inflation (foreign) pidopts % reset price inflation (foreign) psigmas % price dispersion (foreign) fs % numerator of reset price inflation (foreign) gprs % denominator of reset price inflation (foreign) xs % total consumption (foreign) cs % total habit adjusted consumption (foreign) cds % domestic consumption (foreign) cms % imported consumption (foreign) ns % labor supply (foreign) ws % real wage rate (foreign) rs % real rate of return on capital (foreign) ks % stock of future capital (foreign) is % investment (foreign) yds % domestic output supplied/demanded (foreign) lambda2s % multiplier of investment/ marginal utility of investment (foreign) mcs % marginal cost (foreign) lambda1s % real marginal utility of money (foreign) ydnats % natural rate of domestic output (foreign) num % monetary policy shock (cu) icu % nominal interest rate (cu) ygapcu % output gap (cu) picugap % inflation gap (cu) picu % inflation (cu) picunat % natural steady state inflation (cu) tau % terms of trade between two countries (cu) ycu % currency union output ycun % natural rate of currency union output ; % ----- -------------------------------------------------------------- % Exogenous Variables % ----- -------------------------------------------------------------- varexo ebeta % discount factor shock error (domestic) en % labor supply shock error (domestic) ecshock % taste shock error (domestic) ei % investment shcok error (domestic) ep % productivity shock error (domestic) //etauyd //etaun //etauk //etaud //etaum epsg ebetas ens ecshocks eis eps //etauyds //etauns //etauks //etauds //etaums epsgs em ; predetermined_variables k ks; parameters rhob rhon rhoc rhoi rhop rhotauyd rhotaun rhotauk rhotaud rhotaum el eg ej om omg rhog xi mun pidss % steady state domestic inflation rate (domestic) pidsss % steady state domestic inflation rate (foreign) muc ec betta h kappa zeta // population proportion (domestic) zetas // population proportion (foreign) zeta + zetas = 1 alppha delta egs oms omgs rhom psiygap psipi rhoicu taudss taumss taukss taunss tauydss gss sigmataud sigmataum sigmatauk sigmataun sigmatauyd sigmag sigmam sigmac sigman sigmabeta sigmai sigmap taudsss sigmatauds taumsss sigmataums tauksss sigmatauks taunsss sigmatauns tauydsss sigmatauyds gsss sigmags sigmacs sigmans sigmabetas sigmais sigmaps tauss icuss ; // Parameters Values (Calibrated/Estimated Parameters) rhob = 0.9 ; rhon = rhob ; rhoc = rhob ; rhoi = rhob ; rhop = rhob ; rhotauyd = rhob ; rhotaun = rhob ; rhotauk = rhob ; rhotaud = rhob ; rhotaum = rhob ; el = 1.1 ; %// substitution of goods across countries eg = el ; ej = 10 ; om = 0.3 ; omg = 0.3 ; rhog = 0.8 ; xi = 0.9 ; mun = 10; pidss = 0.02 ; % Steady state inflation rate is at 2% for domestic country. pidsss = 0.02 ; % steady state inflation rate is at 2% for foreign country. muc = 1; ec = 2.000; %// Intertemporal