<p>You are trying to find the solution of $K$ and $L$ in terms of $Q$ to derive the total cost function with respect to quantity.</p>

<p><a href="https://homepage.univie.ac.at/matan.tsur/courses/lecture7.pdf" rel="nofollow">Page 13 of this lecture</a> has the <strong>exact</strong> question you are interested in. (Google is your friend.)</p>

<ul>
<li><p>From the Lagrangian you'll have $\frac{K}{L}=\frac{w}{r}$</p></li>
<li><p>From the production constraint you'll have $50K^{\frac{1}{2}}L^{\frac{1}{2}} = 1000$ and you have already derived the solution to this.</p></li>
</ul>

<p>The first implies that $4K = L$ given your solution.</p>

<p>Then since,
$50K^{\frac{1}{2}}L^{\frac{1}{2}} = Q$</p>

<p>$$\implies 50K^{\frac{1}{2}}(4K)^{\frac{1}{2}} = Q \\
\implies 100K = Q \\
\implies K(Q) = \frac{Q}{100}$$</p>

<p>A similar solution can be found for $L(Q)$, and then it should be smooth sailing for you from there.</p>

CC BY-SA 3.0
2016-06-22T18:10:02.317Z
2016-06-22T18:10:02.317Z
Kitsune Cavalry
<blockquote>
  <p>Suppose $F(K,L)= 50L^{\frac{1}{2}}K^{\frac{1}{2}}$, the wage is $w = 5$
   (euros) and rent is $r = 20$ (euros). What is the cost of producing
  $1000$ units? Derive the cost function $TC(Q)$.</p>
</blockquote>

<p>I know how to find out the cost. </p>

<p>$L = 40$ and $K = 10$. So the total cost is $400$.</p>

<p>But now I am stuck. The solution for the cost function should be: $$TC(Q) = 5\cdot \dfrac{4Q}{100} + 20\cdot \dfrac{Q}{100} = \dfrac{2}{5}Q$$
How do I get to this function?</p>

<microeconomics><production-function>
Derive the Cost Function TC(Q)
Maria
# Derive the Cost Function TC(Q)

(Asked by: username_0 on 2016-06-21)

> Suppose $F(K,L)= 50L^{\frac{1}{2}}K^{\frac{1}{2}}$, the wage is $w = 5$
> (euros) and rent is $r = 20$ (euros). What is the cost of producing
> $1000$ units? Derive the cost function $TC(Q)$.

I know how to find out the cost.

$L = 40$ and $K = 10$. So the total cost is $400$.

But now I am stuck. The solution for the cost function should be: $$TC(Q) = 5\cdot \dfrac{4Q}{100} + 20\cdot \dfrac{Q}{100} = \dfrac{2}{5}Q$$
How do I get to this function?

**Tags:** microeconomics, production-function

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## 1 answers

### Answer by username_1 on 2016-06-22. Score: 2

You are trying to find the solution of $K$ and $L$ in terms of $Q$ to derive the total cost function with respect to quantity.

[Page 13 of this lecture](https://homepage.univie.ac.at/matan.tsur/courses/lecture7.pdf) has the **exact** question you are interested in. (Google is your friend.)

* From the Lagrangian you'll have $\frac{K}{L}=\frac{w}{r}$
* From the production constraint you'll have $50K^{\frac{1}{2}}L^{\frac{1}{2}} = 1000$ and you have already derived the solution to this.

The first implies that $4K = L$ given your solution.

Then since,
$50K^{\frac{1}{2}}L^{\frac{1}{2}} = Q$

$$\implies 50K^{\frac{1}{2}}(4K)^{\frac{1}{2}} = Q \\
\implies 100K = Q \\
\implies K(Q) = \frac{Q}{100}$$

A similar solution can be found for $L(Q)$, and then it should be smooth sailing for you from there.


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