<p>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 <strong>log-linearization</strong>, and is fairly common.</p>

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

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

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

<p>and applying this rule gives</p>

<p>$r_t-\delta=\theta+\sigma \eta$</p>

<p>which gives the desired result.</p>

CC BY-SA 3.0
2016-10-08T12:27:12.913Z
Thanks Chris! This author is never careful in most of his deductions...
An old man in the sea.
CC BY-SA 3.0
2016-10-08T10:18:01.640Z
2016-10-08T13:55:51.780Z
2016-10-08T13:55:51.780Z
Chris tie
<p>In Wickens' Macroeconomic Theory book, in page 48(1st edition), the author states that by doing a taylor approximation we get the following result.</p>
<p><a href="https://i.sstatic.net/r26gg.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/r26gg.png" alt="enter image description here" /></a></p>
<p>Why is that? What approximation did he use?</p>
<p>I've tried several to linearize the fraction <span class="math-container">$\frac{(1+\eta)^{\sigma}}{\beta}$</span> around those values, but I don't get the same...</p>

<macroeconomics><mathematical-economics>
What Taylor approximation is used in this equation?
An old man in the sea.
# What Taylor approximation is used in this equation?

(Asked by: username_0 on 2016-10-04)

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

[![enter image description here](https://i.sstatic.net/r26gg.png)](https://i.sstatic.net/r26gg.png)

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...

**Tags:** macroeconomics, mathematical-economics


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## 1 answers

### Answer by username_1 on 2016-10-08. Score: 4 | Accepted answer

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.

> **Comments:**
> **username_0**: Thanks username_1! This author is never careful in most of his deductions...  

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