 Stationarity of Nominal Exchange Rate

Hi there,

I work with a 2-country NK DSGE model. I work with the non-linear form. According to Sims’ Notes, I express all prices in terms of inflation rates (because stationarity). In the system of optimality conditions, the nominal exchange rate does not disappear (for example, it is not possible to express the nominal exchange rate in terms of the change of the nominal exchange rate).

Im my model, there is only a 1% technology shock. Can I assume that the nominal exchange rate is equal to 1 in steady state? I mean, when there are no differences between the 2 countries in the deterministic steady-state, it is reasonable to assume a nominal exchange rate = 1?

Many thanks!

Write down the model in terms of the real exchange rate instead of the nominal exchange rate. That will imply that home and foreign expected inflations will turn up in the UIP condition.
You can back out nominal exchange rate depreciation from the real exchange rate depreciation and the two inflations. And then you can get the level of the nominal exchange rate by cumulating IRFs when necessary.

Reuben

I already use real exchange rates. The nominal exchange rate appears in terms of Schmitt-Grohé & Uribe’s convex foreign bond adjustment cost to close the economy. The cost depends on foreign bond holdings (in excess to steady-state bond holdings, which are zero) which have to be multiplied by the nominal exchange rate and the adjustment cost parameter. It also appears in the resource constraint:

[name='Variation of the nominal exchange rate']
delta_e(+1) = (r_h/r_f)*(1+e*chi_B*b_h);

[name='Home resource constraint (RC)']
y_h = c_h + i_h + ((b_h*e)^2)*chi_B*0.5;
[name='Foreign resource constraint (RC)']
y_f = c_f + i_f + ((-b_h*(1/e))^2)*chi_B*0.5;

where ‘e’ denotes the nominal exchange rate from home-country’s point of view and b_h stands for foreign bond holdings.

I don’t know how to treat ‘e’ in steady state? So far, I use e = 1 (define ‘e’ as free parameter) in steady-state as b_h = 0 anyway.

are r_h and r_f nominal or real?

If you can redefine net foreign assets in real, home currency terms, and e will disappear from the model. Here it looks like b_h is still in foreign currency…
What does your current account equation look like?

Reuben

‘r_h’ (‘r_f’) is home (foreign) nominal interest rate set by the central bank.

The net foreign assets (b_h) are not a problem because only the change in nominal exchange rate appears (delta_e):

[name='Home net foreign assets']
b_h = delta_e*(r_f(-1)/Infl_H)*b_h(-1) + (Infl_x_H/Infl_H)*(x_h-y_h);

For the current account I have:

[name='Home current account (CA)']
ca = b_h - b_h(-1);

It is not clear to me whether you are working with a linearized model or the original nonlinear version.
For example, in the non-linear model, one would not get the equation

delta_e(+1) = (r_h/r_f) *(1+e* chi_B*b_h);

The left hand side and the right hand side would be expressed in utility terms, i.e. the stochastic discount factor would enter your first order conditions. Since you get UIP by combining two FOCs, the expected growth in marginal utility will affect both sides of the equation, but cannot be cancelled out.

Do you also defined b_h as a proportion of output?

On another front, why are the terms of trade missing from the b_h equation?

Cheers
Reuben

I work with the non-linear model.

No, so far I did not.

I tried to express all price levels in terms of inflation. I think I did a mistake in the b_h equation, that is why the terms of trade disappear. How can I handle the non-stationary price levels in the b_h equation?

Therefore, the proper condition has to be:

(beta*lambda(+1)*delta_e(+1)*(r_f/Infl_H(+1)) / (beta*lambda(+1)*(r_h/Infl_H(+1)) = (1+e*chi_B*b_h);

Is that correct?

How can I handle the non-stationary price levels in the b_h equation?

The terms of trade itself is stationary, just like the real ex rate.

For the correct UIP equation, look at the Adolfson paper. your new specification seems correct, but it is good to check.

Ah ok, I see!

Therefore, I don’t have to reformulate for example the intermediate goods market equilibria, which are formulated in prices relative to CPI?

