# 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"? Commented Nov 3, 2018 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. Commented Nov 4, 2018 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$(\cdot)$. Commented Nov 2, 2018 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. Commented Nov 2, 2018 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. Commented Nov 3, 2018 at 6:32
• I am shocked at your $k$ can be anything comment. The other one is well taken. Commented Nov 3, 2018 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. Commented Nov 4, 2018 at 22:16