<blockquote>
<p>why formula #3 may be more appropriate than formula #2?</p>
</blockquote>
<p>Actually, #2 is a linear approximation of #3.</p>
<p>Note that</p>
<p><span class="math-container">$\partial \log Z$</span></p>
<p>can be written/seen as</p>
<p><span class="math-container">$\log Z_2 - \log Z_1 = \log\frac{Z_2}{Z_1} (\approx 0)$</span></p>
<p>where the above difference is <em>infinitesimal</em>, hence the use of <span class="math-container">$\partial$</span>.
In such a case, #1 can be written as</p>
<p><span class="math-container">$\epsilon = \frac{\frac{\Delta Y}{Y}}{\frac{\Delta X}{X}}= \frac{\log (Y_2/Y_1)}{\log (X_2/X_1)}$</span></p>
<p>Equivalently, the above equality is only true because we are dealing with infinitesimal changes. Which means that if, say, <span class="math-container">$Y_2 &gt;&gt; Y_1$</span>, the above equality implies an approximation, i.e.</p>
<p><span class="math-container">$\epsilon = \frac{\log (Y_2/Y_1)}{\log (X_2/X_1)} \approx \frac{\frac{\Delta Y}{Y}}{\frac{\Delta X}{X}}$</span></p>
<p>Which leads to #2:</p>
<p><span class="math-container">$\frac{\Delta Y}{Y} \approx \epsilon \frac{\Delta X}{X}$</span></p>
<p>Actually, the smaller <span class="math-container">$\Delta Y$</span> and <span class="math-container">$\Delta X$</span>, the better the approximation.</p>
<hr>
<blockquote>
<p>how does one get mathematically from formula #1 to formula #3?</p>
</blockquote>
<p><span class="math-container">$\epsilon = \frac{\log (Y_2/Y_1)}{\log (X_2/X_1)}$</span></p>
<p><span class="math-container">$\Leftrightarrow \log \frac{Y_2}{Y_1} = \epsilon \log \frac{X_2}{X_1} =$</span>
<span class="math-container">$\log\left[\left(\frac{X_2}{X_1}\right)^\epsilon\right]$</span></p>
<p><span class="math-container">$\Leftrightarrow \frac{Y_2}{Y_1} = \left(\frac{X_2}{X_1}\right)^\epsilon$</span></p>
<p><span class="math-container">$\Leftrightarrow \frac{Y_2}{Y_1} -1 = \frac{Y_2 - Y_1}{Y_1} = \frac{\Delta Y}{Y}= \left(\frac{X_2}{X_1}\right)^\epsilon -1$</span></p>

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2019-10-17T11:36:16.933Z
Any question @Anon ?
keepAlive
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2018-07-17T10:21:17.117Z
2018-07-17T10:43:11.703Z
2020-06-17T09:40:54.643Z
keepAlive
<p>Often in textbooks and online I see thte definition of an elasticity to be as follows:</p>

<blockquote>
  <p>1) $\epsilon = \frac{\frac{\Delta Y}{Y}}{\frac{\Delta X}{X}}= \frac{\partial \log Y}{\partial \log X}$</p>
</blockquote>

<p>Thus, when using an elasticity along with a percentage change in X to estimate a percentage change in Y, it would seem to make sense that one would multiply the percentage change in X by the elasticity as below:</p>

<blockquote>
  <p>2) ${\frac{\Delta Y}{Y}}=\epsilon\cdot(\frac{\Delta X}{X})$</p>
</blockquote>

<p>However, in one instance I've seen mention that it is more accurate to estimate the percentage change in Y as follows:</p>

<blockquote>
  <p>3) ${\frac{\Delta Y}{Y}}=(\frac{X_2}{X_1})^\epsilon-1$</p>
</blockquote>

<p>I'm seeking to apply elasticities derived by estimating log-log regression models and would appreciate some guidance as to whether formula #2 or #3 is more appropriate when evaluating impacts on Y resulting from changes in X.</p>

<p>Can someone please explain why formula #3 may be more appropriate than formula #2? And how does one get mathematically from formula #1 to formula #3?</p>

<econometrics><elasticity>
Using an elasticity to evaluate changes in a dependent variable
Anon
# Using an elasticity to evaluate changes in a dependent variable

(Asked by: username_0 on 2018-07-17)

Often in textbooks and online I see thte definition of an elasticity to be as follows:

> 1) $\epsilon = \frac{\frac{\Delta Y}{Y}}{\frac{\Delta X}{X}}= \frac{\partial \log Y}{\partial \log X}$

Thus, when using an elasticity along with a percentage change in X to estimate a percentage change in Y, it would seem to make sense that one would multiply the percentage change in X by the elasticity as below:

> 2) ${\frac{\Delta Y}{Y}}=\epsilon\cdot(\frac{\Delta X}{X})$

However, in one instance I've seen mention that it is more accurate to estimate the percentage change in Y as follows:

> 3) ${\frac{\Delta Y}{Y}}=(\frac{X\_2}{X\_1})^\epsilon-1$

I'm seeking to apply elasticities derived by estimating log-log regression models and would appreciate some guidance as to whether formula #2 or #3 is more appropriate when evaluating impacts on Y resulting from changes in X.

Can someone please explain why formula #3 may be more appropriate than formula #2? And how does one get mathematically from formula #1 to formula #3?

**Tags:** econometrics, elasticity


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## 1 answers

### Answer by username_1 on 2018-07-17. Score: 4

> why formula #3 may be more appropriate than formula #2?

Actually, #2 is a linear approximation of #3.

Note that

$\partial \log Z$

can be written/seen as

$\log Z\_2 - \log Z\_1 = \log\frac{Z\_2}{Z\_1} (\approx 0)$

where the above difference is *infinitesimal*, hence the use of $\partial$.
In such a case, #1 can be written as

$\epsilon = \frac{\frac{\Delta Y}{Y}}{\frac{\Delta X}{X}}= \frac{\log (Y\_2/Y\_1)}{\log (X\_2/X\_1)}$

Equivalently, the above equality is only true because we are dealing with infinitesimal changes. Which means that if, say, $Y\_2 >> Y\_1$, the above equality implies an approximation, i.e.

$\epsilon = \frac{\log (Y\_2/Y\_1)}{\log (X\_2/X\_1)} \approx \frac{\frac{\Delta Y}{Y}}{\frac{\Delta X}{X}}$

Which leads to #2:

$\frac{\Delta Y}{Y} \approx \epsilon \frac{\Delta X}{X}$

Actually, the smaller $\Delta Y$ and $\Delta X$, the better the approximation.

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> how does one get mathematically from formula #1 to formula #3?

$\epsilon = \frac{\log (Y\_2/Y\_1)}{\log (X\_2/X\_1)}$

$\Leftrightarrow \log \frac{Y\_2}{Y\_1} = \epsilon \log \frac{X\_2}{X\_1} =$
$\log\left[\left(\frac{X\_2}{X\_1}\right)^\epsilon\right]$

$\Leftrightarrow \frac{Y\_2}{Y\_1} = \left(\frac{X\_2}{X\_1}\right)^\epsilon$

$\Leftrightarrow \frac{Y\_2}{Y\_1} -1 = \frac{Y\_2 - Y\_1}{Y\_1} = \frac{\Delta Y}{Y}= \left(\frac{X\_2}{X\_1}\right)^\epsilon -1$

> **Comments:**
> **username_1**: Any question @username_0 ?  

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