“Home’s intermediate good production has to be equal the sum of optimal home demand for home intermediate goods and optimal foreign demand for home intermediate goods”:

x_h = (1-XiO)*(( p_x_h )^(-psi))*y_h + (XiO)*(( (1/RER)*p_x_h )^(-psi))*y_f ;

where p_x_h = P_x_h / P_h. Therefore, p_x_h terms are stationary (equal to 1 in steady-state) anyway? And also:

[name='Home net foreign assets']
b_h = delta_e*(r_f(-1)/Infl_H)*b_h(-1) + (p_x_h)*(x_h-y_h);

Thank you very much for a confirmation. It helps me a lot.

Nonetheless, when the terms of trade are stationary and the real exchange rate as well, then the nominal exchange rate has to be stationary too? Thats why I would fix it in steady-state equal to 1.

RER = (P_f*e)/(P_h);

Relative prices do not necessarily have to be zero in steady-state. Yes, they could be, but that depends on the markups that the producers charge.

I use a fix cost component to eliminate steady-state gains of intermediate producers. Therefore, I think I can assume relative producer price divided by CPI (p_x_h) equal to 1 in steady-state… (?)

Nonetheless, is it allowed to set the nominal exchange rate ‘e’ as a free parameter and set e=1 in steady-state? I have no other idea to formulate the nonlinear model with the nominal exchange rate. I cannot eliminate the nominal exchange rate because I need portfolio adjustment costs to close the model. In the resource constraint the nominal exchange rate appears unavoidable … Why is it the case?

I think the confusion can be avoided if you express the net foreign assets in real, home currency terms.

nfa=e*b_h/p_h

The cost of adjusting debt can also be expressed in terms of nfa. Similarly, wherever the nominal exchange rate appears, e.g. in the exporter/importer’s FOCs, you can use the real exchange rate instead. You can back out the nominal from the RER changes and the two inflations if you are interested.

In terms of your model, I am not sure if the cost of adjusting nfa will appear in the goods market clearing conditions of both the home and foreign country. Have you seen the working paper version of Bergin (JIMF 2006) who also uses portfolio adjustment costs like you do?

Reuben

I’m reading now the Bergin 2006 paper. So far, it seems that the adjustment cost does not appear in the resource constraint. This fact would already solve my problem: I can express all nominal exchange rates in terms of changes in nominal exchange rates (equal 1 in steady-state).
Nonetheless, I will read the paper in detail.

Best,
Cyrill

Not sure if Bergin shows the full non-linear model. You could also use a risk premium like in the Adolfson et al JIE paper to get rid of the non-stationarity of the nfa. That does not influence the goods market clearing conditions (but will influence the balance of payments).

Hi Reuben,

why is it that the stochastic discount factor does not cancel out, although it appears on both sides of the equation? I do not get this point Could you please explain it again?

Best,
Ben

It is because on both sides of the equation, you have expectations of products, and not products of expectations. One cancel, of course, cancel out the time t variables, like the lambda in the denominator and beta the constant discount factor. But the expectations of t+1 variables cannot be cancelled out in the UIP equation.

Many thanks for the explanation. My UIP condition is given by

UC(1)(1+NEER(1))/(1+CPI(1))(1+R_F) = (1+R)*UC(1)/(1+CPI(1));

Where UC(1) is the lagrange multiplier in Period 1, 1+NEER is the change in the exchange rate, 1+CPI is the change in the domestic CPI, R is the domestic and R_F is the foreign interest rate faced by domestic consumers (including the small premium charged with respect to the level of foreign bonds held by all national households). My question here is, will this equation in dynare be any different from (for instance) this equation, where I cancelled the lagrange multiplier.

(1+NEER(1))/(1+CPI(1))*(1+R_F) = (1+R)/(1+CPI(1));

These are different equations for dynare. Mathematically, we cannot cancel out UC(1) from both sides. I am not sure what difference cancelling would make in practical terms. But clearly, these two equations do not have identical implications.

I agree that these two have different implications (given your explanation above). I was just curious if it actually makes a difference in practice. I will test it tomorrow. Thanks for your help Ben

For a first order approximation, i think the two equations would make no difference. For higher order approximations, it would matter.