<p>Using the following definition of a homotheic function (taken from my <em>Mathematical Economics</em> course pack).</p>
<ul>
<li>A function <span class="math-container">$f: \mathbb{R^{n+}} \to \mathbb{R}$</span> is <strong>homothetic</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>For example: Let the following be a production function:
<span class="math-container">$f(x, y) = x^a y^b$</span> with <span class="math-container">$a+b &gt; 0$</span> We can rite this as:</p>
<p><span class="math-container">$f(x,y) = h(r(x,y))= (x^qy^p)^{a+b}$</span></p>
<p>Where <span class="math-container">$q = \frac{a}{a+b}$</span> <span class="math-container">$p = \frac{b}{a+b}$</span></p>
<p>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>. And <span class="math-container">$h$</span> is strictly increasing.</p>
<p><strong>Question 1):</strong></p>
<ul>
<li>I want to clarify here that &quot;<em>Monotonic transformation</em>&quot; is related but different to <em>&quot;Montonic function&quot;</em>.</li>
<li><strong>Montonic Function:</strong> A non-decreasing function (It could be constant?)</li>
<li><strong>Monotonic transformation:</strong> Preserves the ordering of the function e.g. If <span class="math-container">$f(h(x_2,y_2))$</span> is a monotonic transformation of <span class="math-container">$h$</span>, and <span class="math-container">$h$</span> is strictly increasing, e.g. <span class="math-container">$h(x_1, y_1) &gt; h(x_2,y_2)$</span> <span class="math-container">$\forall (x_i,y_i)&gt;(x_j,y_j)$</span> i.e. then <span class="math-container">$f(h(x_1,y_1)) &gt; f(h(x_2,y_2))$</span></li>
<li>This would mean that a monotonic transformation would preserve the <strong>strictness</strong> I.e. if <span class="math-container">$h$</span> is strictly increasing <span class="math-container">$f(h(x_2,y_2))$</span> is a monotonic transformation and is strictly increasing?</li>
</ul>
<p><strong>Question 2):</strong></p>
<p>As a result of the discussion above regarding the necessity for a strictly increasing function. How does the following definition from <em>Hal R. Varian, &quot;Microeconomic Analysis,&quot; 3rd Edition.</em> fit with this? Why the subtle difference? Note, he uses <strong>&quot;monotonic&quot;</strong> rather than <strong>&quot;strictly increasing:&quot;</strong></p>
<blockquote>
<p>A  homothetic  function  is  a  monotonic  transformation of
a  function  that  is  homogeneous  of  degree  1.  In  other  words,  <span class="math-container">$f(x)$</span> is  homothetic if  and only  if  it  can  be  written  as <span class="math-container">$f(x) = g(h(x))$</span> where  <span class="math-container">$h(.)$</span> is homogeneous of  degree 1 and <span class="math-container">$g(.)$</span> is a  <strong>monotonic function</strong></p>
</blockquote>
<p><em>Essentially this is a hybrid between <strong>definition 2</strong> and <strong>definition 2</strong> in a longer post i have made about <a href="https://economics.stackexchange.com/questions/55178/homotheic-function-definitions">homothetic function definitions</a>.</em></p>

CC BY-SA 4.0
2023-04-23T14:02:56.263Z
Hello again, thanks for pointing that out. I've tired to make it more clear. If anything else isn't clear it might be more my lack of understanding, which needs clarifying in answer. But let me know anyway, cheers!
CormJack
<mathematical-economics><consumer-theory><production-function>
Homothetic Functions and Monotonic Transformations
CormJack
# Homothetic Functions and Monotonic Transformations

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

Using the following definition of a homotheic function (taken from my *Mathematical Economics* course pack).

* A function $f: \mathbb{R^{n+}} \to \mathbb{R}$ is **homothetic** 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**

For example: Let the following be a production function:
$f(x, y) = x^a y^b$ with $a+b > 0$ We can rite this as:

$f(x,y) = h(r(x,y))= (x^qy^p)^{a+b}$

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 $r$. And $h$ is strictly increasing.

**Question 1):**

* I want to clarify here that "*Monotonic transformation*" is related but different to *"Montonic function"*.
* **Montonic Function:** A non-decreasing function (It could be constant?)
* **Monotonic transformation:** Preserves the ordering of the function e.g. If $f(h(x\_2,y\_2))$ is a monotonic transformation of $h$, and $h$ is strictly increasing, e.g. $h(x\_1, y\_1) > h(x\_2,y\_2)$ $\forall (x\_i,y\_i)>(x\_j,y\_j)$ i.e. then $f(h(x\_1,y\_1)) > f(h(x\_2,y\_2))$
* This would mean that a monotonic transformation would preserve the **strictness** I.e. if $h$ is strictly increasing $f(h(x\_2,y\_2))$ is a monotonic transformation and is strictly increasing?

**Question 2):**

As a result of the discussion above regarding the necessity for a strictly increasing function. How does the following definition from *Hal R. Varian, "Microeconomic Analysis," 3rd Edition.* fit with this? Why the subtle difference? Note, he uses **"monotonic"** rather than **"strictly increasing:"**

> A homothetic function is a monotonic transformation of
> a function that is homogeneous of degree 1. In other words, $f(x)$ is homothetic if and only if it can be written as $f(x) = g(h(x))$ where $h(.)$ is homogeneous of degree 1 and $g(.)$ is a **monotonic function**

*Essentially this is a hybrid between **definition 2** and **definition 2** in a longer post i have made about [homothetic function definitions](https://economics.stackexchange.com/questions/55178/homotheic-function-definitions).*

**Tags:** mathematical-economics, consumer-theory, production-function
> **Comments:**
> **username_0**: Hello again, thanks for pointing that out. I've tired to make it more clear. If anything else isn't clear it might be more my lack of understanding, which needs clarifying in answer. But let me know anyway, cheers!  

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## 0 answers