Elasticity for Consumption, Sims 2.47 betta = 0.998; h = 0.8 ; kappa = 4.00 ; % Sims (2016) zeta = 1/3 ; % Population in Home Country zetas = 1 - zeta ; alppha = 1/3 ; delta = 0.025; egs = eg ; oms = 0.3 ; omgs = oms ; rhom = 0.7 ; psiygap = 0.000; psipi = 0.000; %//psiygap = 1.000; %//psipi = 2.500; rhoicu = 0.900; taudss = 0.01 ; taumss = 0.01 ; taukss = 0.01 ; taunss = 0.01 ; tauydss = 0.01 ; gss = 0.20 ; sigmataud = 0.1 ; sigmataum = 0.1 ; sigmatauk = 0.1 ; sigmataun = 0.1 ; sigmatauyd = 0.1 ; sigmag = 0.1 ; sigmam = 0.1 ; sigmac = 0.100; sigman = 0.100; sigmabeta = 0.100; sigmai = 0.1 ; sigmap = 0.1 ; taudsss = 0.01 ; sigmatauds = 0.1 ; taumsss = 0.01 ; sigmataums = 0.1 ; tauksss = 0.01 ; sigmatauks = 0.1 ; taunsss = 0.01 ; sigmatauns = 0.1 ; tauydsss = 0.01 ; sigmatauyds = 0.1 ; gsss = 0.20 ; sigmags = 0.1 ; sigmacs = 0.1 ; sigmans = 0.1 ; sigmabetas = 0.1 ; sigmais = 0.1 ; sigmaps = 0.1 ; tauss = 1.0 ; %// steady state terms of trade % ----- -------------------------------------------------------------- % Model Block % ----- -------------------------------------------------------------- model; % Consumers First Order Conditions % 1 Marginal Utility of Money Definition for consumer lambda1 = (x - muc * nuc)^(-ec); % 2 Consumption Definition for domestic consumer x = c - h * c(-1); % 3 Eulers Equation for consumer (nubeta *lambda1 *(1+pi))/(betta* nubeta(+1)* lambda1(+1)) = (nubetas * lambda1s * (1+pis))/(betta * nubetas(+1) * lambda1s(+1)); % 4 Fischer Equation for consumer mun * n * nun = lambda1 * w * (1-taun); % 5 International Risk Sharing on account of complete markets for consumers lambda1*(1+pi(+1))*nubeta = lambda1s * (1+pis(+1))*nubetas; % 6 Investment foc for consumer lambda1 = lambda2 * nui* ( 1 - ( (kappa/2) * ((i/i(-1)) - 1)^2 ) - (((nui * i * kappa)/(i(-1)) ) * ((i/i(-1)) - 1)) ) + (betta * (nubeta(+1)/nubeta) * nui(+1) * lambda2(+1) * kappa * ((i(+1)/i) - 1)*(i(+1)/i)^2); %7 Future capital stock for consumer lambda2 = betta * (nubeta(+1)/nubeta) * ((r(+1) *lambda1(+1) * (1-tauk(+1))) + (lambda2 * (1-delta))); % 8 Capital Accumulation equation k(+1) = k*(1-delta) + (nui * (1- (kappa/2) *((i/i(-1)) - 1)^2) * i ) ; % Economy Wide Prices % 9 Domestic price index composition between calvo and non calvo firms (1+pid)^(1-ej) = ( xi*(1+pidss)^(1-ej) + (1-xi)*(1+pidopt)^(1-ej) ); % 10 CPI between domestic price and foreign price (via tot) (1+pi)^(1-el) = ( ((1-om)*(1+taud)^(1-el)+ om * (1+taum)^(1-el) *tau^(1-el) )/((1-om)*(1+taud(-1))^(1-el) + om * (1+taum(-1))^(1-el) * tau(-1)^(1-el) ) ) * (1+pid)^(1-el); % 11 Price Dispersion on account of Monopolistic competition psigma = (1-tauyd)^(ej)*((1-xi) * ((1+pidopt)/(1+pid))^(-ej) + xi * ((1+pid)/(1+pidss))^(ej) * psigma(-1)); % 12 Numerator for domestic price evolution for Calvo firms as a function of average f = nubeta*lambda1*(1-tauyd)^(-ej)*mc*yd + xi*betta*((1+pid(+1))/(1+pidss))^(ej)*f(+1); % 13 Denominator for domestic price evolution for Calvo firms as a function of average gpr = nubeta*lambda1*(1-tauyd)^(-ej)*yd + xi*betta*((1+pid(+1))/(1+pidss))^(ej-1)*gpr(+1); % 14 Domestic Price evolution for Calvo Firms (1+pidopt) = (ej/(ej-1))*(f/gpr)*(1+pid); % 15 Terms of trade between the economies tau = ((1+pids)/(1+pid))* tau(-1); % Firms Optimality % 16 Optimal Input Ratio economy-wide (w*alppha)/(r*(1-alppha)) = (k/n); % 17 real marginal cost mc = (1/nup)*((n/k)^(alppha))*(w/(1-alppha)); % 18 Input markets and total tradable output (used for natural rate purposes) nup*(k/n)^(alppha) * n = yd*(1-tauyd)^(ej)*psigma^(-1); % Aggregate Market Clearing % 19 Total output demand yd = cd + gd + i + (zetas/zeta)*(cms + gms); % 20 Domestic consumption of home countrys final good by consumer cd = (((1-om)* x)/((1+taud)^(el))) * ((1-om)*(1+taud)^(1-el) + om * (1+taum)^(1-el) * tau^(1-el))^(el/(1-el)); % 21 Import by home country consumers (demand for foreign goods) cm = (((om)* x)/((1+taum)^(el))) * ((1-om)*(1+taud)^(1-el) * tau^(-(1-el)) + om*(1+taum)^(1-el))^(el/(1-el)); % 22 Domestic consumption of home countrys final good by government gd = (((1-omg)* g)/((1+taud)^(eg))) * ((1-omg)*(1+taud)^(1-eg)+ omg*(1+taum)^(1-eg) * tau^(1-eg))^(eg/(1-eg)); % 23 Import by home country government (demand for foreign goods) gm = (((omg)* g)/((1+taum)^(eg))) * ((1-omg)*(1+taud)^(1-eg)* tau^(-(1-eg)) + omg*(1+taum)^(1-eg) )^(eg/(1-eg)); % Fiscal Policy Processes % 24 Taxes on domestic consumption taud = (1- rhotaud) * taudss + rhotaud * taud(-1) ; %taud = (1- rhotaud) * taudss + rhotaud * taud(-1) + sigmataud * etaud; % 25 Taxes on Imported consumption taum = (1- rhotaum) * taumss + rhotaum * taum(-1) ; %taum = (1- rhotaum) * taumss + rhotaum * taum(-1) + sigmataum * etaum; % 26 Tases on Rental Income from Capital tauk = (1- rhotauk) * taukss + rhotauk * tauk(-1); %tauk = (1- rhotauk) * taukss + rhotauk * tauk(-1) + sigmatauk * etauk; % 27 Taxes on Labor Income taun = (1- rhotaun) * taunss + rhotaun * taun(-1); %taun = (1- rhotaun) * taunss + rhotaun * taun(-1)+ sigmataun * etaun; % 28 Taxes on Selling Final output to the market on Firms tauyd = (1- rhotauyd) * tauydss + rhotauyd * tauyd(-1) ; %tauyd = (1- rhotauyd) * tauydss + rhotauyd * tauyd(-1) + sigmatauyd * etauyd; % 29 exogenous government spending, equation is in logs g = (1- rhog) * gss*yd + rhog * g(-1) + sigmag * epsg; % Monetary Policy and Processes % 30 ZLB version of Taylors Rule commented out. icu = (1-rhoicu) * ( psiygap *ygapcu + psipi * picugap + icuss) + rhoicu * icu(-1) + log(num); % icu = max(0, psiygap * (1-rhoicu)* ygapcu + psipi * picu + rhoicu * icu(-1) + num); % 31 Monetary policy shock processes log(num) = rhom * log(num(-1)) + sigmam * em; % Exogenous Processes % 32 Taste Shock process