<p>A homothetic function can be characterized as follows:
Let <span class="math-container">$f(\mathbf x)$</span>, <span class="math-container">$\mathbf x \in \mathbb R^n$</span> be a function homogeneous of degree <span class="math-container">$r$</span>. Let <span class="math-container">$g()$</span> be a function with <span class="math-container">$g'\neq 0$</span>. Then </p>

<p><span class="math-container">$$G(\mathbf x) = g[f(\mathbf x)]$$</span>
is homothetic. Since <span class="math-container">$f(\mathbf x)$</span> is homogeneous of degree <span class="math-container">$r$</span> we have that </p>

<p><span class="math-container">$$f(\lambda \mathbf x) = \lambda ^ rf(\mathbf x) $$</span></p>

<p>Then </p>

<p><span class="math-container">$$G(\lambda \mathbf x) = g[\lambda ^ r f(\mathbf x)]$$</span> and so </p>

<p><span class="math-container">$$\frac{\partial G(\lambda x)}{\partial x_i}=g'[f(\lambda  x)]\cdot \lambda ^r \frac{f(x)}{\partial x_i}=\lambda^r \cdot\frac{g'[f(\lambda x)]}{g'[f(x)]}\frac{\partial G(x)}{\partial x_i}$$</span></p>

<p>Evidently, we will also have </p>

<p><span class="math-container">$$\frac{\partial G(\lambda x)}{\partial x_j}=g'[f(\lambda  x)]\cdot \lambda ^r \frac{f(x)}{\partial x_j}=\lambda^r \cdot\frac{g'[f(\lambda x)]}{g'[f(x)]}\frac{\partial G(x)}{\partial x_j}$$</span></p>

<p>which leads to the "constant MRS along rays" characterization of homothetic functions,</p>

<p><span class="math-container">$$\frac{\partial G(\lambda x) / \partial x_i}{\partial G(\lambda x) / \partial x_j} = \frac{\partial G (x) / \partial x_i}{\partial G( x) / \partial x_j}$$</span></p>

<p>(see Simon and Blume 1994, p. 503).</p>

CC BY-SA 4.0
2018-11-02T23:56:38.717Z
Can you please back up your characterization of a homotethic function? Seems like $g$ should be strictly increasing or decreasing, not just never constant like abs$(\cdot)$.
Giskard
CC BY-SA 4.0
2018-11-02T23:57:34.360Z
Can you please also clarify if you think the original claim is true, to make sure we are on the same page.
Giskard
CC BY-SA 4.0
2018-11-03T06:32:00.407Z
@denesp Simon and Blume (1994) p. 500 define a homothetic function indeed as a monotonic transformation of a homogeneous function. But as Chiang (1984) p. 423-424 notes, from a mathematical standpoint we only need a non-zero derivative, although in economics we restrict the definition to a monotonic transformation (and usually to a strictly positive derivative of $g$) so that it continues to be useful for representing economic phenomena.
Alecos Papadopoulos
CC BY-SA 4.0
2018-11-03T06:37:35.870Z
@denesp What is the original claim here? If it is the expression after the OP saying "the ordering has the property..." then my post shows that it holds. If it is the title of the OP's post, and by this title it is meant "constant MRS along rays", it is also true.
Alecos Papadopoulos
CC BY-SA 4.0
2018-11-03T08:04:55.370Z
Well, the property claimed is
$$
\frac{\partial f}{\partial x_i}(\lambda x)= k \frac{\partial f}{\partial x_i }(x)
$$
which is not what you show. (What is $k$?)  And the title says constant marginal product, not constant MRS.
Giskard
CC BY-SA 4.0
2018-11-03T18:37:45.137Z
@denesp $k$ is some number. Then it can be $\lambda ^r$ as I show.
Alecos Papadopoulos
CC BY-SA 4.0
2018-11-03T18:39:41.283Z
@denesp The title says "constant _relative_ marginal products", which I concluded that it was a sloppy way to say MRS.
Alecos Papadopoulos
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2018-11-03T19:56:02.417Z
I am shocked at your $k$ can be anything comment. The other one is well taken.
Giskard
CC BY-SA 4.0
2018-11-04T16:28:50.487Z
I think this is exactly what I was looking for, I am confused about the math, though.  By the chain rule:
$\frac{∂G(λx)}{∂x_i }=g′(f(λ x)) λ^r\frac{∂f(x)}{∂x_i}$

