# Indeterminacy in basic NK model with an exchange rate peg

Dear all,
I have a question regarding Gali and Monacelli’s (2002,2005) basic open economy NK model.

We know that in order to close the model, we need a monetary policy rule. We can have either a Taylor rule (actually, two types of Taylor rules depending on whether we target home goods inflation or CPI inflation, but that’s not the point of my question). Alternatively, we can have an exchange rate peg so e-e(-1)=0 i.e. no change in the nominal exchange rate. My question is the following:

Whenever we have a standard Taylor rule i.e. a floating nominal exchange rate, in the analytical derivation we substitute the nominal exchange rate with whatever Taylor rule we’ve got and we end up with a system of two difference equations and two variables: output gap and domestic goods inflation and then we find the stability conditions. All clear.

However, whenever we have a nominal exchange rate peg, then the analytical derivation isn’t so clear to me. That’s why I tried solving it first with DYNARE and I have the following cases:

1. Whenever I set e-e(-1)=0 meaning q-q(-1) = picpi* - picpi, where e is the nominal exchange rate, q is the real exchange rate, picpi* is world CPI inflation and picpi is home CPI inflation, then things work fine and the domestic nominal exchange rate doesn’t move unless the foreign one moves (which is the idea of a peg).

2. However, when I do it the other way around i.e. set the nominal exchange rate to zero (R=0), then I get indeterminacy. I’d like to do it like that since I can then solve the same system of difference equations whereby I will just delete R from the so called “Dynamic IS curve” equation. Otherwise, I’m not sure how I can substitute the nominal exchange rate or maybe I shouldn’t and solve a larger system of equations…

q-q(-1) = picpi* - picpi is equivalent to sigmaa*((y-y(-1))-(ystar-ystar(-1)) )= -pih according to Gali’s notation whenever all foreign variables are exogenous, but this doesn’t help me much since the nominal interest rate isn’t in there…

Any suggestions?

P.S. please find attached my code for the basic open economy NK model where it’s equation 7 that determines what monetary policy rule the Central Bank follows.

Best,
Peter
opennk_big.mod (1.6 KB)

I am not sure I understand what you are doing. You write

``set the nominal exchange rate to zero (R=0), then I get indeterminacy``

R is the nominal interest rate in your mod-file, not the nominal exchange rate. Why should a interest rate peg work? In the background of the model is the uncovered interest rate parity. If you fix the change in nominal exchange rates, this uniquely determines the nominal interest rate and inflation. Fixing the interest rate in contrast does not seem to be sufficient, to determine inflation and the nominal exchange rate

Hi Johannes,
Sorry, made a mistake. When I set the nominal exchange rate to zero, then I get the indeterminacy. But it should be actually that way since there’s no interest rate to clear the market. And yes, if we fix the “change” in the nominal interest rate, this does uniquely determine inflaiton and the interest rate. So all good. Many thanks.

Best,
Peter

The only thing that I get puzzled about is how to get an analytical solution and analytical determinacy conditions since it isn’t as straightforward as with a simple Taylor rule to substitute in the Euler equation (or how it’s called the dynamic IS curve, although I don’t like this name so much). Then I think I’ll end up with a second-order difference equation for the output gap so ytil(+1), ytil and ytil(-1) and I guess I will have to make use of auxiliary variables to solve it…not sure yet actually.

If you get a second order difference equation, you might indeed need an auxiliary variable (if you use the method of undetermined coefficients, that should not be the case, because you plug in the same solution repeatedly)