# The BGG Model with Bubbles

I am trying to replicate

The BGG model with Bubbles from

Bernanke & Gertler “Monetary Policy & Asset Price Volatility” paper (2000).

The link to the paper is https://www.nber.org/system/files/working_papers/w7559/w7559.pdf

However, I came across a problem,

which is that there are 18 equations but 17 endogenous variables in the model.

I would like to solve this problem.

Any help would be greatly appreciated.

The following are my code:

var c ce i g r s k q rq mc y rs n z l pi rn rk;

varexo e_g e_z;

parameters C_Y Ce_Y, I_Y, G_Y sigma phi delta b beta R Rq Y_N N_K alpha vtheta psi chi theta_f theta_b K_N rho_g rho_z kappa tau Rk;
C_Y = 0.61;
Ce_Y = 0.04;
I_Y = 0.18;
G_Y = 0.2;
sigma = 1.0;
phi = 0.25;
delta = 0.025;
b = 0.98*(1-0.025);
beta = 0.99;
R = 1/beta;
Rq = R+0.01;
Y_N = 0.28;
N_K = 0.5;
alpha = 0.98;
vtheta = (1-delta)/(alphaY_NN_K + 1 - delta);
psi = 0.05;
chi = 1.33;
theta_f = 0.5;
theta_b = 0.5;
K_N = 1/N_K;
rho_g = 0.95;
rho_z = 1.0;
rnss = 1.02; // Target nominal interest
kappa = 0.086;
tau = 0.95;
Rk = R + 0.02;

model;

// Aggregate Demand

y = C_Yc + Ce_Yce + I_Yi + G_Yg; // A1 Aggregate Demand
c = -sigmar + c(+1); // A2 Euler Condition for Household Consumption
ce = s + k(+1); // A3 Assumption: Entrepreneurial Cons is prop. to stock value
q(+1) = phi
(i(+1) - k(+1)); // A4 Investment is prop. to fundamental value of capital

//Returns to Stock and Capital

s - q = (1-delta)/(bRq)(s(+1)- q(+1)); // A5 The expected evolution of the bubble
rs(+1) = rq(+1) - (1-b)(s-q); // A8 Relation between stock return and fundamental return
rs(+1) = r - psi

// Aggregate Supply

y = z + alphak + (1-alpha)l; // A10 Cobb Douglas Production Function
y - l + mc - c = (chi-1)l; // A11 Household Labor-Leisure Decision
pi = kappa
mc + theta_f
pi(+1) + theta_b
pi(-1); // A12 Evolution of inflation (sticky prices) The Gali n Gertler NK phillip curve

// Evolution of state variables and shock process

k(+1) = delta*i + (1-delta)*k; // A13 Capital

n = Rq * ( (K_N) * (rs - rs(-1)) + (1-taurk)/(tau)y + n(-1) ); // A14 Internal Equity*

g = rho_gg(-1) + e_g; // A15 Government Spending AR(1) process
z = rho_z
z(-1) + e_z; // A16 Total Factor Productivity AR(1) process (the technology obey BGG 4.27)

// Monetary Policy Rule

rn = beta* pi(+1); // A17 Policy Rule Accommodative***
r = rn - pi(+1); // A18 Fisher Equation***

end;

check;

shocks;
var e_g; stderr 0.1;
var e_z; stderr 0.1;
//var e_rn; stderr 1.0;

end;

stoch_simul(irf=24);

See the discussion in

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