<p>Recall the following equivalent definitions for luxury goods and necessities:</p>
<ul>
<li><p>A good <span class="math-container">$x$</span> is considered a necessity if <span class="math-container">$e_{(x,I)}&lt;1$</span>.</p>
</li>
<li><p>A good <span class="math-container">$x$</span> is considered a luxury good if <span class="math-container">$e_{(x,I)}&gt;1$</span>.</p>
<p>As you can see, these definition do not encompass all possible scenarios, so any specific good does not have to be either a luxury or a necessity.</p>
</li>
</ul>
<p>In the case of a Cobb-Douglas utility function <span class="math-container">$U(x,y)=x^\alpha \cdot y^\beta$</span> we get <span class="math-container">$x^* = \frac {\alpha I}{P_x}$</span>. One can easily verify that <span class="math-container">$e_{(x,I)}=1$</span>. In other words, the demand does for <span class="math-container">$x$</span> does not change with <span class="math-container">$I$</span>. This means that in this case, <span class="math-container">$x$</span> is neither a luxury good nor a necessity.</p>

CC BY-SA 3.0
2016-05-29T15:00:27.833Z
I would add that for necessity good the elasticity is more precisely between 0 and 1 otherwise for a negative elasticity it would be a Giffen good !
Alexis L.
CC BY-SA 3.0
2016-05-29T07:22:26.007Z
2016-05-29T08:26:23.400Z
2020-06-17T09:40:54.643Z
galoosh33
<p>Since the concavity of the Engel Curve determines whether it is a necessity or luxury (i.e. how fast quantity demand changes in relation to changes in income), and since the second derivative of a Cobb-Douglas Engel Curve is 0, does that mean it is neither category?</p>

<p>Edit:  In simple terms, if an Engel Curve is a straight positively-sloped line, it is obviously a normal good.  But if the curve is represented by a function like so:</p>

<p>$I = 10 * P_x * x.$</p>

<p>Then the curve is ambiguously sloped.  If the price of $x$ ($P_x$) happens to be greater than 1/10 then the good is a luxury, and vice versa for a good with $P_x &lt; 1/10$.  But if it's ambiguous like this, then the second derivative (which indicates concavity and therefore what direction the curve is increasing/decreasing in) then is it impossible to tell?</p>

<p>For reference, this Engel Curve was derived from the Cobb-Douglas utility function:</p>

<p>$U(x,y) = x^{(1/10)}y^{(9/10)}.$</p>

<microeconomics><consumer-theory><supply-and-demand><elasticity>
If the Engel Curve of a Cobb-Douglas utility function is positive and linear, than does that mean it is neither a necessity nor a luxury good?
bloopton
# If the Engel Curve of a Cobb-Douglas utility function is positive and linear, than does that mean it is neither a necessity nor a luxury good?

(Asked by: username_0 on 2016-02-14)

Since the concavity of the Engel Curve determines whether it is a necessity or luxury (i.e. how fast quantity demand changes in relation to changes in income), and since the second derivative of a Cobb-Douglas Engel Curve is 0, does that mean it is neither category?

Edit: In simple terms, if an Engel Curve is a straight positively-sloped line, it is obviously a normal good. But if the curve is represented by a function like so:

$I = 10 \* P\_x \* x.$

Then the curve is ambiguously sloped. If the price of $x$ ($P\_x$) happens to be greater than 1/10 then the good is a luxury, and vice versa for a good with $P\_x < 1/10$. But if it's ambiguous like this, then the second derivative (which indicates concavity and therefore what direction the curve is increasing/decreasing in) then is it impossible to tell?

For reference, this Engel Curve was derived from the Cobb-Douglas utility function:

$U(x,y) = x^{(1/10)}y^{(9/10)}.$

**Tags:** microeconomics, consumer-theory, supply-and-demand, elasticity


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## 1 answers

### Answer by username_1 on 2016-05-29. Score: 6 | Accepted answer

Recall the following equivalent definitions for luxury goods and necessities:

* A good $x$ is considered a necessity if $e\_{(x,I)}<1$.
* A good $x$ is considered a luxury good if $e\_{(x,I)}>1$.

  As you can see, these definition do not encompass all possible scenarios, so any specific good does not have to be either a luxury or a necessity.

In the case of a Cobb-Douglas utility function $U(x,y)=x^\alpha \cdot y^\beta$ we get $x^\* = \frac {\alpha I}{P\_x}$. One can easily verify that $e\_{(x,I)}=1$. In other words, the demand does for $x$ does not change with $I$. This means that in this case, $x$ is neither a luxury good nor a necessity.

> **Comments:**
> **username_2**: I would add that for necessity good the elasticity is more precisely between 0 and 1 otherwise for a negative elasticity it would be a Giffen good !  

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