<blockquote>
  <p>Should I solve for $L^∗$ by separating $K^∗$ from the equation and plugging into $pMP_{L}$</p>
</blockquote>

<p>Yep, that's about it.</p>

<blockquote>
  <p>Wouldn't this yield a very complicated solution?</p>
</blockquote>

<p>Somewhat. The math is available at many places, like section 4 <a href="https://www3.nd.edu/~agervais/documents/Cobb-Douglas.pdf" rel="nofollow noreferrer">here</a>. But you can surely do it yourself!</p>

CC BY-SA 3.0
2017-10-08T23:41:55.523Z
2017-10-08T23:41:55.523Z
android16.5
<p>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 $\alpha + \beta \neq 1$.</p>

<hr>

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

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

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

<p>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?</p>

CC BY-SA 3.0
2017-10-08T20:13:24.073Z
How about actually maximizing with respect to the decision variables? (I.e. take the derivatives, set to equal zero, etc.)
Giskard
CC BY-SA 3.0
2017-10-08T21:49:37.407Z
@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?
pril
CC BY-SA 3.0
2017-10-09T06:20:18.483Z
I'm voting to close this question as off-topic because based on the comments the OPs problems are with math, not economics.
Giskard
CC BY-SA 3.0
2017-10-09T08:13:02.620Z
What is the actual problem? How much further you want to go? Why? You already got the solution in terms of MRTS=MRT. You found the point where the isoquants of $F(L,K)$ match the relative factor prices. If you want to proceed "further", you need a general equilibrium approach, where you define labour supply, capital supply, consumption, maybe utility maximisation. As the problem stands, you have got as far as you can get.
luchonacho
CC BY-SA 3.0
2017-11-08T08:30:29.287Z
@luchonacho It seems prices are given, there is no need for a general equilibrium approach.
Giskard
<self-study><cobb-douglas><profit-maximization>
Profit maximization with Cobb-Douglas function
pril
# Profit maximization with Cobb-Douglas function

(Asked by: username_0 on 2017-10-08)

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?

**Tags:** self-study, cobb-douglas, profit-maximization
> **Comments:**
> **username_1**: How about actually maximizing with respect to the decision variables? (I.e. take the derivatives, set to equal zero, etc.)  
> **username_0**: @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?  
> **username_1**: I'm voting to close this question as off-topic because based on the comments the OPs problems are with math, not economics.  
> **username_2**: What is the actual problem? How much further you want to go? Why? You already got the solution in terms of MRTS=MRT. You found the point where the isoquants of $F(L,K)$ match the relative factor prices. If you want to proceed "further", you need a general equilibrium approach, where you define labour supply, capital supply, consumption, maybe utility maximisation. As the problem stands, you have got as far as you can get.  
> **username_1**: @luchonacho It seems prices are given, there is no need for a general equilibrium approach.  

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## 1 answers

### Answer by username_3 on 2017-10-08. Score: 3

> 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](https://www3.nd.edu/~agervais/documents/Cobb-Douglas.pdf). But you can surely do it yourself!


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