but $g'$ still depends on lambda, and does not directly give $G'(x)$
Chris tie
CC BY-SA 4.0
2018-11-04T18:22:08.627Z
@Chris tie The whole point of writing $g'$ is to indicate that it is the derivative of $g$ with respect to its argument, _irrespective of which argument is_. I added one more step to make this more clear.
Alecos Papadopoulos
CC BY-SA 4.0
2018-11-04T22:16:10.667Z
I would argue that it should look something like this:

$$\frac{\partial G(\lambda x)}{\partial x_i}=g'[f(\lambda  x)]\cdot \lambda ^r \frac{f(x)}{\partial x_i}=\lambda^r \cdot\frac{g'[f(\lambda x)]}{g'[f(x)]}\frac{\partial G(x)}{\partial x_i} $$

The function argument has to be identical in order to revert the chain rule. The current solution implies that every homothetic function is homogeneous, which is not the case.
Chris tie
CC BY-SA 4.0
2018-11-05T01:41:29.387Z
@Chris tie Correct. Answer adjusted.
Alecos Papadopoulos
CC BY-SA 4.0
2018-11-02T21:48:33.393Z
2018-11-05T01:39:33.787Z
2018-11-05T01:39:33.787Z
Alecos Papadopoulos
<p>A homothetic ordering is defined as</p>

<p><span class="math-container">$x \succeq y \Rightarrow \lambda x \succeq \lambda y \qquad \forall \lambda &gt;0$</span></p>

<p>where <span class="math-container">$x,y \in \mathbb{R}^n$</span></p>

<p>Then, any differentiable function representing the ordering has the property</p>

<p><span class="math-container">$\frac{\partial f}{\partial x_i}(\lambda x)= k \frac{\partial f}{\partial x_i }(x)$</span></p>

<p>with <span class="math-container">$k,\lambda &gt;0$</span></p>

<p>How is this results derived? </p>

<p>I can see how we can derive properties of the function values from homothety, but have no idea how we can say anything about its derivatives.</p>

CC BY-SA 4.0
2018-11-03T19:56:53.537Z
Could you please clarify your question? What is $k$ and what exactly do you mean by "relative marginal products"?
Giskard
CC BY-SA 4.0
2018-11-04T15:48:39.257Z
By constant relative marginal products I mean the MRS, essentially. I changed "relative" to "ratio" for more clarity. I assume $k$ to be a constant, but I am not sure. The property in question stems from the Palgrave Dictionary of Economics.
Chris tie
<microeconomics><production-function><producer-theory>
Why does a homothetic function have constant ratio of marginal products along rays?
Chris tie
# Why does a homothetic function have constant ratio of marginal products along rays?

(Asked by: username_0 on 2018-11-02)

A homothetic ordering is defined as

$x \succeq y \Rightarrow \lambda x \succeq \lambda y \qquad \forall \lambda >0$

where $x,y \in \mathbb{R}^n$

Then, any differentiable function representing the ordering has the property

$\frac{\partial f}{\partial x\_i}(\lambda x)= k \frac{\partial f}{\partial x\_i }(x)$

with $k,\lambda >0$

How is this results derived?

I can see how we can derive properties of the function values from homothety, but have no idea how we can say anything about its derivatives.

**Tags:** microeconomics, production-function, producer-theory

> **Comments:**
> **username_1**: Could you please clarify your question? What is $k$ and what exactly do you mean by "relative marginal products"?  
> **username_0**: By constant relative marginal products I mean the MRS, essentially. I changed "relative" to "ratio" for more clarity. I assume $k$ to be a constant, but I am not sure. The property in question stems from the Palgrave Dictionary of Economics.  

