Predetermined Variables - Steady State- Deterministic models

[msh855]

Hi,

I need some help/clarification with the following.

I have a deterministic model, with capital and debt, the model is very simple. I could solve it by hand, however there some subtle details about stability and the choice of predetermined variables. I present the model below

Code: Select all

var k b r w in;
varexo g;
predetermined_variables k;
parameters beta alpha A n k0;

beta=0.3;
alpha=1/3;
A=6.35;
n=1.097;

k0=0.228771;
b0=(((beta)/(beta+1))*(1-alpha)*A*k0^alpha)/(1+n)-k0;
r0=alpha*A*k0^(alpha-1);
w0=(1-alpha)*A*k0^alpha;
in0=(r0*(k0+b0))/w0;

model;
(1+n)*k(+1) = beta*(1-alpha)*A*k^alpha/(1+beta)-(alpha*A*k^(alpha-1))*b- g;
(1+n)*b(+1) =(alpha*A*k^(alpha-1))*b+ g;
r = alpha*A*k^(alpha-1);
w = (1-alpha)*A*k^alpha;
in= r*(k+b)/w;
end;

initval;
k     = k0;
b     = b0;
in    = in0;
r     = r0;
w     = w0;
g   =-0.2;
end;

endval;
g=-0.1;
end;

steady;
check;

simul(periods=20,maxit=1000);

Now, to get a bite from the model, this is a conventional OLG model with debt. First, I solved the model and used some nonlinear equation solver. I know that there are two steady states, the first is the one i mention in the code, and actually the one I am mostly interested. In this steady state the one eigenvalue is larger than one and the other is lower. So, according to the theory this describes a saddle point. In theory therefore, convergence will happen iff the relationship between my state variables are along the stable eigenvector. Now, as you can see i have two law of motions, one describes capital and the other debt accumulation. Agents choose both, capital and debt in period t (i.e the k(+1) and b(+1) are choosen by agents in period t). However, k(+1) is going to be used by firms in period t+1, while b(+1), is used at period t from the government to finance spending and pay the interest rate on debt issued at t. My first question, in the way i wrote my model, i could obtain convergence, since the rank condition is satisfied, but I am confused from the Dynare manual on whether i should also include debt as a predetermined variable, i.e write the relevant section as:

Code: Select all

predetermined_variables k b;

If i indeed do this, the output is rubbish for me. So, either is indeed the model rubbish unless the clarification about the predetermined variable is correct in the way i did in the code, for the choice of agents described in the text above.

Now, for the second steady state which i also know, that is stable according to theory. repeated complex eigenvalues with real part less than one. However, if i keep the choice of predetermined variables as in the code above, the rank condition is not satisfied but i obtain convergence. Only if i change to define both variables as predetermined i can in addition satisfy the rank condition. Here is for instance the code with the second steady state.

Code: Select all

var k b r w in;
varexo g;
predetermined_variables k;
parameters beta alpha A n k0 kss b0;

beta=0.3;
alpha=1/3;
A=6.35;
n=1.097;

kss=1.24566;

k0=kss;
b0=(((beta)/(beta+1))*(1-alpha)*A*k0^alpha)/(1+n)-k0;
r0=alpha*A*k0^(alpha-1);
w0=(1-alpha)*A*k0^alpha;
in0=(r0*(k0+b0))/w0;

model;
(1+n)*k(+1) = beta*(1-alpha)*A*k^alpha/(1+beta)-(alpha*A*k^(alpha-1))*b- g;
(1+n)*b(+1) =(alpha*A*k^(alpha-1))*b+ g;
r = alpha*A*k^(alpha-1);
w = (1-alpha)*A*k^alpha;
in= r*(k+b)/w;
end;

initval;
k     = k0;
b     = b0;
in    = in0;
r     = r0;
w     = w0;
g=-0.2;
end;

endval;
g=-0.1;
end;

steady;
check;

simul(periods=20,maxit=1000);

In the second case, qualitatively i have the same results no matter what my predetermined variables are but quantitatively there are slightly different depending or not i define my debt as predetermined or not.

So from all the above, i am confused about what the dynare needs from the rank condition, for the first example is necessary to obtain convergence since my steady state is saddle point, but is not necessary in the second case. What kind of strategy should i use here ?

Apologies for my long message but i believe that i had to be as specific as possible. I would really appreciate a swift response.

Best,
Moustafa

[jpfeifer]

From the way it looks, your model has standard timing. Both capital and bonds should be predetermined. The bonds with which the government enters the current period and which have to be repaid have been decided upon in the previous period and cannot be changed anymore. Thus, bonds are clearly predetermined.

However, in the first model equation, capital tomorrow does not depend on the bonds savings chosen today, but on the ones chosen last period. This might be the source of your problems.

[msh855]

Thank you for the reply. I believe the source of my problem is the saddle point property in one of the steady states, which i do not know how to handle in dynare for the model as it is. Since one of my eigenvalues are larger than one, i cannot obtain converge in dynare unless i define one predetermined variable and not two. Although clearly both, capital and bonds are predetermined. Correct ?

[jpfeifer]

For an outsider, it looks like your model is misspecified in having two predetermined variables, but your described saddle path property that is only compatible with having one predetermined variable.

[msh855]

I understand your point, but the model is fine. Is a standard OLG model with debt and constant deficit. Actually, is taken from a book. This particular model has two steady states, and only one is stable. I think if someone wants to be consistent i have to restrict my analysis in the stable one, where putting two predetermined variables is fine. For the other steady state (the saddle one) which is the interesting one, I do not think i can do something as it is in dynare. Thank you for your help

[jpfeifer]

Where exactly is your problem? Finding the steady state or working with the model afterwards? For the steady state, the timing should not matter.

[msh855]

No, basically is for the "afterwords" part. Let me be specific. In the "stable" steady state i have positive capital and negative debt. This is "ok" but i would rather prefer positive debt. This is why i was interested with the other steady state where both capital and debt are positive numbers, but the problem there as said before, is that this steady state is a saddle. So convergence is not obtained unless I define only one predetermined variable instead of two.