// Ramsey problem for Laffer curve // Mattia Ricci // Glasgow University, February 2015 // Dynare version 4.4.3 //---------------------------------------------------------------- // 0. Housekeeping //---------------------------------------------------------------- close all; warning off; //---------------------------------------------------------------- // 1. Defining variables //---------------------------------------------------------------- // 1.1 Endogenous variables of the model var c // Consumption k // Capital l // Labour m // Capital capacity utilization xi1 xi2 xi3 k_ratio ; // 1.2 Exogenous variables of the model varexo Wbar ; //---------------------------------------------------------------- // 2. Parameters of the model //---------------------------------------------------------------- parameters beta $\beta$ // Discount rate chi0 $\chi_0$ // depreciation funct. parameter chi1 $\chi_1$ // depreciation funct. parameter alpha $\alpha$ // Elasticity of output with respect to private capital gamma $\gamma$ // Exogenous rate of labour augmenting technical process sigma $\sigma$ // Risk adversion coefficient a // Leisure preference eta $\eta$ // Coefficient of capital adjustment cost deltabar $\bar{\delta}$ // Capital devaluation z // Steady state ratio i/k g // PO/initial/final level of consumption //INITIAL CONDITIONS: c0 // initial consumption l0 // initial labour W0 // initial Welfare k0 // initial capital xi10 // initial lambda1 xi20 // initial lambda2 xi30 // initial lambda3 //TERMINAL CONDITIONS: css // terminal consumption lss // terminal labour Wss // terminal Welfare kss // terminal capital xi1ss // terminal lambda1 xi2ss // terminal lambda2 xi3ss // terminal lambda3 k_ratioss ; //---------------------------------------------------------------- // 3. Calibration of parameters //---------------------------------------------------------------- // we borrow from Mendoza, see table 2 pag. 41 of "Saving Europe..." beta=0.9942; alpha=0.61; gamma=0.0022; sigma=2; a=2.675; eta=2; deltabar=0.0164; z=gamma+deltabar; //Set it so as adjustment cost are zero in SS chi1=(1+gamma-beta)/beta*(1/deltabar)+1; //Set it so as deltabar= calibration target. chi0=(1+gamma-beta)/beta*chi1/(chi1-1); //Set it so as m=1 in SS. g=0.3901; //SETTING THE EXERCISE: W0=-569.9679; //initial Welfare //Wss=-580; //terminal Welfare Wss=-580; //terminal Welfare k_ratioss=(1/(1-alpha)*((1+gamma)/beta-(1-deltabar)))^(-1/alpha); //Initial conditions: [Endo0,fval] =fsolve(@RamseyInitial,0.5*ones(1,7),[],W0,beta,chi0,chi1,sigma,eta,z,alpha,gamma,a,g); xi10=Endo0(1); xi20=Endo0(2); xi30=Endo0(3); c0=Endo0(4); l0=Endo0(5); k0=Endo0(6); m0=Endo0(7); //Terminal conditions [Endoss,fval] = fsolve(@RamseySS,0.5*ones(1,7),[],Wss,c0,l0,beta,chi0,chi1,sigma,eta,z,alpha,gamma,a,g); xi1ss=Endoss(1); xi2ss=Endoss(2); xi3ss=Endoss(3); css=Endoss(4); lss=Endoss(5); kss=Endoss(6); mss=Endoss(7); //---------------------------------------------------------------- // 4. Model //---------------------------------------------------------------- model; #Transf_lead=exp(-m(+1))/(1+exp(-m(+1))); #Transf_lag=exp(-m(-1))/(1+exp(-m(-1))); #Transf=exp(-m)/(1+exp(-m)); (xi3*(1 - l)^a)/((c*(1 - l)^a)^sigma*(beta - 1)) - xi1 + (c0^sigma*sigma*(1 - l0)^(a*(sigma - 1))*(l^alpha*(k(-1)*Transf)^(1 - alpha) - chi0*k(-1)*Transf^chi1 + (a*c*l)/(l - 1) + chi0*eta*k(-1)*Transf*Transf^(chi1 - 1)*(z - (k*(gamma + 1) + k(-1)*((chi0*Transf^chi1)/chi1 - 1))/k(-1))))/(c^(sigma + 1)*(1 - l)^(a*(sigma - 1))) - (a*c0^sigma*l*(1 - l0)^(a*(sigma - 1)))/(c^sigma*(1 - l)^(a*(sigma - 1))*(l - 1)) - (c(+1)^sigma*sigma*xi2*(1 - l(+1))^(a*(sigma - 1))*(gamma - eta*(gamma + 1)*(z - (k*(gamma + 1) + k(-1)*((chi0*Transf^chi1)/chi1 - 1))/k(-1)) + 1))/(beta*c^(sigma + 1)*(1 - l)^(a*(sigma - 1))) + (c^(sigma - 1)*sigma*xi2(-1)*(1 - l)^(a*(sigma - 1))*(gamma - eta*(gamma + 1)*(z - (k(-1)*(gamma + 1) + k(-1)*((chi0*Transf_lag^chi1)/chi1 - 1))/k(-1)) + 1))/(beta^2*c(-1)^sigma*(1 - l(-1))^(a*(sigma - 1))) =0; alpha*l^(alpha - 1)*xi1*(k(-1)*Transf)^(1 - alpha) - (c0^sigma*(1 - l0)^(a*(sigma - 1))*(alpha*l^(alpha - 1)*(k(-1)*Transf)^(1 - alpha) + (a*c)/(l - 1) - (a*c*l)/(l - 1)^2))/(c^sigma*(1 - l)^(a*(sigma - 1))) - (a*c*xi3*(1 - l)^(a - 1))/((c*(1 - l)^a)^sigma*(beta - 1)) - (a*c0^sigma*(1 - l0)^(a*(sigma - 1))*(sigma - 1)*(l^alpha*(k(-1)*Transf)^(1 - alpha) - chi0*k(-1)*Transf^chi1 + (a*c*l)/(l - 1) + chi0*eta*k(-1)*Transf*Transf^(chi1 - 1)*(z - (k*(gamma + 1) + k(-1)*((chi0*Transf^chi1)/chi1 - 1))/k(-1))))/(c^sigma*(1 - l)^(a*(sigma - 1) + 1)) + (a*c(+1)^sigma*xi2*(1 - l(+1))^(a*(sigma - 1))*(sigma - 1)*(gamma - eta*(gamma + 1)*(z - (k*(gamma + 1) + k(-1)*((chi0*Transf^chi1)/chi1 - 1))/k(-1)) + 1))/(beta*c^sigma*(1 - l)^(a*(sigma - 1) + 1)) - (a*c^sigma*xi2(-1)*(1 - l)^(a*(sigma - 1) - 1)*(sigma - 1)*(gamma - eta*(gamma + 1)*(z - (k(-1)*(gamma + 1) + k(-1)*((chi0*Transf_lag^chi1)/chi1 - 1))/k(-1)) + 1))/(beta^2*c(-1)^sigma*(1 - l(-1))^(a*(sigma - 1))) =0; xi2*(eta*k*(((chi0*Transf_lead^chi1)/chi1 - 1)/k - (k(+1)*(gamma + 1) + k*((chi0*Transf_lead^chi1)/chi1 - 1))/k^2)^2 - 2*eta*(((chi0*Transf_lead^chi1)/chi1 - 1)/k - (k(+1)*(gamma + 1) + k*((chi0*Transf_lead^chi1)/chi1 - 1))/k^2)*(z - (k(+1)*(gamma + 1) + k*((chi0*Transf_lead^chi1)/chi1 - 1))/k) + eta*k*((2*((chi0*Transf_lead^chi1)/chi1 - 1))/k^2 - (2*(k(+1)*(gamma + 1) + k*((chi0*Transf_lead^chi1)/chi1 - 1)))/k^3)*(z - (k(+1)*(gamma + 1) + k*((chi0*Transf_lead^chi1)/chi1 - 1))/k) - chi0*eta*Transf_lead*Transf_lead^(chi1 - 1)*(((chi0*Transf_lead^chi1)/chi1 - 1)/k - (k(+1)*(gamma + 1) + k*((chi0*Transf_lead^chi1)/chi1 - 1))/k^2) + (c(+1)^sigma*eta*(gamma + 1)^2*(1 - l(+1))^(a*(sigma - 1)))/(beta*c^sigma*k(-1)*(1 - l)^(a*(sigma - 1)))) + beta*(xi1(+1)*((l(+1)^alpha*Transf_lead)/(k*Transf_lead)^alpha - eta*z - (eta*z^2)/2 - eta/2 + (eta*k(+1)^2)/(2*k^2) - (chi0*Transf_lead^chi1)/chi1 + (chi0*eta*Transf_lead^chi1)/chi1 + (eta*gamma^2*k(+1)^2)/(2*k^2) - (alpha*l(+1)^alpha*Transf_lead)/(k*Transf_lead)^alpha + (eta*gamma*k(+1)^2)/k^2 - (chi0^2*eta*Transf_lead^(2*chi1))/(2*chi1^2) + (chi0*eta*Transf_lead^chi1*z)/chi1 + 1) + (c0^sigma*(1 - l0)^(a*(sigma - 1))*(chi0*Transf_lead^chi1 - (l(+1)^alpha*Transf_lead)/(k*Transf_lead)^alpha - chi0*eta*Transf_lead^chi1 + (alpha*l(+1)^alpha*Transf_lead)/(k*Transf_lead)^alpha - chi0*eta*Transf_lead^chi1*z + (chi0^2*eta*Transf_lead^(2*chi1))/chi1))/(c(+1)^sigma*(1 - l(+1))^(a*(sigma - 1))) - (eta*k(+1)*xi2(+1)*(gamma + 1)^2)/(beta*k^2)) - xi1*(gamma - eta*(gamma + 1)*(z - (k*(gamma + 1) + k(-1)*((chi0*Transf^chi1)/chi1 - 1))/k(-1)) + 1) - (eta*xi2(-1)*(gamma + 1)*(k + gamma*k + chi0*k(-1)*Transf^chi1))/(beta*k(-1)^2) + (c0^sigma*chi0*eta*Transf*Transf^(chi1 - 1)*(gamma + 1)*(1 - l0)^(a*(sigma - 1)))/(c^sigma*(1 - l)^(a*(sigma - 1))) =0; (c0^sigma*(1 - l0)^(a*(sigma - 1))*((k(-1)*l^alpha*(alpha - 1))/(k(-1)*Transf)^alpha + chi0*chi1*k(-1)*Transf^(chi1 - 1) + chi0^2*eta*k(-1)*Transf*Transf^(2*chi1 - 2) - chi0*eta*k(-1)*Transf^(chi1 - 1)*(z - (k*(gamma + 1) + k(-1)*((chi0*Transf^chi1)/chi1 - 1))/k(-1)) - chi0*eta*k(-1)*Transf*Transf^(chi1 - 2)*(chi1 - 1)*(z - (k*(gamma + 1) + k(-1)*((chi0*Transf^chi1)/chi1 - 1))/k(-1))))/(c^sigma*(1 - l)^(a*(sigma - 1))) - xi1*(chi0*k(-1)*Transf^(chi1 - 1) + (k(-1)*l^alpha*(alpha - 1))/(k(-1)*Transf)^alpha - chi0*eta*k(-1)*Transf^(chi1 - 1)*(z - (k*(gamma + 1) + k(-1)*((chi0*Transf^chi1)/chi1 - 1))/k(-1))) - (chi0*Transf^chi1*xi2(-1)*(chi1^2*k(-1) - chi1*k(-1) + chi1*eta*k(-1) + chi1^2*eta*k - chi1^2*eta*k(-1) + chi1*eta*k(-1)*z + chi1^2*eta*gamma*k - chi1^2*eta*k(-1)*z - chi0*eta*k(-1)*Transf^chi1 + 2*chi0*chi1*eta*k(-1)*Transf^chi1))/(beta*chi1*k(-1)*Transf) + (chi0*c(+1)^sigma*eta*Transf^(chi1 - 1)*xi2*(gamma + 1)*(1 - l(+1))^(a*(sigma - 1)))/(beta*c^sigma*(1 - l)^(a*(sigma - 1))) =0; l^alpha*(k(-1)*Transf)^(1 - alpha) - g - c - k*(gamma + 1) - k(-1)*((chi0*Transf^chi1)/chi1 - 1) - (eta*k(-1)*(z - (k*(gamma + 1) + k(-1)*((chi0*Transf^chi1)/chi1 - 1))/k(-1))^2)/2 =0; (eta*(z - (k(+1)*(gamma + 1) + k*((chi0*Transf_lead^chi1)/chi1 - 1))/k)^2)/2 - chi0*Transf_lead^chi1 + (chi0*Transf_lead^chi1)/chi1 - eta*k*(((chi0*Transf_lead^chi1)/chi1 - 1)/k - (k(+1)*(gamma + 1) + k*((chi0*Transf_lead^chi1)/chi1 - 1))/k^2)*(z - (k(+1)*(gamma + 1) + k*((chi0*Transf_lead^chi1)/chi1 - 1))/k) + chi0*eta*Transf_lead*Transf_lead^(chi1 - 1)*(z - (k(+1)*(gamma + 1) + k*((chi0*Transf_lead^chi1)/chi1 - 1))/k) + (c(+1)^sigma*(1 - l(+1))^(a*(sigma - 1))*(gamma - eta*(gamma + 1)*(z - (k*(gamma + 1) + k(-1)*((chi0*Transf^chi1)/chi1 - 1))/k(-1)) + 1))/(beta*c^sigma*(1 - l)^(a*(sigma - 1))) - 1 =0; Wbar - (c*(1 - l)^a)^(1 - sigma)/((beta - 1)*(sigma - 1)) =0; k_ratio=k(-1)/l; end; write_latex_dynamic_model; write_latex_static_model; model_info; //---------------------------------------------------------------- // 5. Initial conditions //---------------------------------------------------------------- initval; c = c0; l = l0; k = k0; m = log(1/m0-1); Wbar = W0; xi1 = xi10; xi2 = xi20; xi3 = xi30; k_ratio= k_ratioss; end; endval; c = css; l = lss; k = kss; m = log(1/mss-1); Wbar = Wss; xi1 = xi1ss; xi2 = xi2ss; xi3 = xi3ss; k_ratio= k_ratioss; end; //---------------------------------------------------------------- // 6. Deterministic Simulation //---------------------------------------------------------------- shocks; var Wbar; periods 1:1; values (Wss); end; steady(solve_algo=3); model_diagnostics; check; resid; simul(periods=10, maxit=2000);