Bubble, indeterminacy and rank condition failure

Stochastic simulations
1

I’ve been recently working on the replication of the paper “MONETARY POLICY AND RATIONAL ASSET PRICE BUBBLES”(Gali,2013). In this paper, the author uses a simple OLG framework to study bubbly assets. However, when entering the linear system into dynare, it reports that there’s a indeterminacy, even if I’ve already treated the indeterminacy problem by introducing the sunspot shock like what is done in this paper. I’ve replicating the flexible price model and everything is normal. However, if I turn to sticky price model, change the equation: wt=0 into E(t-1){wt}=0, the problem of indeterminacy (rank condition unsatisfied) emerges. Can anyone help me solve this problem? Thanks!

The code is below:

var c1 c2 r d b q b w pi we;
varexo e_u e_xi;
parameters eps beta  PI_ss U_ss B_ss R_ss QB_ss Gamma phi_pi phi_b;
eps=4;
beta=0.75;
PI_ss=1;
U_ss=0.04;
params=[eps,beta,U_ss];
B_ss=solve_B(params);
QB_ss=B_ss+U_ss;
R_ss=(1/eps+B_ss)/(beta*(1-1/eps-B_ss));
Gamma=eps*B_ss/(eps*B_ss+1);
phi_pi=4;
phi_b=-2;

model(linear);
0=c1+beta*R_ss*c2;
c1=c2(+1)-r;
c2=(1-Gamma)*d+Gamma*b;
q=R_ss*b+(1-R_ss)*e_u;
b=q(-1)+r(-1)+e_xi;
we=w(+1);
we(-1)=0;
w=d;
r=phi_pi*pi+phi_b*q;
end;

shocks;
var e_u=0.0001;
var e_xi=0.0001;
end;

model_diagnostics;
check;
stoch_simul(irf=40,periods=2000);
2

The m.file is below:

function ANS=solve_B(params)
% params=[eps,beta,U_ss]
% U=(beta*C1/C2-1)*B
% C1=W-B; C2=D+B;
% (1+beta)*B^2+(D+U-beta*W)*B+U*D=0;
% the smaller root is the stable s.s
% easy to show for B>0, it must be 
% (1) delta>=0 (2) U<beta*(1-1/eps)-1/eps
eps=params(1); beta=params(2); U_ss=params(3);
W=1-1/eps; D=1/eps; U=U_ss;
delta=(D+U-beta*W)^2-4*(1+beta)*U*D;
if delta>=0
    ANS=1/(2*(1+beta))*(beta*W-D-U-delta^0.5);
else
    ANS=-1;
end
end
3

You mean equation w=d; in the model you posted? Or lemme ask, which equation in the model block corresponds to E(t-1){wt}=0?

4

we=w(+1); we(-1)=0;
this means E(t-1)(wt)=0;

5

Oh I see, lemme ask again though so

  1. in your flexible model, you have wt=0 (i.e., current wage = 0), and no issues.
  2. but in your sticky model, you want to change the condition wt=0 to E(t-1){wt}=0 (i.e., future wages are zero), so you introduce another variable, ‘we’, so that E(t-1)(wt)=0 is captured by we=w(+1); we(-1)=0;?

Btw, how did you write the condition wt=0 in your flexible model? Maybe, you don’t need to introduce the variable ‘we’. Or perhaps ‘we’ is also in your flexible model? Sorry, just trying to understand your problem.

Of course, if wt = 0 works in your flexible model and you change it to E(t-1){wt}=0 (meaning E{w_{t+1}}=0), you will now have more forward-looking variables than eigenvalues…causing indeterminacy issues.

6

Indeed, there is a difference between something always being 0 and only its average being 0.