<p>There are a number of different definitions of Homothetic functions i have come across. I have used each of them to prove that a  function <span class="math-container">$f(x, y) = x^a y^b$</span> with <span class="math-container">$a+b &gt; 0$</span> is homothetic. But i have some questions.</p>
<ul>
<li><strong>Question 1):</strong> Compared to definition one, why the stricter conditions on definition two and three, if definition one is so simple?</li>
<li><strong>Question 2):</strong>  Why are Definitions  <strong>two</strong> and <strong>three</strong>  equivalent despite subtle differences?</li>
<li><strong>Question3):</strong> An exam question used <span class="math-container">$f$</span> as a production function i.e. <span class="math-container">$x = k$</span> Capital, <span class="math-container">$y = l$</span> Labour. The answer then used <em>definition 3</em> to prove it was homothetic. While not complicated, it seems the most complicated out of the alternatives, can anyone give insight into why this approach might have been used? is there anything special about the function as a production function? Is defintion 3 a tighter definition?</li>
</ul>
<p><strong>Definition 1</strong>
Source: My Mathematical Economics course pack</p>
<ul>
<li><strong>Any</strong> homogeneous function  of degree <span class="math-container">$d&gt;0$</span> is homothetic .</li>
</ul>
<p><span class="math-container">$f(x,y) = x^a y^b$</span> which is homogenous to degree <span class="math-container">$a+b &gt; 0$</span> and is therefore homothetic.</p>
<p><strong>Definition 2</strong>
<a href="https://dergipark.org.tr/tr/download/article-file/772422#:%7E:text=A%20homothetic%20function%20is%20a,is%20a%20monotonically%20increasing%20function." rel="nofollow noreferrer">Source</a>:</p>
<ul>
<li>A function <span class="math-container">$f$</span> is <strong>homothetic</strong> if:</li>
<li><span class="math-container">$f(x,y) = q(r(x,y))$</span></li>
</ul>
<ol>
<li><span class="math-container">$r$</span> is a <strong>homogenous function of <em>any degree</em></strong></li>
<li><span class="math-container">$q$</span> is <strong>monotonically increasing</strong> <em>(Which i believe means non-decreasing)</em></li>
</ol>
<p><span class="math-container">$f(x,y) = h(r(x,y))= (x^{a/2}y^{b/2})^2$</span>
Where i have assumed <span class="math-container">$x,y &gt;0$</span> and so <span class="math-container">$h^2$</span> is <strong>monotonically increasing</strong>
<span class="math-container">$r$</span> is homogenous to degree</p>
<p><strong>Definition 3</strong>
Source: My Mathematical Economics course pack</p>
<ul>
<li>A function <span class="math-container">$f: \mathbb{R^{n+}} \to \mathbb{R}$</span> is <strong>homotheic</strong> if it has the form:</li>
<li><span class="math-container">$f(x,y) = q(r(x,y))$</span></li>
</ul>
<ol>
<li>Where <span class="math-container">$r$</span> is a function that is <strong>homogenous of degree 1</strong></li>
<li><span class="math-container">$q$</span> is <strong>strictly increasing</strong></li>
</ol>
<p><span class="math-container">$f(x,y) = h(r(x,y))= (x^qy^p)^{a+b}$</span>
Where <span class="math-container">$q = \frac{a}{a+b}$</span> <span class="math-container">$p+\frac{b}{a+b}$</span>
Hence we have written <span class="math-container">$f$</span> in terms of a monotonic transformation of a homogenous to degree <span class="math-container">$d = 1$</span> function <span class="math-container">$r$</span>.</p>
<blockquote>
<p><strong>Question 3):</strong> This was the technique used in an exam answer scheme, where <span class="math-container">$x = k$</span> Capital, <span class="math-container">$y = l$</span> Labour, therefore we can assume <span class="math-container">$x\ge0$</span> and <span class="math-container">$y\ge0$</span>
I don't understand why they would use all this when surely just the simple first definition would suffice? Noting that the <span class="math-container">$f$</span> is homogenous of degree d &gt; 0. Or at least definition 4 which is super simple? <strong>Is there any reason one of these other approaches would have been invalid</strong>?</p>
</blockquote>
<p><strong>Definition 4</strong>
<a href="https://dergipark.org.tr/tr/download/article-file/772422#:%7E:text=A%20homothetic%20function%20is%20a,is%20a%20monotonically%20increasing%20function." rel="nofollow noreferrer">Source</a>:</p>
<p>A function <span class="math-container">$f: \mathbb{R}^n_+ \to \mathbb{R}$</span> is homothetic if its marginal rate of substitution (MRS) between any two variables is constant along rays from the origin, i.e., the MRS depends only on the ratio of the variables. I.e. MRS is homogeneous of degree 0.</p>
<p>Using <span class="math-container">$x = k$</span> Capital, <span class="math-container">$y = l$</span> Labour</p>
<p><span class="math-container">$MRTS(k,l) = \frac{al}{bk}$</span> which is clearly homogenous of degree 0.</p>

