I am currently trying to implement a second bond in the framework of Kirchner and van Wijnbergen (2012). However, I fail to find a steady state. Has anyone some guidance or experience in this?
Thank you in advance!
var
Y % final output
Ym % intermediary output
L % labor hours
w % real wage
C % consumption
U_c % marginal utility of consumption
Lambda % stochastic discount factor of households
I % investment
K % capital
Q % price of capital
a % technology
ksi % capital quality
Pm % price of intermediary goods (real marginal cost for final goood producer)
infl % inflation
inflstar % optimal price divided by past price level
F % auxiliary variable for price setting
Z % auxiliary variable for price setting
Dis % price dispersion
Rd % real deposit rate
i % nominal interest rate
Rk % real rate of capital
Rb % real rate of short-term bonds held by banks
Rlb % real rate of long-term bonds held by banks
Rp % real rate of portfolio by banks
ERk % expected real rate of capital
ERb % expected real rate of short-term bonds held by banks
ERlb % expected real rate of long-term bonds held by banks
prem % premium of return on capital - return on deposits
prem2 % premium of return on bonds - return on deposits
prem3 % premium of return on long term bonds - return on deposits
Phi % Leverage ratio of banks
portf_B % portfolio of banks
N % net worth of banks
Om % portfolio share of capital assets (short term bonds and loans)
Zet % portfolio share of long term bonds
D % economy-wide deposits
nu_k % value of having another unit of capital assets
nu_b % value of having another unit of bonds (for banks)
nu_blb % value of having another unit bonds (for banks)
nu_n % value of having another unit of net worth
G % government spending
g % government spending (shock process)
Gy % government spending share of output
T % taxes
B % government debt
Blb % long term government debt
Om_B
;
%capital quality, government spending, interest rate, technology, and net worth shock
varexo e_ksi e_g e_i e_a; //e_n e_b;
parameters
beta hh delta varphi eta_i alpha gam epsilon kappa_pi kappa_y G_over_Y B_over_Y theta lambda chi
rho_i rho_ksi rho_a rho_g sigma_i sigma_ksi sigma_a sigma_g sigma_n
Y_ss L_ss w_ss U_c_ss Y_over_K L_over_K K_ss Ym_ss G_ss I_ss C_ss Z_ss F_ss B_ss N_ss D_ss T_ss
a_ss ksi_ss g_ss Lambda_ss Pm_ss Phi_ss Q_ss Dis_ss infl_ss inflstar_ss Rd_ss Rp_ss Rb_ss Rk_ss i_ss
prem_ss nu_b_ss nu_n_ss nu_k_ss portf_B_ss Om_ss Gy_ss Rlb_ss term_premium depr_debt nu_blb_ss
omega_0 omega_1 omega_2 rho_b sigma_b alp Zet_ss Om_B_ss;
% standard Parameters
beta=0.99; % discount factor
hh=0.815; % habit formation
delta = 0.04; % depreciation rate
varphi = 0.276; % inverse of Frisch elasticity
eta_i = 1.728; % inv. adjustment cost parameter
alpha = 0.33; % output elasticity of capital
gam=0.779; % Calvo parameter
epsilon=4.167; % elasiticity of substitution between varieties of final goods
G_over_Y=0.2; % stst. government spending share of output
//B_over_Y=2.4;
B_over_Y= 0.8;
