# Elasticity of Substitution between 2 factors, without knowing relative prices

I'm trying to figure out the elasticity of substitution between input $$s$$ and input $$v$$. I know that the marginal rate of substitution between these two inputs are $$\frac{v^2}{s(v+k)}$$, where $$k$$ is another input.

Then, the elasticity of substitution is,

$$E_{sv}=\dfrac{dln\big(v/s\big)}{dln\bigg(\frac{v^2}{s(v+k)}\bigg)}$$

$$=\dfrac{dln\big(v\big)}{dln\bigg(\frac{v^2}{s(v+k)}\bigg)}-\dfrac{dln\big(s\big)}{dln\bigg(\frac{v^2}{s(v+k)}\bigg)}$$

But I'm not sure how to proceed to get the input elasticity of substitution between factors $$s$$ and $$v$$? Do I need to know the prices of both?

• Sorry the $x$ was a mistake, it should not have been there. Yes, I'm having mathematical difficulties w.r.t. differentiation. Also, conceptually, I'm not sure whether I can even proceed without having the relative prices of both $v$ and $s$. – pafnuti Sep 24 '18 at 16:11
• @denesp, sorry about the mistake. – pafnuti Sep 24 '18 at 17:43