<p>Applications to different data render the two different definitions useful. Put differently, the distribution of the variables--whether DV/y or IV/x--determines the analytical strategy and choice of definition.</p>
<blockquote>
<p><span class="math-container">$\epsilon_1 = \dfrac{\partial y/y}{\partial x}$</span></p>
</blockquote>
<blockquote>
<p>Which is the percentage change of <span class="math-container">$y$</span> in response to a (non percentage) change of <span class="math-container">$x$</span>.</p>
</blockquote>
<p>This is the semielasticity of y.</p>
<blockquote>
<p><span class="math-container">$\epsilon_2 = \dfrac{\partial y}{\partial x/x}$</span></p>
</blockquote>
<blockquote>
<p>Which is the (non percentage) change in <span class="math-container">$y$</span> in response to a percentage change in <span class="math-container">$x$</span>.</p>
</blockquote>
<p>This is the semielasticity of x.</p>
<p>I have experience referring to <span class="math-container">$\epsilon_1$</span> in the context of <a href="https://stats.stackexchange.com/questions/520434/is-a-signficiant-interaction-term-in-poisson-really-statistically-significant">Poisson</a> regressions where the DV (a count) is log-transformed at compute while the IV is not log-transformed. The response in the count is proportional to the increase in the IV. &quot;Proportional&quot; I take to mean that the association is nonlinear. Perhaps an increase at the low end of the range of x1 is associated with a modest absolute increase in counts, while an increase at the high end of the range of x1 is associated with a large increase in the counts.</p>
<p>Perhaps in a context where an IV is log-transformed the semielasticity of x may be useful.</p>

CC BY-SA 4.0
2024-10-23T11:37:38.327Z
Really? We have one name for two very different definitions? Isn't that confusing?
robertspierre
CC BY-SA 4.0
2024-11-13T09:25:22.943Z
They have the same name in being called semielasticities. But they are, as you almost said yourself, different semielasticities.
Johan
CC BY-SA 4.0
2024-10-23T11:25:37.843Z
2024-11-13T10:07:47.450Z
2024-11-13T10:07:47.450Z
Johan
<p>The &quot;<span class="math-container">$x$</span>-elasticity of <span class="math-container">$y$</span>&quot; is defined as <span class="math-container">$\epsilon = \dfrac{\partial y/y}{\partial x/x}$</span>.</p>
<p>The <span class="math-container">$x$</span> goes down, and the <span class="math-container">$y$</span> goes up.</p>
<p>But we have two definitions of semi-elasticity:</p>
<p><span class="math-container">$\epsilon_1 = \dfrac{\partial y/y}{\partial x}$</span></p>
<p>Which is the percentage change of <span class="math-container">$y$</span> in response to a (non percentage) change of x, and:</p>
<p><span class="math-container">$\epsilon_2 = \dfrac{\partial y}{\partial x/x}$</span></p>
<p>Which is the (non percentage) change in <span class="math-container">$y$</span> in response to a percentage change in <span class="math-container">$x$</span>.</p>
<p>How do we distinguish between the two?</p>
<p>That is, what are they called?</p>
<p>When we say &quot;the <span class="math-container">$x$</span>-semi-elasticity of <span class="math-container">$y$</span>&quot;, what do we refer to?</p>
<p><strong>REFERENCES</strong></p>
<p><a href="https://www.stata.com/manuals/rmargins.pdf#page=26" rel="noreferrer">STATA manual for rmargins</a>, page 26.</p>
<hr />
<p>Note: The actual definition is on <a href="https://www.stata.com/manuals/rmargins.pdf#page=47" rel="noreferrer">page 47</a>.</p>

CC BY-SA 4.0
2024-10-23T15:32:51.740Z
I've grown accustomed to semi elasticity referring to the first definition. I don't think I've ever come across the second used in practice.

I can imagine situations in which you would though. E.g. if I increase treatment uptake by 10% how many more people attend university? Though that could just as easily be done with just a standard elasticity with clearer interpretation.
Brennan
<elasticity><definition><economic-terms>
Definition of semi-elasticity
robertspierre
# Definition of semi-elasticity

(Asked by: username_0 on 2023-11-30)

The "$x$-elasticity of $y$" is defined as $\epsilon = \dfrac{\partial y/y}{\partial x/x}$.

The $x$ goes down, and the $y$ goes up.

But we have two definitions of semi-elasticity:

$\epsilon\_1 = \dfrac{\partial y/y}{\partial x}$

Which is the percentage change of $y$ in response to a (non percentage) change of x, and:

$\epsilon\_2 = \dfrac{\partial y}{\partial x/x}$

Which is the (non percentage) change in $y$ in response to a percentage change in $x$.

How do we distinguish between the two?

That is, what are they called?

When we say "the $x$-semi-elasticity of $y$", what do we refer to?

**REFERENCES**

[STATA manual for rmargins](https://www.stata.com/manuals/rmargins.pdf#page=26), page 26.

---

Note: The actual definition is on [page 47](https://www.stata.com/manuals/rmargins.pdf#page=47).

**Tags:** elasticity, definition, economic-terms

> **Comments:**
> **username_1**: I've grown accustomed to semi elasticity referring to the first definition. I don't think I've ever come across the second used in practice.  I can imagine situations in which you would though. E.g. if I increase treatment uptake by 10% how many more people attend university? Though that could just as easily be done with just a standard elasticity with clearer interpretation.  

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## 1 answers

### Answer by username_2 on 2024-10-23. Score: 1

Applications to different data render the two different definitions useful. Put differently, the distribution of the variables--whether DV/y or IV/x--determines the analytical strategy and choice of definition.

> $\epsilon\_1 = \dfrac{\partial y/y}{\partial x}$

> Which is the percentage change of $y$ in response to a (non percentage) change of $x$.

This is the semielasticity of y.

> $\epsilon\_2 = \dfrac{\partial y}{\partial x/x}$

> Which is the (non percentage) change in $y$ in response to a percentage change in $x$.

This is the semielasticity of x.

I have experience referring to $\epsilon\_1$ in the context of [Poisson](https://stats.stackexchange.com/questions/520434/is-a-signficiant-interaction-term-in-poisson-really-statistically-significant) regressions where the DV (a count) is log-transformed at compute while the IV is not log-transformed. The response in the count is proportional to the increase in the IV. "Proportional" I take to mean that the association is nonlinear. Perhaps an increase at the low end of the range of x1 is associated with a modest absolute increase in counts, while an increase at the high end of the range of x1 is associated with a large increase in the counts.

Perhaps in a context where an IV is log-transformed the semielasticity of x may be useful.

> **Comments:**
> **username_0**: Really? We have one name for two very different definitions? Isn't that confusing?  
> **username_2**: They have the same name in being called semielasticities. But they are, as you almost said yourself, different semielasticities.  

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