Blb_over_Y = 1.6; % stst. debt/GDP
term_premium = 0.6; % term premium on long-term bonds
depr_debt = 0.05;
alp = 0.15;
% Debt issuance parameters
omega_0 = 0.85; % 85% baseline preference for long-term bonds
omega_1 = -0.5; % Negative response to higher long-term rates
omega_2 = 0.2; % Higher debt leads to more long-term issuance
rho_b = 0.6; % Moderate persistence in debt composition
sigma_b = 0.10; % 10% standard deviation for composition shocks
//rho_tp = 0.8; % High persistence in term premium
//sigma_tp = 0.01; % 1% standard deviation for term premium shocks
Phi_ss=4; % stst. Leverage ratio of banks
theta=1-1/16; % survival rate of banks
kappa_pi=1.50000000;% Taylor rule coefficient on inflation
kappa_y =0.12500000;% Taylor rule coefficient on output growth
%shock parameters
rho_i=0.8; % interest rate smoothing
rho_ksi=0.66; % persistence of capital quality shock
rho_a=0.95; % persistence of technology shock
rho_g=0.80; % persistence of government spending shock
sigma_ksi=0.05; % std. dev. of capital quality shock
sigma_a=0.025; % std. dev. of technology shock
sigma_g=0.05; % std. dev. of government spending shock
sigma_i=0.0025; % std. dev. of interest rate shock
sigma_n=0.1; % std. dev. of net worth shock
%%% steady state values of variables labelled above
Lambda_ss = 1; %
Q_ss=1; %
Dis_ss=1;
infl_ss=1;
inflstar_ss=1;
a_ss=1;
g_ss=1;
ksi_ss=1;
Pm_ss = (epsilon-1)/epsilon; %
prem_ss = 0.02/4;
prem2_ss = prem_ss;
prem3_ss = prem_ss;
Phi_ss=6; //4
Rd_ss=1/beta;
i_ss=Rd_ss;
Rk_ss=Rd_ss+prem_ss;
Rb_ss=Rk_ss;
Rlb_ss = Rb_ss+term_premium;
ERk_ss=Rk_ss;
ERb_ss=Rk_ss;
ERlb_ss = Rb_ss+term_premium;
nu_n_ss = (1-theta)*beta*(Rd_ss)/(1-theta*beta);
nu_k_ss = (1-theta)*beta*(Rk_ss-Rd_ss)/(1-theta*beta);
nu_b_ss = nu_k_ss;
nu_blb_ss = nu_k_ss;
lambda = nu_k_ss + nu_n_ss/Phi_ss; % tightness parameter on incentive constraint
chi = 1-theta*((Rk_ss-Rd_ss)*Phi_ss+Rd_ss); % share of old net worth that is given to new bankers
w_ss = (alpha^(alpha)*(1-alpha)^(1-alpha)*Pm_ss*(Rk_ss-1+delta)^(-alpha))^(1/(1-alpha));
Y_over_K = (Rk_ss-1+delta)/(alpha*Pm_ss);
L_over_K = (Y_over_K)^(1/(1-alpha));
I_over_Y = delta*(Y_over_K)^(-1);
C_over_Y = 1-G_over_Y-I_over_Y;
habit = (1-beta*hh)/(1-hh);
Y_ss = (L_over_K)^(-alpha*varphi/(1+varphi))*(habit*w_ss*C_over_Y^(-1))^(1/(1+varphi));
Ym_ss= Y_ss;
C_ss = Y_ss*C_over_Y;
G_ss = Y_ss*G_over_Y;
Gy_ss = G_over_Y;
I_ss = Y_ss*I_over_Y;
K_ss = I_ss/delta;
L_ss = L_over_K*K_ss;
U_c_ss = habit/C_ss;
Z_ss = 1/(1-beta*gam)*Y_ss;
F_ss = Z_ss*Pm_ss;
B_ss = (Y_ss*B_over_Y);
Blb_ss = (Y_ss*Blb_over_Y);
N_ss = (K_ss+B_ss+Blb_ss)/Phi_ss;
Om_ss = K_ss/(K_ss+B_ss+Blb_ss);
Zet_ss = Blb_ss/(K_ss+B_ss+Blb_ss);
Om_B_ss = B_ss/(K_ss+B_ss+Blb_ss);
Rp_ss = Om_ss*(Rk_ss)+(Om_B_ss)*Rb_ss+Zet_ss*(Rlb_ss);
T_ss = (Rb_ss-1)*B_ss+G_ss+(Rlb_ss-1)*Blb_ss;
portf_B_ss = Phi_ss*N_ss;
D_ss = K_ss + B_ss + Blb_ss - N_ss;
Rp_ss=Om_ss*Rk_ss + Om_B_ss*Rb_ss + Zet_ss*Rlb_ss;
model;
% Households