log(nuc) = rhoc*log(nuc(-1)) + sigmac * ecshock; % 33 Labor Supply Shock log(nun) = rhon*log(nuc(-1)) + sigman * en; % 34 Discount Factor Shock: a demand shock log(nubeta) = rhob* log(nubeta(-1)) + sigmabeta * ebeta; % 35 Investment supply shock log(nui) = rhoi* log(nui(-1)) + sigmai * ei; % 36 Productivity Shock log(nup) = rhop * log(nup(-1)) + sigmap * ep; % Consumers First Order Conditions (Foreign Country) % 37 Marginal Utility of Money Definition for consumer lambda1s = (xs - muc * nucs)^(-ec); % 38 Consumption Definition for foreign consumer xs = cs - h * cs(-1); % 39 Eulers Equation for consumer (combined in equation 3, result of international risk sharing) %nubetas * lambda1s * (1+pis) = betta * nubetas(+1) * (1+icu) * lambda1s(+1); % 39 Fischer Equation for consumer mun * ns * nuns = lambda1s * ws * (1-tauns); % 40 Investment foc for consumer lambda1s = lambda2s * nuis * ( 1 - ( (kappa/2) * ((is/is(-1)) - 1)^2 ) - (((nuis * is * kappa)/(is(-1)) ) * ((is/is(-1)) - 1)) ) + (betta * (nubetas(+1)/nubetas) * nuis(+1) * lambda2s(+1) * kappa * ((is(+1)/is) - 1)*(is(+1)/is)^2); % 41 Future capital stock for consumer lambda2s = betta* (nubetas(+1)/nubetas)*(rs(+1)*lambda1s(+1)*(1-tauks(+1))+lambda2s*(1-delta)); % 42 Capital Accumulation equation ks(+1) = ks*(1-delta) + (nuis * (1- (kappa/2) *((is/is(-1)) - 1)^2) * is ) ; % Economy Wide Prices (Foreign Country) % 43 Domestic price index composition between calvo and non calvo firms (1+pids)^(1-ej) = ( xi*(1+pidsss)^(1-ej) + (1-xi)*(1+pidopts)^(1-ej) ); % 44 CPI between domestic price and foreign price (via tot) (1+pis)^(1-el) = ( (((1-oms)*(1+tauds)^(1-el)+oms*(1+taums)^(1-el)*tau^(-1+el)))/(((1-oms)*(1+tauds(-1))^(1-el)+oms*(1+taums(-1))^(1-el)*tau(-1)^(-1+el))) ) * (1+pids)^(1-el); % 45 Price Dispersion on account of Monopolistic competition psigmas = (1-tauyds)^(ej)*((1-xi)*((1+pidopts)/(1+pids))^(-ej) + xi* ((1+pids)/(1+pidsss))^(ej)*psigmas(-1)); % 46 Numerator for domestic price evolution for Calvo firms as a function of average fs = nubetas*lambda1s*(1-tauyds)^(-ej)*mcs*yds + xi*betta*((1+pids(+1))/(1+pidsss))^(ej)*fs(+1); % 47 Denominator for domestic price evolution for Calvo firms as a function of average gprs = nubetas*lambda1s*(1-tauyds)^(-ej)*yds + xi*betta*((1+pids(+1))/(1+pidsss))^(ej-1)*gprs(+1); % 48 Domestic Price evolution for Calvo Firms (1+pidopts) = (ej/(ej-1))*(fs/gprs)*(1+pids); % Firms Optimality (Foreign Country) % 49 Optimal Input Ratio economy-wide (ws*alppha)/(rs*(1-alppha)) = (ks/ns); % 50 real marginal cost mcs = (1/nups)*((ns/ks)^(alppha))*(ws/(1-alppha)); % 51 Input markets and total tradable output (used for natural rate purposes) nups *(ks/ns)^(alppha) * ns = yds * (1-tauyds)^(ej)* psigmas^(-1); % Aggregate Market Clearing (Foreign Country) % 52 Total output demand yds = cds + gds + is + (zeta/zetas)*(cm + gm); % 53 Domestic consumption