# Real interest rate vs nominal interest rate

Dear Sirs:

When we set up a DSGE model, may we convert the nominal interest rate to real interest rate?That is, can we set r=nr/pie(+1), and using r to replace all the terms of nr/pie(+1) ? where r is gross real interest rate, nr is gross nominal interest rate, and pie denotes the inflation rate?

Thank you very much

Paul

Yes, you could do that. But why do you want to substitute it out?

Thank you Professor Pfeifer.

Because I find a strange situation. I give a exercise (please see attached file) to state my doubt.

When I replace nr(-1)/pie to r(-1), the irfs change.

Attached file is based on Rubio(2015) with fixed markups, equation (5a) adopts c2+q*h2+r(-1)*b2(-1)=w2*n2+q*h2(-1)+b2;
i.e. using the real gross interest rate r(-1) in the budget constraint. We could see the neutrality of money when facing a interest rate shock.

But after disabling equation (5a) and enabling equation (5b):
c2+q*h2+nr(-1)/pie*b2(-1)=w2*n2+q*h2(-1)+b2;
i.e. using the nominal gross interest rate(-1)/ inflation rate to replace the gross interest rate.
One could see that facing a interest rate shock, we could not see the neutrality of money. And the responses to the technology shock also change.

I can not figure out the reason…Is this situation correct?

Thank you.

PaulRubio2015.mod (2.7 KB)

You did not correctly do the replacement. The real interest rate at time t-1 is (ignoring Jensen’s Inequality as we are working at first order)
\frac{R_{t-1}}{E_{t-1}(\pi_t)}\neq \frac{R_{t-1}}{\pi_t}
You need to use

c4+q*h2+nr(-1)/EXPECTATION(-1)(pie)*b2(-1)=w2*n2+q*h2(-1)+b2;


Thank you Professor Pfeifer.

There exits another question.
Generally, we use the specification nr(-1)/pie*b2(-1) to calculate the last period loan which needs to repay, see e.g. Iacoviello (2005).

Shall this specification nr(-1)/EXPECTATION(-1)(pie)b2(-1) be correct one, rather nr(-1)/pieb2(-1)?

Paul

That depends on the source of equation. The model setup should tell you whether what is relevant is the ex-ante or ex-post real interest rate. In the equation you mention I guess it is the latter. Agents are payed the nominal interest rate. The inflation rate comes from expressing the equation in real terms, i.e. deflating. In this case, what matters is the realized real return, not what you expected it to be earlier on.

Thank you very much Professor Pfeifer.

Best Regards

Paul

Professor Pfeifer:

Sorry about bothering again… Because it is still something confusing me.

When it is in a RBC Model, the the budget constraint is:

c2+q*h2+r(-1)*b2(-1)=w2*n2+q*h2(-1)+b2;


where r(-1) has already represented the ex-post real interest rate.

But according your mention, r(-1) should be expressed by following equation when we consider the nominal price.

r(-1)=nr(-1)/EXPECTATION(-1)(pie).


This term r(-1) changes to a ex-ante term. Does it seem some odd?

Or, in the RBC model, actually we consider the ex-ante real interest rate, rather than the ex-post real interest rate?

Best Regards

Paul

In the basic RBC model there are no bonds. It all depends on your setup. You need to approach this with economic intuition. It will tell you whether it is the ex-ante or ex-post real interest rate that matters.

Thank you very much Professor Pfeifer.

Best Regards

Paul