% The RBC Model with informal sector % Informal goods: consumption only % Formal goods: consumption and investment % GHH Preference close all; %---------------------------------------------------------------------- %1. Defining Variables %---------------------------------------------------------------------- %%var tau kappa lY lY_f lY_i lC_f lC_i d lK lH lh_f lh_i llambda lx r A_f A_i tb ca G p lC ; var tau kappa lY lY_f lY_i lC_f lC_i ld lK lH lh_f lh_i llambda lx r A_f A_i lG tb ca lp lC ; %varexo_det kappa tau; varexo ef ei eg ekappa etau; parameters alpha etta beta theta gamma sigma r_star R rho_f rho_i sigma_f sigma_i sigma_g sigma_kappa d_bar phi delta G_ss rho_G omega rho_tau sigma_tau rho_kappa tau_ss kappa_ss; %---------------------------------------------------------------------- % 2. Calibration % Taken from Ihrig & Moe 2004 and Mitra 2013 %---------------------------------------------------------------------- alpha = 0.70; %formal good share of utility beta = 0.96; %discount rate %elasticity of formal output wrt laborgov theta = 0.43; etta = 0; %elasticity of informal output wrt labor gamma = .495; sigma = 1.5; r_star = 0.04; R = 0.00742; %data from J. Fernandez's model rho_f = 0.90; rho_i = 0.64; sigma_f = 0.10; sigma_i = 0.10; sigma_g = 0.10; sigma_kappa = 0.10; phi = 0.028 ; %coefficient of capital adjustment delta = 0.1; %capital depreciation omega = 1.4556789; G_ss = 0.20; rho_G = 0.8; kappa_ss = 0.25; rho_kappa = .8; rho_tau = .8; sigma_tau = .10; tau_ss = .2; %---------------------------------------------------------------------- %3. Model %---------------------------------------------------------------------- model; %C = C_f^(alpha)*C_i^(1-alpha); exp(lC) = exp(lC_f)^(alpha)*exp(lC_i)^(1-alpha); %Y = Y_f + p*Y_i; exp(lY)= exp(lY_f)+exp(lp)*exp(lY_i); %lambda = lambda(+1)*beta*(1+r); exp(llambda) = exp(llambda(+1))*beta*(1+r); %alpha*(C_i/C_f)^(1-alpha)*(C_f^(alpha)*C_i^(1-alpha)-((h_f + h_i)^omega)/omega)^(-sigma) = lambda; alpha*(exp(lC_i)/exp(lC_f))^(1-alpha)*(exp(lC_f)^(alpha)*exp(lC_i)^(1-alpha)-((exp(lh_f)+exp(lh_i))^(omega))/omega)^(-sigma) = exp(llambda); %(1-alpha)*(C_f/C_i)^(alpha)*(C_f^(alpha)*C_i^(1-alpha)-((h_f + h_i)^omega)/omega)^(-sigma) = p*lambda; (1-alpha)*(exp(lC_f)/exp(lC_i))^(alpha)*(exp(lC_f)^(alpha)*exp(lC_i)^(1-alpha)-((exp(lh_f) + exp(lh_i))^(omega))/omega)^(-sigma) = exp(lp)*exp(llambda); %(1-alpha)*(C_f/C_i)^(alpha)*(C_f^(alpha)*C_i^(1-alpha)-((h_f + h_i)^omega)/omega)^(-sigma) = p*lambda; (1-alpha)*(exp(lC_f)/exp(lC_i))^(alpha)*(exp(lC_f)^(alpha)*exp(lC_i)^(1-alpha)-((exp(lh_f) + exp(lh_i))^(omega))/omega)^(-sigma) = exp(lp)*exp(llambda); %(h_f + h_i)^(omega-1)*(C_f^(alpha)*C_i^(1-alpha)-((h_f + h_i)^omega)/omega)^(-sigma) = (1-tau*kappa)*lambda*p*gamma*Y_i/h_i; (exp(lh_f) + exp(lh_i))^(omega-1)*(exp(lC_f)^(alpha)*exp(lC_i)^(1-alpha)-((exp(lh_f) + exp(lh_i))^(omega))/omega)^(-sigma) = (1-(tau)*(kappa))*exp(llambda)*exp(lp)*gamma*exp(lY_i)/exp(lh_i); %lambda*(1+phi*(K-K(-1))) = lambda(+1)*beta*((1-tau)*(1-theta)*Y_f(+1)/K+1-delta+phi*(K(+1)-K)); exp(llambda)*(1+phi*(exp(lK)-exp(lK(-1)))) = exp(llambda(+1))*beta*((1-(tau))*(1-theta)*exp(lY_f(+1))/exp(lK)+1-delta+phi*exp((lK(+1))-exp(lK))); %r = r_star + R*(exp(d-d_bar)- 1); r = r_star + R*(exp(exp(ld)-d_bar)- 1); %(d) = (1+(r(-1)))*(d(-1))-Y_f-p*Y_i+C_f+p*C_i + x+(phi/2)*(K-K(-1))^2+(1-etta)*G; exp(ld) = (1+(r(-1)))*exp(ld(-1))-exp(lY_f)-exp(lp)*exp(lY_i)+exp(lC_f)+exp(lp)*exp(lC_i) + exp(lx)+(phi/2)*(exp(lK)-exp(lK(-1)))^(2)+(1-etta)*exp(lG); %Y_f = exp(A_f)*h_f^theta*K(-1)^(1-theta) +etta*G ; exp(lY_f) = exp(A_f)*exp(lh_f)^(theta)*exp(lK(-1))^(1-theta) ; %Y_i = exp(A_i)*h_i^(gamma); exp(lY_i) = exp(A_i)*exp(lh_i)^(gamma); %x = K - (1-delta)*K(-1); exp(lx) = exp(lK) - (1-delta)*exp(lK(-1)); %tau*kappa*p*1.5*Y_i + tau*Y_f = G; tau*kappa*exp(lp)*exp(lY_i) + tau*exp(lY_f) = exp(lG); %h_f + h_i = H; exp(lh_f) + exp(lh_i) = exp(lH); %Y_i = C_i ; exp(lY_i) = exp(lC_i) ; %tb = Y_f - C_f - (1-etta)*G - x - (phi/2)*(K)^2+phi*K*K(-1)-(phi/2)*K(-1)^2; exp(tb) = exp(lY_f) - exp(lC_f) - (1-etta)*exp(lG) - exp(lx) - (phi/2)*(exp(lK))^2+phi*exp(lK)*exp(lK(-1))-(phi/2)*exp(lK(-1))^2; %ca = -(d-d(-1)); exp(ca) = -(exp(ld)-exp(ld(-1))); %A_f = rho_f*A_f(-1)+ ef; %A_i = rho_i*A_i(-1) + ei; (A_f) = rho_f*(A_f(-1))+ ef; (A_i) = rho_i*(A_i(-1)) + ei; %G-G_ss = rho_G*(G(-1)-G_ss)+eg; kappa-kappa_ss = rho_kappa*(kappa(-1)-kappa_ss)+ekappa; tau - tau_ss = rho_tau*(tau(-1)-tau_ss) - etau; end; %------------------------------------------------------------------ % 4. Computation %------------------------------------------------------------------ G_ss = .1; p_ss = log(1.5); h_ss = log(1.1); d_bar = log(0.007442); d_ss = d_bar; initval; tau = 0.2; kappa =0.25; %U = -1.28132; lY = log(2.87263); lY_f = log(2.87263); lY_i = log(0.374144); lC_f = log(1.10319); lC_i = log(0.374144); ld = log(d_ss); lK = log(7.72465); lH = log(0.646782); lh_f = log(0.509551); lh_i = log(.137231); llambda = log(1.77451); lx = log(0.772465); r = 0.0416667; A_f = 0; A_i = 0; %%tb = 0.00875296; %ca = 0; lG = log(G_ss); lp = log(p_ss); lC = log(0.67592); end; G_ss = .1; p_ss = 1.5; h_ss = 1.1; d_bar = 0.007442; d_ss = d_bar shocks; %var ef = (sigma_f)^2; %var ei = (sigma_i)^2; %var eg = (sigma_g)^2; var ekappa = (sigma_kappa)^2; var etau = (sigma_tau)^2; %var kappa; %periods 10:19; %values .5; end; steady; check; stoch_simul(hp_filter = 100, order = 1, irf = 100) lY lY_f lY_i lC lC_f lC_i lH lh_f lh_i ld r lK lx lp tau kappa; %forecast (periods = 100);