Given Cobb-Douglas production functions for 2 factories (same owner), how will the owner produce $y$?

So my question is this: A company owns two factories, A and B, each with the following production functions:

$f_A(x_1,x_2)=x_1^{\alpha}x_2^{1-\alpha}$

$f_B(x_1,x_2)=x_1^{\beta}x_2^{1-\beta}$

Now assuming that $\alpha=1/2$, $\beta=3/4$ and $w_1=w_2=1$ (prices on the input market), how will the company choose to produce $y$?

I've solved the two production functions for the following demand functions:

$\textbf{x}_A(\textbf{w},y)=\left(yr_1^{\alpha},yr_1^{\alpha-1}\right)$ with $r_1=\dfrac{w_1(1-\alpha)}{w_2\alpha}$

$\textbf{x}_B(\textbf{w},y)=\left(yr_2^{\beta},yr_2^{\beta-1}\right)$ with $r_2=\dfrac{w_1(1-\beta)}{w_2\beta}$

And the expenditure functions:

$c_A(\textbf{w},y)=w_1yr_1^{\alpha}+w_2yr_1^{\alpha-1}$

$c_B(\textbf{w},y)=w_1yr_2^{\beta}+w_2yr_2^{\beta-1}$

Now, I am no sure how to determine how the company will produce $y$. If I insert the values above I get some weird numbers.

Any help would be much appreciated!

Thanks.

• There seems to be some problem with your input demand functions. Demand for $x_1$ should be decreasing in $w_1$ but now it is increasing. My guess is that $r_1$ should be $\frac{1}{r_1}$. – Giskard May 31 '15 at 14:43
• Thanks, that could be the problem I guess. I'll try solving it again, see if I end up with $\dfrac{1}{r_1}$. Many thanks :) – Phillip Bredahl May 31 '15 at 14:53
• Try to use your intuition wherever possible. Since the payments to each input type are the same regardless of which factory it is used at, consider whether it is optimal for the marginal productivity of one input to be larger at one factory than the other, and what this would imply for adjusting the inputs at each factory. Consider setting the problem up as a cost minimization problem if that doesn't work for you. – Hessian May 31 '15 at 14:58