<ol>
<li><p>Homothetic goods is not a widely used term (as far as I know).</p>
</li>
<li><p>It is not true that when preferences are homothetic
<span class="math-container">$$
\frac{\Delta x}{\Delta m} = 1
$$</span>
always holds. Instead
<span class="math-container">$$
\frac{\Delta x}{\Delta m} = \text{constant} \cdot \frac{1}{p_x}
$$</span>
(where the constant is the share of income spent on <span class="math-container">$x$</span>)
<br>or
<span class="math-container">$$
\frac{\Delta x}{\Delta m} \frac{m}{x} = 1
$$</span>
hold in this case.</p>
</li>
<li><p>The slope of the Engel-curve <span class="math-container">$\frac{\text{d} x}{\text{d} m}$</span> and income-elasticity are related, as the slope appears in the (point) income elasticity formula:
<span class="math-container">$$
\eta_x(m) = \frac{\text{d} x}{\text{d} m}\frac{m}{x}.
$$</span></p>
</li>
</ol>

CC BY-SA 4.0
2021-05-11T05:17:05.017Z
2021-05-11T06:05:29.987Z
2021-05-11T06:05:29.987Z
Giskard
<h3>Income elasticity of demand</h3>
<p>Let <span class="math-container">$q(y)$</span> be the Engel curve for a good, i.e. it gives the demanded quantity for a given level of income <span class="math-container">$y$</span> (keeping prices fixed). The income elasticity of demand is then given by:
<span class="math-container">$$
\varepsilon^y_q = \frac{\partial q}{\partial y} \frac{y}{q}
$$</span>
It measures the percentage point change in demand <span class="math-container">$q(y)$</span>, due to a 1<span class="math-container">$\%$</span> increase in income, <span class="math-container">$y$</span>.</p>
<p>If <span class="math-container">$\varepsilon^y_q \ge 0$</span> the good is normal, if <span class="math-container">$\varepsilon^y_q &lt; 0$</span>, it is inferior. If <span class="math-container">$\varepsilon^y_q &gt; 1$</span> it is a luxury good, while if <span class="math-container">$\varepsilon^y_q &lt; 1$</span> the good is called a necessity.  The demand for luxury goods increases more than proportionally compared to income. Necessary goods increase less than proportionally.</p>
<p>The advantage of using the elasticity <span class="math-container">$\varepsilon^y_q$</span>,  instead of the slop <span class="math-container">$\frac{\Delta q}{\Delta y}$</span> is that it is unit-independent
We can see this as follows:
<span class="math-container">$$
\varepsilon^y_q = \underbrace{\dfrac{\partial q}{\partial y}}_{\dfrac{kilos}{Euros}} \underbrace{\dfrac{1}{q}}_{\dfrac{1}{kilos}} \underbrace{y}_{\dfrac{Euros}{1}}
$$</span></p>
<p>This means that <span class="math-container">$\varepsilon^y_q$</span> does not change if we express the goods or the income in different units. For example the elasticity <span class="math-container">$\varepsilon^y_q$</span> does not change whether you express income in Euros or Dollars. Also, the elasticity does not change whether you express demand in kilos or pounds. As such, elasticities can easily be compared across countries and over different time periods. The slope <span class="math-container">$\frac{\Delta q}{\Delta y}$</span> does not have this advantage.</p>
<h3>Homothetic preferences</h3>
<p>If preferences are homothetic, the demand function is linear in income:
<span class="math-container">$$
q(y) = c y,
$$</span>
where <span class="math-container">$c$</span> is a constant. In fact, substituting <span class="math-container">$y = 1$</span> into this equation gives:
<span class="math-container">$$
q(1) = c,
$$</span>
so <span class="math-container">$c$</span> is the unit income demand (the amount that you would buy if you would have 1 Euro). This means that we can also write:
<span class="math-container">$$
q(y) = q(1) y.
$$</span>
Engel curves are straight lines through the origin and with slope <span class="math-container">$q(1)$</span>:
<span class="math-container">$$
\frac{\partial q(y)}{\partial y} = q(1).
$$</span>
As such:
<span class="math-container">$$
\varepsilon^y_q = \frac{\partial q}{\partial y}\frac{y}{q(y)} = \frac{q(1)y}{q(y)} = \frac{q(y)}{q(y)} = 1.
$$</span>
So demand curves arising from homothetic preferences have unit income elasticity.</p>

