Profit maximization with Cobb-Douglas function I'm trying to maximize a firm's profit given the production function $F(L,K)=L^\alpha K^\beta$ (where $L$ is labor and $K$ is capital) and that $\alpha + \beta \neq 1$.



So, I know that this maximization problem can be written as $\text{max }pF(L,K)-w_1 L-w_2 K$.

Since $pMP_L (L^*,K^*)=w_1$, $p\alpha(L^*)^{\alpha-1}(K^*)^\beta=w_1$. And since $pMP_K (L^*,K^*)=w_2$, $p\beta(L^*)^{\alpha}(K^*)^{\beta-1}=w_2$.

By dividing these functions and simplifying, we get $\displaystyle\frac{\alpha K^*}{\beta L^*}=\displaystyle\frac{w_1}{w_2}$.

I'm unsure how to proceed from here, though. Should I solve for $L^*$ by separating $K^*$ from the equation and plugging into $pMP_L$? Wouldn't this yield a very complicated solution?
Should I solve for $L^∗$ by separating $K^∗$ from the equation and plugging into $pMP_{L}$


Yep, that's about it.


  Wouldn't this yield a very complicated solution?


Somewhat. The math is available at many places, like section 4 here. But you can surely do it yourself!