% Define whether to use interest rate or money growth rate rule
@#define money_growth_rule=0

var 
    pi ${\\pi}$ (long_name='inflation')
    y_gap ${\\tilde y}$ (long_name='output gap')
    y_nat ${y^{nat}}$ (long_name='natural output')
    y ${y}$ (long_name='output')
    yhat ${\\hat y}$ (long_name='output deviation from steady state')
    r_nat ${r^{nat}}$ (long_name='natural interest rate')
    r_real ${r^r}$ (long_name='real interest rate')
    i ${i}$ (long_name='nominal interest rate')
    n ${n}$ (long_name='hours worked')
    m_real ${m-p}$ (long_name='real money stock')
    m_growth_ann ${\\Delta m}$ (long_name='money growth annualized')
    m_nominal ${m}$ (long_name='nominal money stock')
    a ${a}$ (long_name='AR(1) technology shock process')
    r_real_ann ${r^{r,ann}}$ (long_name='annualized real interest rate')
    i_ann ${i^{ann}}$ (long_name='annualized nominal interest rate')
    r_nat_ann ${r^{nat,ann}}$ (long_name='annualized natural interest rate')
    pi_ann ${\\pi^{ann}}$ (long_name='annualized inflation rate')
    z ${z}$ (long_name='AR(1) preference shock process')
    p ${p}$ (long_name='price level')
    w ${w}$ (long_name='nominal wage')
    c ${c}$ (long_name='consumption')
    w_real ${\\frac{w}{p}}$ (long_name='real wage')
    mu ${\\mu}$ (long_name='markup')
    mu_hat ${\\hat \\mu}$ (long_name='markup gap')
    d ${d}$ (long_name='deposits')
    e ${e}$ (long_name='FX rate between paper currency and CBDC')
    tau ${\\tau}$ (long_name='deposit fee')
    B ${B}$ (long_name='bank loans')
    D ${D}$ (long_name='bank deposits')
    R_b ${R^b}$ (long_name='loan rate')
    R_d ${R^d}$ (long_name='deposit rate')
    CBDC ${CBDC}$ (long_name='central bank digital currency')
    spread ${spread}$ (long_name='interest rate spread')
    bank_capital ${K^b}$ (long_name='bank capital')
    leverage ${lev}$ (long_name='bank leverage')
    bank_profit ${\\Pi^b}$ (long_name='bank profit')
    omega_cbdc ${\\omega_{cbdc}}$ (long_name='CBDC share in money demand')
    M ${M}$ (long_name='money holdings')
    V ${V}$ (long_name='velocity of money')
    lambda ${\\lambda}$ (long_name='marginal utility of consumption')
;

varexo 
    eps_a ${\\varepsilon_a}$ (long_name='technology shock')
    eps_z ${\\varepsilon_z}$ (long_name='preference shock innovation')
    eps_fin ${\\varepsilon_{fin}}$ (long_name='financial shock')
;

parameters
    alppha ${\\alpha}$ (long_name='capital share')
    betta ${\\beta}$ (long_name='discount factor')
    rho_a ${\\rho_a}$ (long_name='autocorrelation technology shock')
    rho_z ${\\rho_{z}}$ (long_name='autocorrelation preference shock')
    siggma ${\\sigma}$ (long_name='inverse EIS')
    varphi ${\\varphi}$ (long_name='inverse Frisch elasticity')
    phi_pi ${\\phi_{\\pi}}$ (long_name='inflation feedback Taylor Rule')
    phi_y ${\\phi_{y}}$ (long_name='output feedback Taylor Rule')
    eta ${\\eta}$ (long_name='semi-elasticity of money demand')
    epsilon ${\\epsilon}$ (long_name='demand elasticity')
    theta ${\\theta}$ (long_name='relative risk aversion')
    tau_ss ${\\tau_{ss}}$ (long_name='steady state deposit fee')
    e_ss ${e_{ss}}$ (long_name='steady state FX rate')
    p_param ${p}$ (long_name='log of beta')
    phi_cbdc ${\\phi_{cbdc}}$ (long_name='CBDC adjustment parameter')
    kappa_b ${\\kappa_b}$ (long_name='bank capital requirement')
    chi ${\\chi}$ (long_name='cost of violating capital requirement')
    delta_b ${\\delta_b}$ (long_name='depreciation rate of bank capital')
    kappa_y ${\\kappa_y}$ (long_name='loan demand parameter')
    rho_nu ${\\rho_{\\nu}}$ (long_name='autocorrelation monetary policy shock')
    pi_target ${\\pi^{target}}$ (long_name='inflation target')
;

