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?


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


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