<p>I think it is simpler to analyze the production function first, and then use the definition of an isoquant.</p>
<h3>Production function</h3>
<p>Let's rewrite the production function in a more convenient way
<span class="math-container">\begin{align*}
f\left( z_1, z_2 \right) 
&amp;= \min\left\lbrace z_1 + 2z_2, 2z_1 + z_2 \right\rbrace \\ 
&amp;= \begin{cases} 
z_1 + 2z_2 &amp;&amp; \text{if } z_1 + 2z_2 \leq 2z_1 + z_2 \\
2z_1 + z_2 &amp;&amp; \text{if } z_1 + 2z_2 &gt; 2z_1 + z_2 \\
\end{cases} \\
&amp;= \begin{cases} 
z_1 + 2z_2 &amp;&amp; \text{if } z_2 \leq z_1 \\
2z_1 + z_2 &amp;&amp; \text{if } z_2 &gt; z_1 \\
\end{cases}
\end{align*}</span>
So the typical isoquant has a kink at <span class="math-container">$z_1 = z_2$</span>, that is on along the 45° line.</p>
<h3>Isoquant</h3>
<p>Let <span class="math-container">$q$</span> be a fixed quantity. The level-q isoquant is given by the <span class="math-container">$z_1$</span> and <span class="math-container">$z_2$</span> that satisfy <span class="math-container">$f(z_1, z_2) = q$</span>. Therefore,
<span class="math-container">\begin{cases}
z_1 + 2z_2 = q &amp;&amp; \text{if } z_2 \leq z_1 \\
2z_1 + z_2 = q &amp;&amp; \text{if } z_2 &gt; z_1 \\
\end{cases}</span>
Then,
<span class="math-container">\begin{cases} 
z_2 = \frac{q - z_1}{2} &amp;&amp; \text{if } z_2 \leq z_1 \\
z_2 = q - 2z_1 &amp;&amp; \text{if } z_2 &gt; z_1 \\
\end{cases}</span>
We end up with an equation
<span class="math-container">$$
z_2 =
\begin{cases} 
\frac{q - z_1}{2} &amp;&amp; \text{if } z_2 \leq z_1 \\
q - 2z_1 &amp;&amp; \text{if } z_2 &gt; z_1 \\
\end{cases}
$$</span></p>
<p>In words, the level-q isoquant is made of:</p>
<ul>
<li>the line <span class="math-container">$z_2 = \frac{q - z_1}{2}$</span> in the space <span class="math-container">$z_2 \leq z_1$</span> (below the 45° line)</li>
<li>the line <span class="math-container">$z_2 = q - 2z_1$</span> in the space <span class="math-container">$z_2 &gt; z_1$</span>
(above the 45° line)</li>
</ul>
<p>That is something we can draw. For example, for <span class="math-container">$q = 9$</span> we get
<a href="https://i.sstatic.net/k5n8A.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/k5n8A.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2022-03-01T10:45:16.767Z
2022-03-01T10:45:16.767Z
Thomas Favory
<p>Consider a firm that can produce <em>q</em> units of good <em>G</em> using two technologies and two production factors, <span class="math-container">$z_1$</span> and <span class="math-container">$z_2$</span>. There are two ways how a firm
can produce the good G: It can use 2 units of <span class="math-container">$z_1$</span> and 1 unit of <span class="math-container">$z_2$</span> to produce
one unit of good G, or it can use 1 unit of <span class="math-container">$z_1$</span> and 2 units of <span class="math-container">$z_2$</span> to also produce
one unit of the same good. Using these proportions of inputs, the firm is able
to replicate any amount of good in question. The production set of the firm
displays (among other properties) free disposal property and is convex.</p>
<p>My question is whether the production function is expressed like this <span class="math-container">$$f(z)=\min\{ z_1 + 2z_2 ; 2z_1 + z_2\}$$</span>
and how would we express the isoquant function? Would it be like this?
<a href="https://i.sstatic.net/c9PDD.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/c9PDD.jpg" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2022-01-25T11:44:08.573Z
Have you tried substituting some numbers?
Giskard
CC BY-SA 4.0
2022-01-25T11:45:56.810Z
Like how many units can the company produce with inputs $z_1 = 3, z_2=3$ according to your formula, and how many according to the text?
Giskard
CC BY-SA 4.0
2022-01-25T11:49:19.413Z
Wouldn't it be 9 in both cases?
Maybeline Lee
CC BY-SA 4.0
2022-01-25T11:51:02.663Z
Can you please elaborate on how you get 9 from the text?
Giskard
CC BY-SA 4.0
2022-01-25T11:55:29.143Z
Seems like you would need at least 9 of each input in both processes outlined?
Giskard
CC BY-SA 4.0
2022-01-25T11:56:54.347Z
Yes, but isn't that the case?
Maybeline Lee
CC BY-SA 4.0
2022-01-25T12:25:54.193Z
I edited my isoquant figure. Would it be like this?
Maybeline Lee
CC BY-SA 4.0
2022-01-25T14:20:11.250Z
It seems from your graph that $z_1=z_2=1.5q$ is enough to produce $q$  units? So how much can you make from 3-3 units?
Giskard
CC BY-SA 4.0
2022-03-01T13:36:10.150Z
Off topic but gotta say it... the handwriting is gorgeous. Something we dont see in economics very often.
KeithRoberts
<microeconomics><production-function><self-study><cost-functions>
isoquant of a leontief production function
Maybeline Lee
# isoquant of a leontief production function

