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In Wickens' Macroeconomic Theory book, in page 48(1st edition), the author states that by doing a taylor approximation we get the following result.

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Why is that? What approximation did he use?

I've tried several to linearize the fraction $\frac{(1+\eta)^{\sigma}}{\beta}$ around those values, but I don't get the same...

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Even though your question does not allow a definite answer, I am pretty sure the author used a Taylor expansion around the logarithm of both sides of the equation. This process is called log-linearization, and is fairly common.

We can approximate (logs of) growth rates as $x \approx log(1+x)$ when $x$ is small. (and by the rules of logarithm: $\log((1+x)^\sigma ) \approx\sigma x$

Rewriting your equation $1+(r_t-\delta)=(1+\eta)^{\sigma}(1+\theta)$

where $r_t=\alpha (k^{\# *})^{\alpha -1}$

and applying this rule gives

$r_t-\delta=\theta+\sigma \eta$

which gives the desired result.

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  • $\begingroup$ Thanks Chris! This author is never careful in most of his deductions... $\endgroup$ – An old man in the sea. Oct 8 '16 at 12:27

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