<p>Cost minimisation problem of the firm is defined as follows:
<span class="math-container">\begin{eqnarray*} \min_{x_1\geq 0, x_2\geq 0} &amp; w_1x_1+w_2x_2 \\ \text{s.t. } &amp; \min(2x_1+x_2, x_1+2x_2) = y \end{eqnarray*}</span>
where <span class="math-container">$w_1&gt;0$</span>, <span class="math-container">$w_2&gt;0$</span>, <span class="math-container">$y&gt;0$</span> are given.
Here is the graph of the constraint set:
<a href="https://i.sstatic.net/pEXl6.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/pEXl6.png" alt="enter image description here" /></a></p>
<p>Observe that the solution to the above problem is given by:
<span class="math-container">\begin{eqnarray*} (x_1^c, x_2^c) \in \begin{cases} \left\{(y,0)\right\} &amp; \text{if } \frac{w_1}{w_2} &lt; \frac{1}{2} \\ \left\{t(y,0)+(1-t)\left(\frac{y}{3},\frac{y}{3}\right)|0\leq t\leq 1\right\} &amp; \text{if } \frac{w_1}{w_2} = \frac{1}{2} \\ \left\{\left(\frac{y}{3},\frac{y}{3}\right)\right\} &amp; \text{if } \frac{1}{2} &lt; \frac{w_1}{w_2} &lt; 2 \\ \left\{t(0,y)+(1-t)\left(\frac{y}{3},\frac{y}{3}\right)|0\leq t\leq 1\right\} &amp; \text{if } \frac{w_1}{w_2} = 2 \\ \left\{(0,y)\right\} &amp; \text{if } \frac{w_1}{w_2} &gt; 2\end{cases} \end{eqnarray*}</span></p>
<p>Consequently, the cost function (which is the optimal cost) is
<span class="math-container">\begin{eqnarray*} C(w_1,w_2,y) = \begin{cases} w_1y &amp; \text{if } \frac{w_1}{w_2} \leq \frac{1}{2}  \\ (w_1+w_2)\frac{y}{3} &amp; \text{if } \frac{1}{2} &lt; \frac{w_1}{w_2} &lt; 2 \\  w_2y &amp; \text{if } \frac{w_1}{w_2} \geq 2\end{cases} \end{eqnarray*}</span></p>

CC BY-SA 4.0
2022-12-03T18:44:36.467Z
Thank you very much for that detailed answer. If possible can I ask did you draw the graph by hand or by using codes in a program like python , or
Tatanik501
CC BY-SA 4.0
2022-12-04T00:50:56.047Z
I used LaTeX Tikz.
Amit
CC BY-SA 4.0
2022-12-20T09:10:34.463Z
Dear Amit, can I ask how should I  approach the question if the second term in min term is not $x_1 + 2 x_2$ but it is $x_3 + 2 x_4$ ?
Tatanik501
CC BY-SA 4.0
2022-12-20T12:29:39.733Z
Please refer to this link: https://qr.ae/pvKJuI
Amit
CC BY-SA 4.0
2022-12-03T15:27:18.330Z
2022-12-03T15:27:18.330Z
Amit
<p>I saw the same type of questions are asked for that but I am new at this subject so I couldn't grasp it totally.</p>
<p>so basically for that production function:</p>
<p><span class="math-container">$$
f(x_1,x_2) = min \{ 2x_1 +x_2 , x_1 +2x_2 \}
$$</span>
what is the way to find <span class="math-container">$C(w,y)$</span> ?</p>

<microeconomics><firm>
How to find cost function from leontief production
Tatanik501
# How to find cost function from leontief production

(Asked by: username_0 on 2022-12-03)

I saw the same type of questions are asked for that but I am new at this subject so I couldn't grasp it totally.

so basically for that production function:

$$
f(x\_1,x\_2) = min \{ 2x\_1 +x\_2 , x\_1 +2x\_2 \}
$$
what is the way to find $C(w,y)$ ?

**Tags:** microeconomics, firm

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## 1 answers

### Answer by username_1 on 2022-12-03. Score: 4 | Accepted answer

Cost minimisation problem of the firm is defined as follows:
\begin{eqnarray\*} \min\_{x\_1\geq 0, x\_2\geq 0} & w\_1x\_1+w\_2x\_2 \\ \text{s.t. } & \min(2x\_1+x\_2, x\_1+2x\_2) = y \end{eqnarray\*}
where $w\_1>0$, $w\_2>0$, $y>0$ are given.
Here is the graph of the constraint set:
[![enter image description here](https://i.sstatic.net/pEXl6.png)](https://i.sstatic.net/pEXl6.png)

Observe that the solution to the above problem is given by:
\begin{eqnarray\*} (x\_1^c, x\_2^c) \in \begin{cases} \left\{(y,0)\right\} & \text{if } \frac{w\_1}{w\_2} < \frac{1}{2} \\ \left\{t(y,0)+(1-t)\left(\frac{y}{3},\frac{y}{3}\right)|0\leq t\leq 1\right\} & \text{if } \frac{w\_1}{w\_2} = \frac{1}{2} \\ \left\{\left(\frac{y}{3},\frac{y}{3}\right)\right\} & \text{if } \frac{1}{2} < \frac{w\_1}{w\_2} < 2 \\ \left\{t(0,y)+(1-t)\left(\frac{y}{3},\frac{y}{3}\right)|0\leq t\leq 1\right\} & \text{if } \frac{w\_1}{w\_2} = 2 \\ \left\{(0,y)\right\} & \text{if } \frac{w\_1}{w\_2} > 2\end{cases} \end{eqnarray\*}

Consequently, the cost function (which is the optimal cost) is
\begin{eqnarray\*} C(w\_1,w\_2,y) = \begin{cases} w\_1y & \text{if } \frac{w\_1}{w\_2} \leq \frac{1}{2} \\ (w\_1+w\_2)\frac{y}{3} & \text{if } \frac{1}{2} < \frac{w\_1}{w\_2} < 2 \\ w\_2y & \text{if } \frac{w\_1}{w\_2} \geq 2\end{cases} \end{eqnarray\*}

> **Comments:**
> **username_0**: Thank you very much for that detailed answer. If possible can I ask did you draw the graph by hand or by using codes in a program like python , or  
> **username_1**: I used LaTeX Tikz.  
> **username_0**: Dear Amit, can I ask how should I  approach the question if the second term in min term is not $x_1 + 2 x_2$ but it is $x_3 + 2 x_4$ ?  
> **username_1**: Please refer to this link: https://qr.ae/pvKJuI  

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