Steady state okay, but no stable equilibrium

[rbcman]

Hello,

I am replicating a fairly simple RBC model with a human capital sector. Particularly, I am trying to replicate this paper: https://www.aeaweb.org/aea/2013conferen ... ?pdfid=191

I have the code below and I try to solve it using logs (in exp). The steady states exist but it gives me error message:

One of the eigenvalues is close to 0/0 (the absolute value of numerator and denominator is
smaller than 1e-006!

I know I have entered the model correctly but dynare give me error message. I checked the other topics related to this but the answer is usually vague. Any help is appreciated. Below is the code I am using.

// Endogenous variables
var Y C N M L IK IH K H V Z S lambda mu YV YK YH YN IHV IHK IHM IHH;

// Exogenous variables
varexo z s;

// Parameters
parameters beta sigma A phi1 phi2 deltak deltah Ag Ah rhoz rhos sigmaz sigmas;
beta = 0.986;
sigma = 1;
A = 1.55;
phi1 = 0.36;
phi2 = 0.11;
deltak = 0.02;
deltah = 0.005;
Ag = 1;
Ah = 0.0461;
rhoz = 0.95;
rhos = 0.95;
sigmaz = 0.0007;
sigmas = 0.0007;

// Model structure
model;
exp(L) = 1-exp(M)-exp(N);
exp(L)^A*(exp(C)*exp(L)^A)^(-sigma) = exp(lambda);
A*exp(C)*exp(L)^(A-1)*(exp(C)*exp(L)^A)^(-sigma)= exp(lambda)*exp(YN);
A*exp(C)*exp(L)^(A-1)*(exp(C)*exp(L)^A)^(-sigma)= exp(mu)*exp(IHM);
exp(lambda)*exp(YV) = exp(mu)*exp(IHV);
exp(lambda) = beta*(exp(lambda(+1))*(exp((YK(+1))+1-deltak))+exp(mu(+1))*exp(IHK(+1)));
exp(mu) = beta*(exp(lambda(+1))*exp(YH(+1))+exp(mu(+1))*(exp(IHH(+1))+1-deltah));
exp(C) = exp(Y) - exp(IK);
exp(K) = exp(IK) + (1-deltak)*exp(K(-1));
exp(H) = exp(IH) + (1-deltah)*exp(H(-1));
exp(Y) = Ag*exp(Z)*(((V))^(phi1))*((exp(K(-1)))^(phi1))*((exp(N))^(1-phi1))*((exp(H))^(1-phi1));
exp(YV) = phi1*Ag*exp(Z)*(((V))^(phi1-1))*((exp(K(-1)))^(phi1))*((exp(N))^(1-phi1))*((exp(H))^(1-phi1));
exp(YK) = phi1*Ag*exp(Z)*(((V))^(phi1))*((exp(K(-1)))^(phi1-1))*((exp(N))^(1-phi1))*((exp(H))^(1-phi1));
exp(YN) = (1-phi1)*Ag*exp(Z)*(((V))^(phi1))*((exp(K(-1)))^(phi1))*((exp(N))^(-phi1))*((exp(H))^(1-phi1));
exp(YH) = (1-phi1)*Ag*exp(Z)*(((V))^(phi1))*((exp(K(-1)))^(phi1))*((exp(N))^(1-phi1))*((exp(H))^(-phi1));
exp(IH) = Ah*exp(S)*((1-(V))^(phi2))*((exp(K(-1)))^(phi2))*((exp(M))^(1-phi2))*((exp(H))^(1-phi2));
exp(IHV) = phi2*Ah*exp(S)*((1-(V))^(phi2-1))*((exp(K(-1)))^(phi2))*((exp(M))^(1-phi2))*((exp(H))^(1-phi2));
exp(IHK) = phi2*Ah*exp(S)*((1-(V))^(phi2))*((exp(K(-1)))^(phi2-1))*((exp(M))^(1-phi2))*((exp(H))^(1-phi2));
exp(IHM) = (1-phi2)*Ah*exp(S)*((1-(V))^(phi2))*((exp(K(-1)))^(phi2))*((exp(M))^(-phi2))*((exp(H))^(1-phi2));
exp(IHH) = (1-phi2)*Ah*exp(S)*((1-(V))^(phi2))*((exp(K(-1)))^(phi2))*((exp(M))^(1-phi2))*((exp(H))^(-phi2));
Z = rhoz*Z(-1)+z;
S = rhos*S(-1)+s;
end;


// Initial values
initval;
Y = 0;
C = 0;
N = 0;
M = 0;
L = 0;
K = 0;
H = 0;
IK = 0;
IH = 0;
YK = 0;
YN = 0;
YH = 0;
YV = 0;
IHK = 0;
IHM = 0;
IHH = 0;
IHV = 0;
lambda = 0;
mu = 0;
V = 0.95;
Z = 0;
S = 0;
end;

steady(solve_algo=4,maxit=1000);

model_diagnostics;
shocks;
var z = sigma;
var s = sigma;
end;
check;
stoch_simul(periods=1000);

[jpfeifer]

Why does the referenced paper have 7 equations while you have 22?

