I know the Cobb-Douglass function which describes the production quantity:
$$Q(K, L) = A \cdot K^\alpha \cdot L^\beta$$
Also I do know multiple assignments of K (Capital) and L (Labour) to Q (Quantity):
Q | 10 | 30 | 60 | 80 | 95 | 108
K | 10 | 10 | 10 | 10 | 10 | 10
L | 1 | 2 | 3 | 4 | 5 | 6
Because we have three unknown variables and we have jumps in Q I decided to choose three different assignments:
$$Q(10, 1) = 10$$ $$Q(10, 3) = 60$$ $$Q(10, 5) = 95$$
Now when I insert the first assignment I can get the value of A:
$$\implies 10 = A \cdot 10^\alpha \cdot 1^\beta \iff A = \frac{10}{10^\alpha \cdot 1^\beta}$$
However when I now use the second assignment to get ⍺ or β I make some mistake:
$$ \implies 30 = \frac{10 \cdot 2^\alpha 10^\beta}{1^\alpha 10^\beta} \iff 3 = \frac{2^\alpha 10^\beta}{1^\alpha 10^\beta} \iff 3 = \frac{2^\alpha}{1^\alpha} \iff 3 = \frac{1^\alpha \cdot 1^\alpha}{1^\alpha} \iff 3 = 1^\alpha \iff \alpha = \frac{log(3)}{log(1)}$$
Which does not seem correct to me.
Where did the mistake(s) occur?
Bodo
There are several mistakes. First, $A=\frac{10}{10^\alpha1^\beta}=10^{1-\alpha}$ (this was correct!). But now, you use this for the second assignment, and I think you mixed up $\alpha$ and $\beta$. You should have had $Q(10,2)=30$ $\Leftrightarrow$ $10^{1-\alpha}\cdot10^\alpha\cdot2^\beta=30$ $\Leftrightarrow$ $10\cdot 2^\beta=30$ $\Leftrightarrow$ $2^\beta=3$ $\Leftrightarrow$ $\beta=\frac{\log3}{\log2}$. There was a further mistake in your calculation: $2^\alpha\neq1^\alpha\cdot1^\alpha$!