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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?

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  • $\begingroup$ How about actually maximizing with respect to the decision variables? (I.e. take the derivatives, set to equal zero, etc.) $\endgroup$
    – Giskard
    Commented Oct 8, 2017 at 20:13
  • $\begingroup$ the system you are trying to solve has two equations namely the derivatives of the profit function wrt to each of the inputs; what you have obtained above is simply the result of dividing your first equation with your second equation; you still have one more equation you are not taking advantage of... $\endgroup$
    – user14471
    Commented Oct 8, 2017 at 20:27
  • $\begingroup$ @user43282 Ah, I see that I'm not taking advantage of the $pF(L,K)-w_1 L - w_2 K$. Is that what you meant to point out? If so, would it make sense to set up a Lagrange function to solve this? $\endgroup$
    – pril
    Commented Oct 8, 2017 at 21:49
  • $\begingroup$ @pril: you are not making use of the second equation you have listed above; setting up a Lagrangian would imply that your optimization problem is constrained, which doesn't seem to be the case here unless you are solving something like max $p F(K,L)-w_1L-w_2K$ s.t. $F(K,L)=c$ $\endgroup$
    – user14471
    Commented Oct 9, 2017 at 6:19
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    $\begingroup$ I'm voting to close this question as off-topic because based on the comments the OPs problems are with math, not economics. $\endgroup$
    – Giskard
    Commented Oct 9, 2017 at 6:20

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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!

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