(Asked by: username_0 on 2022-01-25)

Consider a firm that can produce *q* units of good *G* using two technologies and two production factors, $z\_1$ and $z\_2$. There are two ways how a firm
can produce the good G: It can use 2 units of $z\_1$ and 1 unit of $z\_2$ to produce
one unit of good G, or it can use 1 unit of $z\_1$ and 2 units of $z\_2$ to also produce
one unit of the same good. Using these proportions of inputs, the firm is able
to replicate any amount of good in question. The production set of the firm
displays (among other properties) free disposal property and is convex.

My question is whether the production function is expressed like this $$f(z)=\min\{ z\_1 + 2z\_2 ; 2z\_1 + z\_2\}$$
and how would we express the isoquant function? Would it be like this?
[![enter image description here](https://i.sstatic.net/c9PDD.jpg)](https://i.sstatic.net/c9PDD.jpg)

**Tags:** microeconomics, production-function, self-study, cost-functions

> **Comments:**
> **username_1**: Have you tried substituting some numbers?  
> **username_1**: Like how many units can the company produce with inputs $z_1 = 3, z_2=3$ according to your formula, and how many according to the text?  
> **username_0**: Wouldn't it be 9 in both cases?  
> **username_1**: Can you please elaborate on how you get 9 from the text?  
> **username_1**: Seems like you would need at least 9 of each input in both processes outlined?  
> **username_0**: Yes, but isn't that the case?  
> **username_0**: I edited my isoquant figure. Would it be like this?  
> **username_1**: It seems from your graph that $z_1=z_2=1.5q$ is enough to produce $q$  units? So how much can you make from 3-3 units?  
> **username_2**: Off topic but gotta say it... the handwriting is gorgeous. Something we dont see in economics very often.  

---

## 1 answers

### Answer by username_3 on 2022-03-01. Score: 2 | Accepted answer

I think it is simpler to analyze the production function first, and then use the definition of an isoquant.

### Production function

Let's rewrite the production function in a more convenient way
\begin{align\*}
f\left( z\_1, z\_2 \right)
&= \min\left\lbrace z\_1 + 2z\_2, 2z\_1 + z\_2 \right\rbrace \\
&= \begin{cases}
z\_1 + 2z\_2 && \text{if } z\_1 + 2z\_2 \leq 2z\_1 + z\_2 \\
2z\_1 + z\_2 && \text{if } z\_1 + 2z\_2 > 2z\_1 + z\_2 \\
\end{cases} \\
&= \begin{cases}
z\_1 + 2z\_2 && \text{if } z\_2 \leq z\_1 \\
2z\_1 + z\_2 && \text{if } z\_2 > z\_1 \\
\end{cases}
\end{align\*}
So the typical isoquant has a kink at $z\_1 = z\_2$, that is on along the 45° line.

### Isoquant

Let $q$ be a fixed quantity. The level-q isoquant is given by the $z\_1$ and $z\_2$ that satisfy $f(z\_1, z\_2) = q$. Therefore,
\begin{cases}
z\_1 + 2z\_2 = q && \text{if } z\_2 \leq z\_1 \\
2z\_1 + z\_2 = q && \text{if } z\_2 > z\_1 \\
\end{cases}
Then,
\begin{cases}
z\_2 = \frac{q - z\_1}{2} && \text{if } z\_2 \leq z\_1 \\
z\_2 = q - 2z\_1 && \text{if } z\_2 > z\_1 \\
\end{cases}
We end up with an equation
$$
z\_2 =
\begin{cases}
\frac{q - z\_1}{2} && \text{if } z\_2 \leq z\_1 \\
q - 2z\_1 && \text{if } z\_2 > z\_1 \\
\end{cases}
$$

In words, the level-q isoquant is made of:

* the line $z\_2 = \frac{q - z\_1}{2}$ in the space $z\_2 \leq z\_1$ (below the 45° line)
* the line $z\_2 = q - 2z\_1$ in the space $z\_2 > z\_1$
  (above the 45° line)

That is something we can draw. For example, for $q = 9$ we get
[![enter image description here](https://i.sstatic.net/k5n8A.png)](https://i.sstatic.net/k5n8A.png)


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