I don't understand how one can get from:




by applying the following rule of thumb:





$r_{t+1}$: real interest rate

$i_{t}$: nominal interest rate

$\pi_{t+1}$: inflation rate

I hope someone will help :)


Argument 1:

$$ 1+r_{t+1} = \frac{1+i_{t}}{1+\pi_{t+1}}$$

Take the natural log of both sides.

$$ \log(1+r_{t+1}) = \log(1+i_{t}) - \log(1+\pi_{t+1})$$

Observe that $\log(1 + x) \approx x$ for $x$ near zero because that's the first order taylor expansion. Hence:

$$ r_{t+1} \approx i_{t} - \pi_{t+1}$$

Argument 2:

$$ 1+r_{t+1} = \frac{1+i_{t}}{1+\pi_{t+1}}$$ $$ 1+r_{t+1} + \pi_{t+1} + r_{t+1} \pi_{t+1} = 1+i_{t}$$

And if $r_{t+1} \pi_{t+1}$ is closer to zero

$$ r_{t+1} + \pi_{t+1} \approx i_{t}$$

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