<p>The problem is basically to derive the cost function of a firm. What you need to do is minimize costs given the output. That is
$$
\min\limits_{K,L} w \cdot L + r \cdot K
$$
subject to
$$
K^{\frac{1}{4}} \cdot L^{\frac{3}{4}} = Q = 120.
$$
You can solve this using Lagrange or the implicit function theorem (similar application as the MRS).</p>

CC BY-SA 3.0
2015-06-02T19:23:20.770Z
2015-06-02T19:23:20.770Z
Giskard
<p>I am trying to find <strong>L$^*$</strong> and <strong>K$^*$</strong> (Labour and Capital respectively) given the following:</p>

<blockquote>
  <p>Q = $K^{1/4}$ $L^{3/4}$ where Q = 120, w=$24, and r=$128 (Not sure if you need w and r)</p>
</blockquote>

<p>I know that if I can find either <strong>L$^*$</strong> or <strong>K$^*$</strong>, I can find the other one easily.</p>

<p>I have found several questions relating to Cobbs-Douglas on here, but none where Q is given. A close example is <a href="https://math.stackexchange.com/questions/997041/finding-the-optimal-combination-for-the-cobb-douglas-function-given-a-budget">this</a>, but I don't have a budget constraint. </p>

CC BY-SA 3.0
2015-06-02T18:28:50.037Z
You can have tidy upper indices by putting whatever you want to have in the upper index in {} brackets. For example
"K^{\frac{1}{4}}".
Giskard
<cobb-douglas>
Optimizing Cobbs-Douglas given output
Alex McLean
# Optimizing Cobbs-Douglas given output

(Asked by: username_0 on 2015-06-02)

I am trying to find **L$^\*$** and **K$^\*$** (Labour and Capital respectively) given the following:

> Q = $K^{1/4}$ $L^{3/4}$ where Q = 120, w=$24, and r=$128 (Not sure if you need w and r)

I know that if I can find either **L$^\*$** or **K$^\*$**, I can find the other one easily.

I have found several questions relating to Cobbs-Douglas on here, but none where Q is given. A close example is [this](https://math.stackexchange.com/questions/997041/finding-the-optimal-combination-for-the-cobb-douglas-function-given-a-budget), but I don't have a budget constraint.

**Tags:** cobb-douglas
> **Comments:**
> **username_1**: You can have tidy upper indices by putting whatever you want to have in the upper index in {} brackets. For example "K^{\frac{1}{4}}".  

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## 1 answers

### Answer by username_1 on 2015-06-02. Score: 2 | Accepted answer

The problem is basically to derive the cost function of a firm. What you need to do is minimize costs given the output. That is
$$
\min\limits\_{K,L} w \cdot L + r \cdot K
$$
subject to
$$
K^{\frac{1}{4}} \cdot L^{\frac{3}{4}} = Q = 120.
$$
You can solve this using Lagrange or the implicit function theorem (similar application as the MRS).


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