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I am a bit confused about the concept of natural real interest rate. I've read that it is the level of real interest rate consistent with the output being at its potential or natural level and with the inflation being static. Is that true?

Furthermore, I do not understand how natural real interest rate is able to affect output and natural level of output (then, the output gap) in the usual New Keynesian framework. I mean the dynamic is curve is defined as \begin{gather} x_t=E_t\{x_{t+1}\}-\frac{1}{\sigma}\{i_t-E_t\{\pi_{t+1}\}-r^n_{t}\} \end{gather} where $x_t=y_t-y^n_t$, i.e. the output gap and $r^n_{t}$ is the natural real interest rate. ($\frac{1}{\sigma}$ comes from a CRRA household utility with elasticity $\sigma$).

Thus it seems that an increase in the natural real interest rate leads to an increase in current output gap, but how is this true? Does it mean that natural level of output increases more than output?

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Regarding you first question, the answer is 'yes', at least in your model. For if the real interest rate, $r_t=i_t-E_t\pi_{t+1}$, is equal to its natural level $r^n_t$, then in your model $x_t=E_tx_{t+1}$ and $x_t=0$ for all $t$ is a solution which is consistent with the dynamic equation. (Note that other solutions are consistent with the dynamic equation in the case $r_t=r^n_t$ for all $t$, such as $x_t=1$ for all $t$.)

According to my experience, the natural level of real interest rate is usually defined as the level of real interest consistent with output being at its 'natural', 'long-run', 'equilibrium' or 'potential' level, given stable inflation, absence of shocks to demand, etc. I.e., natural real interest rate, $r^n_t$, is such that whenever $r^t=r^n_t$ and certain stability/regularity conditions are fulfilled, then $x_t=0$. Read a FRBSF Economic Letter for more information.

We may also view the natural level of real interest rate as the real interest rate at which the output gap $x_t$ is following its trend $E_tx_{t+1}$, i.e. $x_t=E_tx_{t+1}$. This definition is similar to the one given by Knut Wicksell.

Regarding your two last questions, I may say the following. In your model, a ceteris paribus increase in $r^n_t$ increases $x_t$ by $\frac{1}{\sigma}$. (Just take the partial derivative with respect to $r^n_t$ on both sides in the dynamic equation, holding everything else constant.) This means that either output, $y_t$, is increasing, or the natural level of output, $y^n_t$, is increasing. However, if such ceteris paribus changes cannot be made, then taking the partial derivative holding all else constant is not meaningful. Indeed, in new keynesian models, the natural real interest rate is often a function of $y^n_t$, and in such cases we cannot increase the natural real interest rate without affecting the natural level of output. One such case is when $r^n_t=\rho+\sigma(E_ty^n_{t+1}-y^n_t).$

More concrete examples may be provided, I will see if I can find some.

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Quoting from K. Wicksell's "Interest and Prices" (1898), ch. 8 p. 102

"There is a certain rate of interest on loans which is neutral in respect to commodity prices, and tends neither to raise nor to lower them. This is necessarily the same as the rate of interest which would be determined by supply and demand if no use were made of money and all lending were effected in the form of real capital goods. It comes to much the same thing to describe it as the current value of the natural rate of interest on capital."

So the "natural rate of interest" is that level compatible with zero inflation (or deflation). But at the same time, always according to the inventor of the concept (p. 106), the natural rate of interest

"depends on the efficiency of production, on the available amount of fixed and liquid capital, on the supply of labour and land, in short on all the thousand and one things which determine the current economic position of a community..."

...and I would say he is right. This leads us to the more broader characterization, that the natural rate of interest is compatible with output being at its natural level also.

Now, re-arrange and use the Fisher equation to get the interest rate gap,

$$E_t(x_{t+1}) - x_t = \frac{1}{\sigma}(r_t-r^n_{t})$$

with $x_t = y_t - y^n_t$. Given the discussion above, if the actual real interest rate $r_t$ is equal to $r^n_t$ (so zero interest rate gap), we also have $x_t = 0$ and so also $E_t(x_{t+1})=0$

Assume that the interest rate gap is negative, $r_t < r^n_t$. Then the output gap is expected to be smaller next period. Etc. Such is the analysis that makes sense here, rather than considering "stand alone" changes in the natural interest rate (because such changes are never "stand-alone", since the natural level is not a decision variable of any economic agent).

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