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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.

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    $\begingroup$ Could you please clarify your question? What is $k$ and what exactly do you mean by "relative marginal products"? $\endgroup$ – Giskard Nov 3 '18 at 19:56
  • $\begingroup$ 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. $\endgroup$ – Chris tie Nov 4 '18 at 15:48
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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).

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  • $\begingroup$ 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$(\cdot)$. $\endgroup$ – Giskard Nov 2 '18 at 23:56
  • $\begingroup$ Can you please also clarify if you think the original claim is true, to make sure we are on the same page. $\endgroup$ – Giskard Nov 2 '18 at 23:57
  • $\begingroup$ @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. $\endgroup$ – Alecos Papadopoulos Nov 3 '18 at 6:32
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    $\begingroup$ I am shocked at your $k$ can be anything comment. The other one is well taken. $\endgroup$ – Giskard Nov 3 '18 at 19:56
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    $\begingroup$ 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. $\endgroup$ – Chris tie Nov 4 '18 at 22:16

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