Production function and elasticity Let $y=x_1^\alpha x_2^\beta$ where $\beta=1-\alpha$ be a Cobb-Douglas production function.
Find the elasticity of the optimal demand functions (for minimizing production cost) for both goods wrt. $w_2/w_1$ ($w_1,w_2$ are the respective prices of the inputs). What does this say for our spending on $x_1$  compared to the total cost?
My try
I found the elasticity of $x_2^*/x_1^*$ wrt $w_2/w_1$ to be 1. Now, I have to find a value for the ammount spend on good $x_1$. How can I do that with my elasticity? I think it has to be constant, but still no clue on how to find such a value even though I have found the elasticity to be 1.
edit: I know it has to be constant.
The first order conditions equate marginal revenue per factor to the price of that factor:
\begin{align}
p\cdot\alpha\frac{y}{x_1} &amp;= w_1\\
p\cdot\beta\frac{y}{x_2} &amp;= w_2,
\end{align}
Where I used the property of power function $(x^n)'_n = n \frac{x^n}{x}$.
Divide the second FOC by the first to get the relation between the relative prices and the relative factor demands:
$$\frac{\beta}{\alpha}\frac{x_1}{x_2} = \frac{w_2}{w_1}. \tag{A}$$
From this relation we can draw two conclusions:

Rewrite (A) in log form:

$$- \ln \frac{x_2}{x_1} + \ln \frac{\beta}{\alpha} =\ln \frac{w_2}{w_1},$$
And using the log definition of elasticity $\epsilon_y^x = \frac{\mathrm{d}\ln y}{\mathrm{d}\ln x}$ we come to the conclusion, that relative factor demand is decreasing in relative factor prices with unit elasticity:
$$\frac{\mathrm{d}\ln x_2/ x_1}{\mathrm{d}\ln w_2/w_1} = -1.$$

Multiply both sides of (A) by $\frac{x_2}{x_1}$ :

$$\frac{\beta}{\alpha} = \frac{w_2x_2}{w_1x_1}. \tag{B}$$
Rearrangement (B) says that the expenditures on different factors are proportional to their respective input elasticities, i.e. if our total spending on factor 1 is $\\\$\alpha$ then we must spend $\\\$\beta$ on factor 2.
The total cost $C$ is allocated in the same proportion, i.e. for a general Cobb-Douglas production function, spending on factor 1 is
$$w_1 x_1 = \frac{\alpha}{\alpha+\beta}C,$$
or simply $w_1 x_1 =\alpha C$ if $\alpha+\beta=1$, i.e. if the production function is homogeneous of degree 1.