of home countrys final good by consumer cds = (((1-oms)* xs)/((1+tauds)^(el))) * ((1-oms)*(1+tauds)^(1-el) + oms*(1+taums)^(1-el) * tau^(-(1-el)))^(el/(1-el)); % 54 Import by home country consumers (demand for foreign goods) cms = (((oms)* xs)/((1+taums)^(el))) * ((1-oms)*(1+tauds)^(1-el) * tau^(1-el) + oms*(1+taums)^(1-el))^(el/(1-el)); % 55 Domestic consumption of home countrys final good by government gds = (((1-omgs)* gs)/((1+tauds)^(eg))) * ((1-omgs)*(1+tauds)^(1-eg) + omgs*(1+taums)^(1-eg) * tau^(-(1-eg)))^(eg/(1-eg)); % 56 Import by home country government (demand for foreign goods) gms = (((omgs)* gs)/((1+taums)^(eg))) * ((1-omgs)*(1+tauds)^(1-eg)*tau^(1-el) + omgs*(1+taums)^(1-eg))^(eg/(1-eg)); % Fiscal Policy Processes (Foreign Country) % 57 Taxes on domestic consumption tauds = (1 - rhotaud) * taudsss + rhotaud * tauds(-1) ; %tauds = (1 - rhotaud) * taudsss + rhotaud * tauds(-1) + sigmatauds * etauds; % 58 Taxes on Imported consumption taums = (1- rhotaum) * taumsss + rhotaum * taums(-1) ; %taums = (1- rhotaum) * taumsss + rhotaum * taums(-1) + sigmataums * etaums; % 59 Tases on Rental Income from Capital tauks = (1- rhotauk) * tauksss + rhotauk * tauks(-1) ; %tauks = (1- rhotauk) * tauksss + rhotauk * tauks(-1) + sigmatauks * etauks; % 60 Taxes on Labor Income tauns = (1- rhotaun) * taunsss + rhotaun * tauns(-1); %tauns = (1- rhotaun) * taunsss + rhotaun * tauns(-1)+ sigmatauns * etauns; % 61 Taxes on Selling Final output to the market on Firms tauyds = (1- rhotauyd) * tauydsss + rhotauyd * tauyds(-1); %tauyds = (1- rhotauyd) * tauydsss + rhotauyd * tauyds(-1) + sigmatauyds * etauyds; % 62 exogenous government spending, equation is in logs gs = (1- rhog) * gsss*yds + rhog * gs(-1) + sigmags * epsgs; % Exogenous Processes (Foreign Country) % 63 Taste Shock process log(nucs) = rhoc * log(nucs(-1)) + sigmacs * ecshocks; % 64 Labor Supply Shock log(nuns) = rhon * log(nuns(-1)) + sigmans * ens; % 65 Discount Factor Shock: a demand shock log(nubetas) = rhob * log(nubetas(-1)) + sigmabetas * ebetas; % 66 Investment supply shock log(nuis) = rhoi * log(nuis(-1)) + sigmais * eis; % 67 Productivity Shock log(nups) = rhop * log(nups(-1)) + sigmaps * eps; % Currency Union Level Output and Inflation % 68 CPI Inflation at the Currency Union picu = zeta * pi + zetas * pis; % 69 Natural Rate of Currency Union Output ycun = zeta * ydnat + zetas * ydnats; % 70 Actual Currency Union Output ycu = zeta * yd + zetas * yds; % 71 Gap between actual and natural output ygapcu = ycu - ycun; % 72 Natural Output for Home economy ydnat = yd; % 73 Natural Output for Foreign Economy ydnats = yds; % 74 Natural Inflation Rate for CU picunat = zeta*pidss + zetas*pidsss; % 75 Inflation Gap for CU picugap = picu - picunat; end; shocks; var epsg; stderr 0.009; end; steady; check; //stoch_simul(order = 1, relative_irf, irf=30) yd cd cm pi;