% Parametrization
siggma = 1;
varphi = 5;
phi_pi = 1.5;
phi_y = 0.125;
theta = 2;
rho_z = 0.5;
rho_a = 0.9;
betta = 0.99;
eta = 3.77;
alppha = 1/4;
epsilon = 9;
tau_ss = 0.01;
e_ss = 1;
p_param = log(betta);
phi_cbdc = 0.5;
kappa_b = 0.08;
chi = 10;
delta_b = 0.1;
kappa_y = 1;
rho_nu = 0.5;
pi_target = 0.02/4; % 2% annual inflation target

model;
#Omega = (1-alppha)/(1-alppha+alppha*epsilon);
#psi_n_ya = (1+varphi)/(siggma*(1-alppha)+varphi+alppha);
#lambda = (1-theta)*(1-betta*theta)/theta*Omega;
#kappa = lambda*(siggma+(varphi+alppha)/(1-alppha));

% New Keynesian Phillips Curve
pi = betta*pi(+1) + kappa*y_gap;

% Cash-in-advance constraint
M(-1)/p + CBDC(-1)/p = c;

% Money demand function
log(M/p) = y - eta*R_d;

% Velocity of money
V = p*y/M;

% Modified budget constraint
c + (M/p) + (CBDC/p) + B/p = w_real*n + (M(-1)/p) + (CBDC(-1)/p) + R_d(-1)*(B(-1)/p);

% Euler equation
c^(-theta) = betta * R_d * (c(+1)^(-theta) / (1 + pi(+1)));

% Output-consumption relationship
y = c;

% Definition natural rate of interest
r_nat = -siggma*psi_n_ya*(1-rho_a)*a + (1-rho_z)*z;

% Definition real interest rate
r_real = R_d - pi(+1);

% Definition natural output
y_nat = psi_n_ya*a;

% Definition output gap
y_gap = y - y_nat;

% Modified Taylor rule (allowing for NIRP)
i = max(-(1-1/betta), phi_pi*(pi - pi_target) + phi_y*y_gap + phi_cbdc*log(CBDC/CBDC(-1)) + z);

% TFP shock
a = rho_a*a(-1) + eps_a;

% Production function
y = a + (1-alppha)*n;

% Preference shock
z = rho_z*z(-1) - eps_z;

% Money growth (derived from money demand equation)
m_growth_ann = 4*(y - y(-1) - eta*(R_d - R_d(-1)) + pi);

% Annualized nominal interest rate
i_ann = 4*i;

% Annualized real interest rate
r_real_ann = 4*r_real;

% Annualized natural interest rate
r_nat_ann = 4*r_nat;

% Annualized inflation
pi_ann = 4*pi;

% Output deviation from steady state
yhat = y - steady_state(y);

% Definition price level
pi = p - p(-1);

% FOC labor
w - p = siggma*c + varphi*n;

% Definition real wage
w_real = w - p;

% Definition nominal money stock
m_nominal = M;

% Average price markup
mu = -(siggma+(varphi+alppha)/(1-alppha))*y + (1+varphi)/(1-alppha)*a;

% Average price markup gap
mu_hat = -(siggma+(varphi+alppha)/(1-alppha))*y_gap;

% Bank profit function
bank_profit = R_b(-1)*B(-1) - R_d(-1)*D(-1) - chi/2*(leverage - kappa_b)^2*bank_capital(-1);

% Bank balance sheet
B = D + bank_capital;

% Bank capital evolution
bank_capital = (1-delta_b)*bank_capital(-1) + bank_profit;

% Leverage ratio
leverage = B/bank_capital;

% Loan demand (from firms)
B = kappa_y*y - n;

% Deposit supply (from households)
D = (1-omega_cbdc)*M;

% CBDC demand
CBDC = omega_cbdc*M;

% Interest rate spread
spread = R_b - R_d;

% Policy rule for CBDC issuance
log(CBDC/CBDC(-1)) = phi_cbdc*(pi - pi_target) + eps_fin;

% Deposit evolution
d = (1-tau)*d(-1)/e + y - c;

% FX rate evolution (simplified)
log(e/e(-1)) = 0.8*log(e(-1)/e(-2));

% Deposit fee evolution (simplified)
tau - tau_ss = 0.8*(tau(-1) - tau_ss);

% CBDC share in money demand (simplified adjustment)
omega_cbdc - 0.5 = 0.9*(omega_cbdc(-1) - 0.5) + 0.1*(R_d - i);

end;

% Define shock variances
shocks;
var eps_a = 1^2;
var eps_z = 0.5^2;
var eps_fin = 0.25^2;
end;

% Check residuals and steady state
steady;
check;

% Stochastic simulation
stoch_simul(order=1, irf=40) y y_gap pi_ann i_ann R_d R_b spread CBDC bank_capital leverage M V;