# What does the elasticity say about the fraction of total cost used on input 1?

A firm have the following production function $$y=x_{1}^{\alpha} x_{2}^{1-\alpha}, \quad 0< \alpha < 1$$ $$w_1>0$$ is the cost of input 1 and $$w_2 > 0$$ is the cost of input 2.

(1.1) Derive the input demand functions.

I have done that and I get

$$x_{1}^{*}\left(w_{1}, w_{2}, y\right)=\left(\frac{\alpha}{1-\alpha}\right)^{1-\alpha} w_{1}^{-(1-\alpha)} w_{2}^{1-\alpha} y$$

$$x_{2}^{*}\left(w_{1}, w_{2}, y\right)=\left(\frac{\alpha}{1-\alpha}\right)^{-\alpha} w_{1}^{\alpha} w_{2}^{-\alpha} y$$

I'm confident in this result.

1.2 What is the elasticity of $$x_2^*/x_1^*$$with respect to $$w_2/w_1$$?

\begin{aligned} E &=\frac{\partial x_{2}^{*} / x_{1}^{*}}{\partial w_{2} / w_{1}} \frac{w_{2} / w_{1}}{x_{2}^{*} / x_{1}^{*}} \\ &=-\frac{1-\alpha}{\alpha}\left(\frac{w_{2}}{w_{1}}\right)^{-2} \frac{w_{2} / w_{1}}{x_{2}^{*} / x_{1}^{*}} \\ &=-\frac{1-\alpha}{\alpha}\left(\frac{w_{2}}{w_{1}}\right)^{-1} \frac{x_{1}^{*}}{x_{2}^{*}} \\ &=-\frac{1-\alpha}{\alpha} \frac{w_{1}}{w_{2}} \frac{w_{2}}{w_{1}} \frac{\alpha}{1-\alpha} \\ &=-1 \end{aligned}

(1.3) What does the above elasticity say about the fraction of total cost used on input 1?

$$\frac{w_1x_1}{w_1x_1+w_2x_2}$$

I'm lost here. I do not know how to answer this question.