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2023-04-23T13:29:00.530Z
@BakerStreet you should have left it as homeopathic functions hahah
CormJack
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2023-04-23T13:36:29.510Z
Yes, it was an original, interesting notion!
BakerStreet
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2023-04-23T13:50:35.967Z
No worries, @solowsupremacy the idea of my post becoming a messaging board sounds fun!. I to am often in a constant battle to steal BakerStreets time hahah lol
CormJack
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2023-04-23T13:52:19.797Z
@solowsupremacy pretty sure him and a few others are like the guardians of the Econ Stack exchange. They should start a Patreon hahahah
CormJack
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2023-04-23T13:53:22.360Z
@ solow supremacy I would be glad to check your question, but I don't know if I can during  these days, I'm very busy and I'm away from home. I will read it as soon as possible, if I've time.
BakerStreet
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2023-04-23T13:59:42.227Z
@ Cormjack @ solow supremacy Am I a guardian? :-) Thank you  both of you for your kindness!
BakerStreet
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2023-04-23T14:05:22.210Z
@BakerStreet Yes. Start a Patreon hahah: https://www.patreon.com/en-GB
CormJack
<mathematical-economics><consumer-theory><production-function>
Homotheic Function Definitions
CormJack
# Homotheic Function Definitions

(Asked by: username_0 on 2023-04-23)

There are a number of different definitions of Homothetic functions i have come across. I have used each of them to prove that a function $f(x, y) = x^a y^b$ with $a+b > 0$ is homothetic. But i have some questions.

* **Question 1):** Compared to definition one, why the stricter conditions on definition two and three, if definition one is so simple?
* **Question 2):** Why are Definitions **two** and **three** equivalent despite subtle differences?
* **Question3):** An exam question used $f$ as a production function i.e. $x = k$ Capital, $y = l$ Labour. The answer then used *definition 3* to prove it was homothetic. While not complicated, it seems the most complicated out of the alternatives, can anyone give insight into why this approach might have been used? is there anything special about the function as a production function? Is defintion 3 a tighter definition?

**Definition 1**
Source: My Mathematical Economics course pack

* **Any** homogeneous function of degree $d>0$ is homothetic .

$f(x,y) = x^a y^b$ which is homogenous to degree $a+b > 0$ and is therefore homothetic.

**Definition 2**
[Source](https://dergipark.org.tr/tr/download/article-file/772422#:%7E:text=A%20homothetic%20function%20is%20a,is%20a%20monotonically%20increasing%20function.):

* A function $f$ is **homothetic** if:
* $f(x,y) = q(r(x,y))$

1. $r$ is a **homogenous function of *any degree***
2. $q$ is **monotonically increasing** *(Which i believe means non-decreasing)*

$f(x,y) = h(r(x,y))= (x^{a/2}y^{b/2})^2$
Where i have assumed $x,y >0$ and so $h^2$ is **monotonically increasing**
$r$ is homogenous to degree

**Definition 3**
Source: My Mathematical Economics course pack

* A function $f: \mathbb{R^{n+}} \to \mathbb{R}$ is **homotheic** if it has the form:
* $f(x,y) = q(r(x,y))$

1. Where $r$ is a function that is **homogenous of degree 1**
2. $q$ is **strictly increasing**

$f(x,y) = h(r(x,y))= (x^qy^p)^{a+b}$
Where $q = \frac{a}{a+b}$ $p+\frac{b}{a+b}$
Hence we have written $f$ in terms of a monotonic transformation of a homogenous to degree $d = 1$ function $r$.

> **Question 3):** This was the technique used in an exam answer scheme, where $x = k$ Capital, $y = l$ Labour, therefore we can assume $x\ge0$ and $y\ge0$
> I don't understand why they would use all this when surely just the simple first definition would suffice? Noting that the $f$ is homogenous of degree d > 0. Or at least definition 4 which is super simple? **Is there any reason one of these other approaches would have been invalid**?

**Definition 4**
[Source](https://dergipark.org.tr/tr/download/article-file/772422#:%7E:text=A%20homothetic%20function%20is%20a,is%20a%20monotonically%20increasing%20function.):

A function $f: \mathbb{R}^n\_+ \to \mathbb{R}$ is homothetic if its marginal rate of substitution (MRS) between any two variables is constant along rays from the origin, i.e., the MRS depends only on the ratio of the variables. I.e. MRS is homogeneous of degree 0.

Using $x = k$ Capital, $y = l$ Labour

$MRTS(k,l) = \frac{al}{bk}$ which is clearly homogenous of degree 0.

**Tags:** mathematical-economics, consumer-theory, production-function

> **Comments:**
> **username_0**: @username_1 you should have left it as homeopathic functions hahah  
> **username_1**: Yes, it was an original, interesting notion!  
> **username_0**: No worries, @solowsupremacy the idea of my post becoming a messaging board sounds fun!. I to am often in a constant battle to steal username_1s time hahah lol  
> **username_0**: @solowsupremacy pretty sure him and a few others are like the guardians of the Econ Stack exchange. They should start a Patreon hahahah  
> **username_1**: @ solow supremacy I would be glad to check your question, but I don't know if I can during  these days, I'm very busy and I'm away from home. I will read it as soon as possible, if I've time.  
> **username_1**: @ Cormjack @ solow supremacy Am I a guardian? :-) Thank you  both of you for your kindness!  
> **username_0**: @username_1 Yes. Start a Patreon hahah: https://www.patreon.com/en-GB  

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## 0 answers