exp(U_c) = (exp(C)-hh*exp(C(-1)))^(-1)-beta*hh*(exp(C(+1))-hh*exp(C))^(-1); % Marginal utility of consumption
exp(Lambda) = exp(U_c)/exp(U_c(-1)); % Stochastic discount rate
beta*exp(Rd(+1))*exp(Lambda(+1)) = 1; % Euler equation
exp(L)^varphi = exp(U_c)*exp(w); % Labor supply
% Intermediate goods producer
exp(Rk) = (exp(Pm)*alpha*exp(Ym)/exp(K(-1))+exp(ksi)*exp(Q)*(1-delta))/exp(Q(-1)); % Return on capital (Assumption: this = E26)
exp(Ym) = exp(a)*(exp(ksi)*exp(K(-1)))^alpha*exp(L)^(1-alpha); % Production function
exp(w) = exp(Pm)*(1-alpha)*exp(Ym)/exp(L); % Real Wages (Assumption: This replaces E28 (Eq. for mc))
% Capital Goods Producer
%exp(Q) = 1 + eta_i/2*(exp(I)/exp(I(-1))-1)^2
% + eta_i*(exp(I)/exp(I(-1))-1)*exp(I)/exp(I(-1))
% - beta*exp(Lambda(+1))*eta_i*(exp(I(+1))/exp(I)-1)*(exp(I(+1))/exp(I))^2; % Optimal investment decision (check out E29)
1/exp(Q) = 1 - eta_i/2*(exp(I)/exp(I(-1))-1)^2
- eta_i*(exp(I)/exp(I(-1))-1)*exp(I)/exp(I(-1))
+ beta*exp(Lambda(+1))*eta_i*(exp(I(+1))/exp(I)-1)*(exp(I(+1))/exp(I))^2*exp(Q)/exp(Q(+1)); //E29
exp(K) = (1-delta)*exp(ksi)*exp(K(-1)) + (1 - eta_i/2*(exp(I)/exp(I(-1))-1)^2)*exp(I); % Capital accumulation equation
% Retailer
exp(Ym) = exp(Y)*exp(Dis); % Retail output
exp(Dis) = gam*exp(Dis(-1))*exp(infl)^epsilon +
(1-gam)*((1-gam*exp(infl)^(epsilon-1))/(1-gam))^(-epsilon/(1-epsilon)); % Price dispersion
exp(F) = exp(Y)*exp(Pm) + beta*gam*exp(Lambda(+1))*exp(infl(+1))^epsilon *exp(F(+1)); % Optimal price choice
exp(Z) = exp(Y) + beta*gam*exp(Lambda(+1))*exp(infl(+1))^(epsilon-1)*exp(Z(+1)); % Optimal price choice
exp(inflstar) = epsilon/(epsilon-1)*exp(F)/exp(Z)*exp(infl); % Optimal price choice
exp(infl)^(1-epsilon) = gam + (1-gam)*(exp(inflstar))^(1-epsilon); % Price index
% Financial Intermediaries
exp(nu_k)=beta*exp(Lambda(+1))*((exp(Rk(+1))-exp(Rd(+1)))*(1-theta)+
theta*exp(Q(+1))*exp(K(+1))/(exp(Q)*exp(K))*exp(nu_k(+1))); % shadow excess value of loans
exp(nu_b)=beta*exp(Lambda(+1))*((exp(Rb(+1))-exp(Rd(+1)))*(1-theta)+
theta*exp(B(+1))/(exp(B))*exp(nu_b(+1))); % shadow value of bonds
exp(nu_blb)=beta*exp(Lambda(+1))*((exp(Rlb(+1))-exp(Rd(+1)))*(1-theta)+
theta*exp(Blb(+1))/(exp(Blb))*exp(nu_blb(+1))); % shadow value of long-term bonds
exp(nu_n)=beta*exp(Lambda(+1))*(exp(Rd(+1))*(1-theta) +
theta*(exp(N(+1))/exp(N))*exp(nu_n(+1))); % shadow value of equity
exp(nu_k)=exp(nu_b); //+exp(nu_blb); % relation between shadow values of loans and bonds
exp(nu_b)=exp(nu_blb); % relation between shadow values of short and long bonds
exp(Phi)=exp(nu_n)/(lambda-exp(nu_k)); % Leverage of Banks
exp(N)= theta*((exp(Rp)-exp(Rd))*exp(Phi(-1)) + exp(Rd))*exp(N(-1)) + chi*exp(N(-1)); % law of motion for net worth
exp(N)+exp(D)= exp(Q)*exp(K)+exp(B)+exp(Blb); % balance sheet identity of banks
exp(portf_B) = exp(Q)*exp(K)+exp(B)+exp(Blb); % Banks' portfolio
exp(Q)*exp(K) = exp(Om) * exp(Phi) * exp(N); % claims on capital assets
exp(B) = exp(Om_B) * exp(Phi) * exp(N); % claims on short bonds
exp(Blb) = exp(Zet) * exp(Phi) * exp(N); % claims on long bonds
