# Capital in terms of labor

I have a question that asks to find $$\frac{\partial K}{\partial L}$$ from $$Q=cL^aK^b$$, when $$Q$$ and $$c$$ are constants. It lists 4 answer choices but I’m just not sure how to approach it. Implicit function theorem? Chain rule... any help or suggestions would be appreciated.

--Update-- Well I thought about this some more and think I got it: The total differential can be written as: $$dQ=\frac{acK^b}{L^{1-a}}dL+\frac{bcL^a}{K^{1-b}}dK=0$$ dividing by $$dL$$ and c and moving terms, gives $$\frac{aK^b}{L^{1-a}}\frac{\partial L}{\partial L}=-\frac{bL^a}{K^{1-b}}\frac{\partial K}{\partial L}$$ Dividing though by $$L$$ and $$K$$ and expressing it in terms $$\frac{\partial K}{\partial L}$$ gives:

$$\frac{\partial K}{\partial L} = -\frac{aK}{bL}$$,

which I hope is correct.

• +1 for effort shown. I obtained the same result by rearranging the equation as $K=Q^{1/b}c^{-1/b}L^{-a/b}$ and then differentiating. – Adam Bailey Jun 24 at 20:16