<p><strong>Hint:</strong> Think how many units of <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span> are needed to produce <span class="math-container">$q=1$</span> in order to minimize the cost. What about producing <span class="math-container">$q=2$</span>? Can you generalize it?</p>
<p>In order to minimize the total cost, you want to use as few units of either input as possible. Imagine you wanted to produce <span class="math-container">$q$</span> units. You would need <em>at least</em> <span class="math-container">$x_1=q$</span> and <span class="math-container">$x_2=q$</span> (otherwise you wouldn't be able to produce <span class="math-container">$q$</span>). Therefore, to produce <span class="math-container">$q$</span>, you choose precisely <span class="math-container">$x_1(q)=q$</span> and <span class="math-container">$x_2(q)=q$</span>. These are the derived factor demand functions. You only need to plug them in the cost identity</p>
<p><span class="math-container">\begin{align}
TC(q)&amp;=w_1 x_1(q)+w_1 x_2(q)\\
&amp;=w_1 q +w_2 q\\
&amp;=(w_1+w_2)q
\end{align}</span></p>

CC BY-SA 4.0
2020-12-28T03:49:15.167Z
this is a non-answer. Should be posed in the comments.
Brennan
CC BY-SA 4.0
2020-12-24T19:02:18.383Z
2020-12-28T14:46:17.263Z
2020-12-28T14:46:17.263Z
Patricio
<p>Imagine I got a production function like it :
<span class="math-container">$$
\min\{x_1, x_2\}
$$</span></p>
<p>How can I find the cost function?</p>

CC BY-SA 4.0
2020-12-26T10:52:06.270Z
This post should be helpful https://economics.stackexchange.com/questions/41479/finding-the-conditional-input-demand-function/41481#41481
Jesper Hybel
<microeconomics><production-function><leontief>
How to find the cost function for perfect complements
victor
# How to find the cost function for perfect complements

(Asked by: username_0 on 2020-12-24)

Imagine I got a production function like it :
$$
\min\{x\_1, x\_2\}
$$

How can I find the cost function?

**Tags:** microeconomics, production-function, leontief

> **Comments:**
> **username_1**: This post should be helpful https://economics.stackexchange.com/questions/41479/finding-the-conditional-input-demand-function/41481#41481  

---

## 1 answers

### Answer by username_2 on 2020-12-24. Score: 0

**Hint:** Think how many units of $x\_1$ and $x\_2$ are needed to produce $q=1$ in order to minimize the cost. What about producing $q=2$? Can you generalize it?

In order to minimize the total cost, you want to use as few units of either input as possible. Imagine you wanted to produce $q$ units. You would need *at least* $x\_1=q$ and $x\_2=q$ (otherwise you wouldn't be able to produce $q$). Therefore, to produce $q$, you choose precisely $x\_1(q)=q$ and $x\_2(q)=q$. These are the derived factor demand functions. You only need to plug them in the cost identity

\begin{align}
TC(q)&=w\_1 x\_1(q)+w\_1 x\_2(q)\\
&=w\_1 q +w\_2 q\\
&=(w\_1+w\_2)q
\end{align}

> **Comments:**
> **username_3**: this is a non-answer. Should be posed in the comments.  

---