[rbcman]

Good one. What I tried to do was to write the derivatives of the production function and human cap production function separately so there were additional equations. The structural equations are all the same.

[jpfeifer]

Please try to replicate the paper as is before adding anything to it.

[rbcman]

Hello,

I did as you instructed and the model now has a stable steady state. There are 9 equations - 7 that is in the appendix plus 2 shocks. But now I am getting the error message:

There are 8 eigenvalue(s) larger than 1 in modulus
for 7 forward-looking variable(s)

The rank condition ISN'T verified!

Does it mean that there is a timing issue? Human capital is pre-determined so I have it with (-1) but it still gives me that same error message. Any help is appreciated.

Below is the code:


// Replication of Real Business Cycles with a Human Capital
// Investment Sector and Endogenous Growth:
// Persistence, Volatility and Labor Puzzles
//
// https://www.aeaweb.org/aea/2013conferen ... ?pdfid=191
//
// Endogenous variables
var C N M Z V K H S P;

// Exogenous variables
varexo z s;

// Parameters
parameters beta sigma A phi1 phi2 deltak deltah Ag Ah rhoz rhos sigmaz sigmas gamma;
beta = 0.986;
sigma = 1;
A = 1.5455;
phi1 = 0.36;
phi2 = 0.11;
deltak = 0.02;
deltah = 0.005;
Ag = 1;
Ah = 0.0461;
rhoz = 0.95;
rhos = 0.95;
sigmaz = 0.0007;
sigmas = 0.0007;
gamma = 0.0042;

// Model structure
model;

// equation 28
A*exp(C)/(1-exp(N)-exp(M))=(1-phi1)*Z*(((exp(V)*exp(K(-1)))/(exp(N)*exp(H(-1))))^phi1)*H(-1);

// equation 29
(1-phi1)*exp(V)/(phi1*exp(N))=(1-phi2)*(1-exp(V))/(phi2*exp(M));

// equation 30
exp(P)=((phi1/phi2)^phi2)*(((1-phi1)/(1-phi2))^(1-phi2))*(exp(V)*exp(K(-1))/(exp(N)*exp(H(-1))))^(phi1-phi2);

// equation 31 - Euler
1/beta=((exp(C)/exp(C(+1)))^sigma)*(1+gamma)^(-sigma)*((1-exp(N(+1))-exp(M(+1)))/(1-exp(N)-exp(M)))^(A*(1-sigma))*(phi1*exp(Z(+1))*((exp(V(+1))*exp(K)/(exp(N(+1))*exp(H)))^(phi1-1)+1-deltak));

// equation 32 - Euler
1/beta=(exp(P(+1)/exp(P)))*((exp(C)/exp(C(+1)))^sigma)*(1+gamma)^(-sigma)*((1-exp(N(+1))-exp(M(+1)))/(1-exp(N)-exp(M)))^(A*(1-sigma))*((exp(N(+1))+exp(M(+1)))*(1-phi2)*exp(S(+1))*((1-exp(V(+1))))*exp(K)/(exp(M(+1))*exp(H))+1-deltah);

// equation 33
exp(C)+(1+gamma)*exp(K)-(1-deltak)*exp(K(-1))=Ag*exp(Z)*((exp(V)*exp(K(-1)))^phi1)*(exp(N)*exp(H(-1)))^(1-phi1);

// equation 34
(1+gamma)*exp(H)-(1-deltah)*exp(H(-1))=Ah*exp(S)*(((1-exp(V))*exp(K(-1)))^phi2)*((exp(M)*exp(H(-1)))^(1-phi2));

// technology shock
Z = rhoz*Z(-1)+ z;

// technology shock
S = rhoz*S(-1)+ s;

end;


// Initial values
initval;
C = 0;
N = 0;
M = 0;
Z = 0;
V = -2;
K = 0;
H = 0;
S = 0;
P = 0;
Z = 1;
S = 1;
end;

steady(solve_algo=4,maxit=1000);

model_diagnostics;
shocks;
var z = sigma;
var s = sigma;
end;
check;
stoch_simul(periods=1000);

[jpfeifer]

Hard to tell. You need to check your equations and the ones in the paper. In equation 32 for example, you have M(+1) in the denominator and the paper just has M. In equation 28, you have Z but it should be exp(Z)