<p><span class="math-container">$z_2$</span> is included in the production function. Please see the production function <span class="math-container">$f:\mathbb{R}^2_+\rightarrow\mathbb{R}$</span>:
<span class="math-container">\begin{eqnarray*} f(z_1,z_2) = \left(\min(\lfloor z_2\rfloor, 1)\right)z_1^\alpha = \begin{cases} z_1^\alpha &amp; \text{if } z_2 \geq 1 \\ 0 &amp; \text{if } z_2 &lt; 1\end{cases} \end{eqnarray*}</span>
where <span class="math-container">$\lfloor z_2\rfloor$</span> is greatest integer less than or equal to <span class="math-container">$z_2$</span>, and <span class="math-container">$\alpha &gt; 0$</span> is given.</p>
<p>Cost minimization problem of a competitive firm is defined as follows:
<span class="math-container">\begin{eqnarray*} \min_{z_1\geq 0,z_2\geq 0} &amp; \ w_1z_1 + w_2z_2 \\ \text{s.t. } &amp; f(z_1,z_2)\geq q\end{eqnarray*}</span>
where <span class="math-container">$q\geq 0$</span>, <span class="math-container">$w_1&gt;0$</span> and <span class="math-container">$w_2&gt;0$</span>.
Solving the problem, we get the conditional input demand as follows:
<span class="math-container">\begin{eqnarray*} (z_1,z_2)(w_1,w_2,q) =  \begin{cases} (0,0) &amp; \text{if } q = 0 \\ \left(q^{\frac{1}{\alpha}},1\right) &amp; \text{if } q &gt; 0\end{cases} \end{eqnarray*}</span>
Therefore, the cost function is
<span class="math-container">\begin{eqnarray*} C(w_1,w_2,q) =  \begin{cases} 0 &amp; \text{if } q = 0 \\ w_2+w_1q^{\frac{1}{\alpha}} &amp; \text{if } q &gt; 0\end{cases} \end{eqnarray*}</span></p>

CC BY-SA 4.0
2023-04-19T08:37:52.777Z
2023-04-19T08:37:52.777Z
Amit
<p>proportions production function as follows:
<a href="https://i.sstatic.net/v3cG8.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/v3cG8.png" alt="production function" /></a></p>
<p>where the price of input is 1 and z2 is supposed to be a fixed factor of production. I've been having trouble finding the cost function because if z2 isn't included in the production function, I'm not sure how I'd approach finding the cost function.</p>

<microeconomics><production-function><producer-theory>
Derive cost function from production function
Lily B
# Derive cost function from production function

(Asked by: username_0 on 2023-04-19)

proportions production function as follows:
[![production function](https://i.sstatic.net/v3cG8.png)](https://i.sstatic.net/v3cG8.png)

where the price of input is 1 and z2 is supposed to be a fixed factor of production. I've been having trouble finding the cost function because if z2 isn't included in the production function, I'm not sure how I'd approach finding the cost function.

**Tags:** microeconomics, production-function, producer-theory


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## 1 answers

### Answer by username_1 on 2023-04-19. Score: 3 | Accepted answer

$z\_2$ is included in the production function. Please see the production function $f:\mathbb{R}^2\_+\rightarrow\mathbb{R}$:
\begin{eqnarray\*} f(z\_1,z\_2) = \left(\min(\lfloor z\_2\rfloor, 1)\right)z\_1^\alpha = \begin{cases} z\_1^\alpha & \text{if } z\_2 \geq 1 \\ 0 & \text{if } z\_2 < 1\end{cases} \end{eqnarray\*}
where $\lfloor z\_2\rfloor$ is greatest integer less than or equal to $z\_2$, and $\alpha > 0$ is given.

Cost minimization problem of a competitive firm is defined as follows:
\begin{eqnarray\*} \min\_{z\_1\geq 0,z\_2\geq 0} & \ w\_1z\_1 + w\_2z\_2 \\ \text{s.t. } & f(z\_1,z\_2)\geq q\end{eqnarray\*}
where $q\geq 0$, $w\_1>0$ and $w\_2>0$.
Solving the problem, we get the conditional input demand as follows:
\begin{eqnarray\*} (z\_1,z\_2)(w\_1,w\_2,q) = \begin{cases} (0,0) & \text{if } q = 0 \\ \left(q^{\frac{1}{\alpha}},1\right) & \text{if } q > 0\end{cases} \end{eqnarray\*}
Therefore, the cost function is
\begin{eqnarray\*} C(w\_1,w\_2,q) = \begin{cases} 0 & \text{if } q = 0 \\ w\_2+w\_1q^{\frac{1}{\alpha}} & \text{if } q > 0\end{cases} \end{eqnarray\*}


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