CC BY-SA 4.0
2021-05-11T05:55:45.067Z
2021-05-11T06:09:42.060Z
2021-05-11T06:09:42.060Z
tdm
<p>I learnt that <span class="math-container">$\frac{\Delta x}{\Delta m} \gt 0$</span> for normal goods, <span class="math-container">$\frac{\Delta x}{\Delta m} \lt 0$</span> for inferior goods, <span class="math-container">$\frac{\Delta x}{\Delta m} \gt 1$</span> for luxury goods and <span class="math-container">$0 \lt \frac{\Delta x}{\Delta m} \lt 1$</span> for necessary goods (where x is the amount of units of some good and m is the income).</p>
<p>Now, looking at the Engel-curve for homothetic preferences (i.e. Cobb-Douglas, perfect substitutes/perfect complements), I understand <span class="math-container">$\frac{\Delta x}{\Delta m} = 1$</span> for homothetic goods (btw is that a term, &quot;homothetic goods&quot;?).</p>
<p>Since the slope of the Engel-curve is <span class="math-container">$\frac{\Delta m}{\Delta x_1}$</span>, I'm wondering if <span class="math-container">$\left(\frac{\Delta m}{\Delta x_1}\right)^{-1}$</span> is a way to determine the elasticity of demand, which, from my understanding, deals with the exact same values to categorise goods.</p>
<p>So I'm wondering if <span class="math-container">$\left(\frac{\Delta m}{\Delta x_1}\right)^{-1}$</span> is equal to the elasticity of income or if there's a way to determine the elasticity of income from the Engel-curve.</p>

CC BY-SA 4.0
2021-05-11T05:56:50.603Z
Whoops, just noticed that I have answered a very similar question years ago :)
Giskard
CC BY-SA 4.0
2021-05-11T05:56:55.607Z
Does this answer your question? [How exactly does elasticity relate to slope?](https://economics.stackexchange.com/questions/19524/how-exactly-does-elasticity-relate-to-slope)
Giskard
CC BY-SA 4.0
2021-05-11T10:41:29.720Z
@Giskard Nearly, actually I think this one comes quite close: https://economics.stackexchange.com/questions/14162/why-is-elasticity-not-defined-simply-as-the-slope-of-the-graph/14163#14163
j3141592653589793238
<microeconomics><utility><elasticity>
Relation of Engel-curve to income elasticity of demand; is the slope of the Engel-curve equal to the elasticity of income?
j3141592653589793238
# Relation of Engel-curve to income elasticity of demand; is the slope of the Engel-curve equal to the elasticity of income?

(Asked by: username_0 on 2021-05-10)

I learnt that $\frac{\Delta x}{\Delta m} \gt 0$ for normal goods, $\frac{\Delta x}{\Delta m} \lt 0$ for inferior goods, $\frac{\Delta x}{\Delta m} \gt 1$ for luxury goods and $0 \lt \frac{\Delta x}{\Delta m} \lt 1$ for necessary goods (where x is the amount of units of some good and m is the income).

Now, looking at the Engel-curve for homothetic preferences (i.e. Cobb-Douglas, perfect substitutes/perfect complements), I understand $\frac{\Delta x}{\Delta m} = 1$ for homothetic goods (btw is that a term, "homothetic goods"?).

Since the slope of the Engel-curve is $\frac{\Delta m}{\Delta x\_1}$, I'm wondering if $\left(\frac{\Delta m}{\Delta x\_1}\right)^{-1}$ is a way to determine the elasticity of demand, which, from my understanding, deals with the exact same values to categorise goods.

So I'm wondering if $\left(\frac{\Delta m}{\Delta x\_1}\right)^{-1}$ is equal to the elasticity of income or if there's a way to determine the elasticity of income from the Engel-curve.