exp(ERk)= exp(Rk(+1)); % Expected Return on Capital
exp(ERb) = exp(Rb(+1)); % Expected Return on Bonds
exp(ERlb) = exp(Rlb(+1)); % Expected Return on long bonds
exp(prem) = exp(Rk(+1))-exp(Rd(+1)); % Loan Premium
exp(prem2) = exp(Rb(+1))-exp(Rd(+1)); % Bond Premium
exp(prem3) = exp(Rlb(+1))-exp(Rd(+1)); % Long bond premium
exp(Rp) = exp(Rk)*exp(Om(-1)) + exp(Rb)*(Om_B(-1)) + exp(Rlb)*exp(Zet(-1)); % Return on portfolio
% Fiscal policy
%G = (1-rho_g)*G_ss + rho_g*G(-1) + eg; %
//alp*exp(B) = exp(Rb(-1))*exp(B(-1))+exp(Rlb(-1))*exp(Blb(-1))+exp(G)-exp(T)-(1-alp)*exp(Blb); % Government Budget Constraint
//exp(G) = G_ss*exp(g); % Government consumption
//exp(T) = T_ss; % Taxes
//exp(Gy) = exp(G)/Y_ss; % Share of gov. consumption in GDP
//# TotalFinancingNeed = exp(Rb(-1))*exp(B(-1)) + exp(Rlb(-1))*exp(Blb(-1)) + exp(G) - exp(T);
//(1-alp)*exp(Blb) = alp*exp(B)-TotalFinancingNeed ;
exp(Rlb) = exp(Rb) + term_premium;
# TotalFinancingNeed = exp(Rb(-1))*exp(B(-1)) + exp(Rlb(-1))*exp(Blb(-1)) + exp(G) - exp(T);
//exp(B) + exp(Blb) = TotalFinancingNeed; % Government budget constraint
alp*exp(B) + (1-alp)*exp(Blb) = TotalFinancingNeed;
% Debt allocation (where alp is the share of short-term bonds) was davon macht mehr sinn????
//exp(Blb) = TotalFinancingNeed - alp*exp(B);
exp(Blb) = (1-alp) * TotalFinancingNeed;
# DebtRatio = exp(B)/exp(Blb);
//Endogenous alp
% Government consumption
exp(G) = G_ss*exp(g);
% Taxes
exp(T) = T_ss;
% Share of government consumption in GDP
exp(Gy) = exp(G)/Y_ss;
% Additional Equations
exp(Y) = exp(C) +exp(G) + exp(I); % Aggregate resource constraint
exp(i(-1)) = exp(Rd)*exp(infl); % Fisher equation
exp(i) = exp(i(-1))^rho_i* (i_ss*exp(infl)^kappa_pi*((Y)/(Y(-1)))^(kappa_y))^(1-rho_i)*exp(-e_i); % Taylor rule
% Shocks
a = rho_a*a(-1)-e_a; %42. TFP shock (negativ wegen -e_a)
ksi= rho_ksi*ksi(-1)-e_ksi; %43. Capital quality shock
g = rho_g*g(-1)+e_g; %44. Government consumption shock
end;
initval;
Y=log(Y_ss);
Ym=log(Ym_ss);
K=log(K_ss);
L=log(L_ss);
I=log(I_ss);
C=log(C_ss);
G=log(G_ss);
Q=log(Q_ss);
U_c=log(U_c_ss);
Lambda=log(Lambda_ss);
Rk=log(Rk_ss);
Rd=log(Rd_ss);
N=log(N_ss);
Pm=log(Pm_ss);
w=log(w_ss);
Dis=log(Dis_ss);
F=log(F_ss);
Z=log(Z_ss);
i=log(i_ss);
prem=log(prem_ss);
a=0.00000000;
ksi=0.00000000;
g=0.00000000;
infl=0.00000000;
inflstar=0.00000000;
e_a=0.00000000;
e_ksi=0.00000000;
e_g=0.00000000;
e_i=0.00000000;
B=log(B_ss);
Blb=log(Blb_ss);
D=log(D_ss);
T=log(T_ss);
Phi=log(Phi_ss);
Rb=log(Rb_ss);
Rlb=log(Rlb_ss);
nu_k=log(nu_k_ss);
nu_b=log(nu_b_ss);
//nu_blb=log(nu_blb_ss);
nu_n=log(nu_n_ss);
prem2=log(prem2_ss);
prem3=log(prem3_ss);
ERk=log(Rk_ss);
ERb=log(Rb_ss);
ERlb = log(Rlb_ss);
Rp=log(Rp_ss);
portf_B=log(portf_B_ss);
Om=log(Om_ss);
Gy=log(Gy_ss);
Zet = log(Zet_ss);
Om_B = log(Om_B_ss);
end;
steady;
check;
shocks;
var e_a=sigma_a^2;
%var e_b=sigma_b^2;
%var e_ksi=sigma_ksi^2;
%var e_g=sigma_g^2;
%var e_g=25;
%var e_i=sigma_i^2;
end;
stoch_simul(order = 1, irf = 40) B Blb Rb Rlb Y infl a i;