---

## 1 answers

### Answer by username_2 on 2018-11-02. Score: 3 | Accepted answer

A homothetic function can be characterized as follows:
Let $f(\mathbf x)$, $\mathbf x \in \mathbb R^n$ be a function homogeneous of degree $r$. Let $g()$ be a function with $g'\neq 0$. Then

$$G(\mathbf x) = g[f(\mathbf x)]$$
is homothetic. Since $f(\mathbf x)$ is homogeneous of degree $r$ we have that

$$f(\lambda \mathbf x) = \lambda ^ rf(\mathbf x) $$

Then

$$G(\lambda \mathbf x) = g[\lambda ^ r f(\mathbf x)]$$ and so

$$\frac{\partial G(\lambda x)}{\partial x\_i}=g'[f(\lambda x)]\cdot \lambda ^r \frac{f(x)}{\partial x\_i}=\lambda^r \cdot\frac{g'[f(\lambda x)]}{g'[f(x)]}\frac{\partial G(x)}{\partial x\_i}$$

Evidently, we will also have

$$\frac{\partial G(\lambda x)}{\partial x\_j}=g'[f(\lambda x)]\cdot \lambda ^r \frac{f(x)}{\partial x\_j}=\lambda^r \cdot\frac{g'[f(\lambda x)]}{g'[f(x)]}\frac{\partial G(x)}{\partial x\_j}$$

which leads to the "constant MRS along rays" characterization of homothetic functions,

$$\frac{\partial G(\lambda x) / \partial x\_i}{\partial G(\lambda x) / \partial x\_j} = \frac{\partial G (x) / \partial x\_i}{\partial G( x) / \partial x\_j}$$

(see Simon and Blume 1994, p. 503).

> **Comments:**
> **username_1**: Can you please back up your characterization of a homotethic function? Seems like $g$ should be strictly increasing or decreasing, not just never constant like abs$(\cdot)$.  
> **username_1**: Can you please also clarify if you think the original claim is true, to make sure we are on the same page.  
> **username_2**: @denesp Simon and Blume (1994) p. 500 define a homothetic function indeed as a monotonic transformation of a homogeneous function. But as Chiang (1984) p. 423-424 notes, from a mathematical standpoint we only need a non-zero derivative, although in economics we restrict the definition to a monotonic transformation (and usually to a strictly positive derivative of $g$) so that it continues to be useful for representing economic phenomena.  
> **username_2**: @denesp What is the original claim here? If it is the expression after the OP saying "the ordering has the property..." then my post shows that it holds. If it is the title of the OP's post, and by this title it is meant "constant MRS along rays", it is also true.  
> **username_1**: Well, the property claimed is $$ \frac{\partial f}{\partial x_i}(\lambda x)= k \frac{\partial f}{\partial x_i }(x) $$ which is not what you show. (What is $k$?)  And the title says constant marginal product, not constant MRS.  
> **username_2**: @denesp $k$ is some number. Then it can be $\lambda ^r$ as I show.  
> **username_2**: @denesp The title says "constant _relative_ marginal products", which I concluded that it was a sloppy way to say MRS.  
> **username_1**: I am shocked at your $k$ can be anything comment. The other one is well taken.  
> **username_0**: I think this is exactly what I was looking for, I am confused about the math, though.  By the chain rule: $\frac{∂G(λx)}{∂x_i }=g′(f(λ x)) λ^r\frac{∂f(x)}{∂x_i}$  but $g'$ still depends on lambda, and does not directly give $G'(x)$  
> **username_2**: @username_0 The whole point of writing $g'$ is to indicate that it is the derivative of $g$ with respect to its argument, _irrespective of which argument is_. I added one more step to make this more clear.  
> **username_0**: I would argue that it should look something like this:  $$\frac{\partial G(\lambda x)}{\partial x_i}=g'[f(\lambda  x)]\cdot \lambda ^r \frac{f(x)}{\partial x_i}=\lambda^r \cdot\frac{g'[f(\lambda x)]}{g'[f(x)]}\frac{\partial G(x)}{\partial x_i} $$  The function argument has to be identical in order to revert the chain rule. The current solution implies that every homothetic function is homogeneous, which is not the case.  
> **username_2**: @username_0 Correct. Answer adjusted.  

---

