# Why does a homothetic function have constant ratio of marginal products along rays?

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.

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

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).

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