**Tags:** microeconomics, utility, elasticity
> **Comments:**
> **username_1**: Whoops, just noticed that I have answered a very similar question years ago :)  
> **username_1**: Does this answer your question? [How exactly does elasticity relate to slope?](https://economics.stackexchange.com/questions/19524/how-exactly-does-elasticity-relate-to-slope)  
> **username_0**: @Giskard Nearly, actually I think this one comes quite close: https://economics.stackexchange.com/questions/14162/why-is-elasticity-not-defined-simply-as-the-slope-of-the-graph/14163#14163  

---

## 2 answers

### Answer by username_2 on 2021-05-11. Score: 5 | Accepted answer

### Income elasticity of demand

Let $q(y)$ be the Engel curve for a good, i.e. it gives the demanded quantity for a given level of income $y$ (keeping prices fixed). The income elasticity of demand is then given by:
$$
\varepsilon^y\_q = \frac{\partial q}{\partial y} \frac{y}{q}
$$
It measures the percentage point change in demand $q(y)$, due to a 1$\%$ increase in income, $y$.

If $\varepsilon^y\_q \ge 0$ the good is normal, if $\varepsilon^y\_q < 0$, it is inferior. If $\varepsilon^y\_q > 1$ it is a luxury good, while if $\varepsilon^y\_q < 1$ the good is called a necessity. The demand for luxury goods increases more than proportionally compared to income. Necessary goods increase less than proportionally.

The advantage of using the elasticity $\varepsilon^y\_q$, instead of the slop $\frac{\Delta q}{\Delta y}$ is that it is unit-independent
We can see this as follows:
$$
\varepsilon^y\_q = \underbrace{\dfrac{\partial q}{\partial y}}\_{\dfrac{kilos}{Euros}} \underbrace{\dfrac{1}{q}}\_{\dfrac{1}{kilos}} \underbrace{y}\_{\dfrac{Euros}{1}}
$$

This means that $\varepsilon^y\_q$ does not change if we express the goods or the income in different units. For example the elasticity $\varepsilon^y\_q$ does not change whether you express income in Euros or Dollars. Also, the elasticity does not change whether you express demand in kilos or pounds. As such, elasticities can easily be compared across countries and over different time periods. The slope $\frac{\Delta q}{\Delta y}$ does not have this advantage.

### Homothetic preferences

If preferences are homothetic, the demand function is linear in income:
$$
q(y) = c y,
$$
where $c$ is a constant. In fact, substituting $y = 1$ into this equation gives:
$$
q(1) = c,
$$
so $c$ is the unit income demand (the amount that you would buy if you would have 1 Euro). This means that we can also write:
$$
q(y) = q(1) y.
$$
Engel curves are straight lines through the origin and with slope $q(1)$:
$$
\frac{\partial q(y)}{\partial y} = q(1).
$$
As such:
$$
\varepsilon^y\_q = \frac{\partial q}{\partial y}\frac{y}{q(y)} = \frac{q(1)y}{q(y)} = \frac{q(y)}{q(y)} = 1.
$$
So demand curves arising from homothetic preferences have unit income elasticity.


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### Answer by username_1 on 2021-05-11. Score: 3

1. Homothetic goods is not a widely used term (as far as I know).
2. It is not true that when preferences are homothetic
   $$
   \frac{\Delta x}{\Delta m} = 1
   $$
   always holds. Instead
   $$
   \frac{\Delta x}{\Delta m} = \text{constant} \cdot \frac{1}{p\_x}
   $$
   (where the constant is the share of income spent on $x$)
     
   or
   $$
   \frac{\Delta x}{\Delta m} \frac{m}{x} = 1
   $$
   hold in this case.
3. The slope of the Engel-curve $\frac{\text{d} x}{\text{d} m}$ and income-elasticity are related, as the slope appears in the (point) income elasticity formula:
   $$
   \eta\_x(m) = \frac{\text{d} x}{\text{d} m}\frac{m}{x}.
   $$


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