# Interpretation of the Cross Partials of the Cobb-Douglas

Consider a Cobb-Douglas Prod. Function $$Y=AL^{a}K^{1-\alpha}$$ This has the cross-partial: $$\frac{\partial^2 Y}{\partial K\partial L}=(1-\alpha)\alpha AL^{\alpha-1}K^{-\alpha}$$

Is the interpretation of this the change in the return to labour, with a change in capital (and by Schwarz's theorem, also the change in the return the capital, with a change in labour)?

Then, is it correct to say that the return to labour increases as $\frac{1}{K^a}$ with capital?

• shouldn't your cross partial be $\frac{\partial^2 Y}{\partial K\partial L}=(1-\alpha)\alpha AL^{\alpha-1}K^{-\alpha}$ – EconJohn May 30 '18 at 4:21
• @EconJohn Yep, corrected. – user526463 May 30 '18 at 16:29

Furthermore, it is not correct to say that the return to labour increases as $\frac{1}{K^a}$ with capital. How did you arrive at this conclusion?
• By simply observing thtat the cross partial scales as $K^{-a}$. – user526463 May 31 '18 at 16:20