Show that the Cobb-Douglas production function, for Labour costs L and Capital costs K, $P(L, K) = AL^{\alpha}K^{1-\alpha}$ satisfies the equation:
$$L\frac{\partial P}{\partial L} + K\frac{\partial P}{\partial K} = P$$
and
$$L^2\frac{\partial^2 P}{\partial L^2} + K^2\frac{\partial^2 P}{\partial K^2} = P$$
Sorry for the bad formatting. I am a bit new to the community and I don't really know how to write the formulas nicer.
I am able to prove the first equation nicely enough: $$ \begin{eqnarray} \frac{\partial P}{\partial L} = \alpha A L^{\alpha-1}K^{1-\alpha},\\ L\frac{\partial P}{\partial L} = \alpha A L^{\alpha}K^{1-\alpha}. \end{eqnarray} $$ and $$ \begin{eqnarray} K\frac{\partial P}{\partial K} &=& A L^{\alpha}K^{1-\alpha}-\alpha AL^{\alpha}K^{1-\alpha},\\ L\frac{\partial P}{\partial L} + K\frac{\partial P}{\partial K} &=& \alpha A L^{\alpha}K^{1-\alpha}+AL^{\alpha}K^{1-\alpha}-\alpha AL^{\alpha}K^{1-\alpha} = AL^{\alpha}K^{1-\alpha} = P \end{eqnarray} $$
Anyway... I am trying equation number two and I have gotten down to:
$$ \begin{eqnarray} L^2\frac{\partial^2P}{\partial L^2} + K^2\frac{\partial^2P}{\partial K^2} &=& \alpha^2AL^{\alpha}K^{1-\alpha}-\alpha AL^{\alpha}K^{1-\alpha}+\alpha^2 AL^{\alpha}K^{1-\alpha}-\alpha AL^{\alpha}K^{1-\alpha}\\ &=& 2\alpha^2 AL^{\alpha}K^{1-\alpha} - 2\alpha AL^{\alpha}K^{1-\alpha}. \end{eqnarray} $$ No idea how to get to a $AL^{\alpha}K^{1-\alpha}$ from here. Any help appreciated.
