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Hi

I have the following code for a basic New Keynesian model in a deterministic set up with one negative technology shock in period 1 (size 0.01) including a restriction on the nominal interest rate (ZLB constrain)
I would like to extend my Taylor rule to allow for an inflation target ("pi_target") but I'm struggling with it I only get one eigenvalue outside the unit circle (need 2 to satisfy the Blanchard-Kahn condition).

Also - I would like to allow for the nominal interest rate to be positive in equilibrium which I also cannot figure out how to do properly. I would like it to be positive in steady state to mimic reality and so that the ZLB is not binding all the time.

Code: Select all

 var x pi i ez;
 varexo epsz;
 parameters betta sigma omega eta phipi phix rhoz pi_target;

 betta  =0.98; %Discount factor
 omega =0.75;  %Calvo parameter price stickiness
 phipi =1.5;   %Taylor rule: feedback on expected inflation
 phix  =0.125;   %Taylor rule: feedback on output gap
 eta = 1;      %Labor supply parameter
 sigma = 2;    %CRRA parameter in consumption
 rhoz = 0.9;  %persistence of technology shock
pi_target=0;
 
 model(linear);
 x = x(+1)-(1/sigma)*(i-pi(+1))-ez;
 pi = betta*pi(+1)+((sigma+eta)*(1-betta*omega)*(1-omega)/omega)*x;
 i = max(0,phipi*(pi-pi_target)+phix*x);
 ez = rhoz*ez(-1)+epsz;
 end;


 steady;

 shocks;
 var epsz;
periods 1;
values 0.01;
end;

 check;
 simul(periods=100);
rplot x;
rplot pi;
rplot i;
rplot ez;

In that case, you need to extend your model. Also note that there is a distinction between the deviation of the nominal interest rate from its mean, which is 0 in steady state and the level of the nominal interest rate, which in steady state is the real interest rate + inflation rate. If you compare with Gali's textbook, he does not use percentage deviations from steady state, but the logarithm of the nominal interest rate (because his model is roughly log-linear)

My model is based on the Walsh textbook but I see your point. So I should base my basic NK model on Gali's textbook in order to get the desired result?

Or, my best guess is that it should be enough to rewrite the max-function to:

max(-(1/beta-1), Taylor-rule)

as the nominal interest rate in steady state must be 1/beta-1 (disregarding the pi_target)

That is basically the same, but only if your Taylor rule contains the constant term, giving you information on the level.

Wasn't that irrelevant if it is deviations from the steady state?

I may have misunderstood you. All I am saying is: if you are not using levels, it depends if you are talking about absolute deviations or percentage deviations. Say the real interest rate in steady state is 1/beta=1.05. In that case, the absolute deviation is at most allowed to be 0.05 percentage points. But the percentage deviation can be at most 0.05/1.05

Hi

That makes perfect sense. Do you by any chance know what is the case in the two textbooks, Walsh and Gali, respectively?

For the basic NK model in the Walsh (2010) textbook I think the nominal interest rate is the absolute deviation from steady state based on the appendix in chapter 2 (page 84)

But to my first question: How come that the Blanchard Condition is not fulfilled?

I would need the mod-file.

See the attached

Your model still does not complete incorporate the inflation target. Looking at the steady states

Code: Select all

STEADY-STATE RESULTS:

x         -0.00154024
pi        -0.0204082
i         -0.0204082
ez        0


there still is a problem.