<p>I have the following production function</p>
<p><span class="math-container">$$f(x_1,x_2,x_3,x_4)=max\{\min\{x_1, x_2), x_3+2x_4\}\}\ge q$$</span></p>
<p>And I want to find the cost function.</p>
<p>What I think</p>
<p>(1) <span class="math-container">$P_1+P_2 &lt;P_3$</span> and <span class="math-container">$P_3/P_4&lt;1/2$</span></p>
<p>then <span class="math-container">$(x_1^*, x_2^*, x_3^*, x_4^*) =(q,q,0,0) $</span></p>
<p>Cost function is <span class="math-container">$c(q,w)=q[w_1+w_2]$</span></p>
<p>(2) <span class="math-container">$P_1+P_2 &lt;P_4$</span> and <span class="math-container">$P_3/P_4&gt;1/2$</span></p>
<p>then <span class="math-container">$(x_1^*, x_2^*, x_3^*, x_4^*) =(q,q,0,0) $</span></p>
<p>Cost function is <span class="math-container">$c(q,w)=q[w_1+w_2]$</span></p>
<p>(3) <span class="math-container">$P_1+P_2 &gt;P_3$</span> and <span class="math-container">$P_3/P_4&lt;1/2$</span></p>
<p>then <span class="math-container">$(x_1^*, x_2^*, x_3^*, x_4^*) =(0,0,q,0) $</span></p>
<p>Cost function is <span class="math-container">$c(q,w)=qw_3$</span></p>
<p>(4) <span class="math-container">$P_1+P_2 &gt;P_4/2$</span> and <span class="math-container">$P_3/P_4&gt;1/2$</span></p>
<p>then <span class="math-container">$(x_1^*, x_2^*, x_3^*, x_4^*) =(0,0,q/2,0) $</span></p>
<p>Cost function is <span class="math-container">$c(q,w)=qw_4/2$</span></p>
<p>I have no idea how to solve such a production function.  But I posted what I tried. Please show me the correct solution and how to solve such a function. Thank you.</p>

CC BY-SA 4.0
2022-09-09T08:19:43.177Z
Someone has deleted the answer. Let me share it here: One of the ways to solve this problem is almost the same as the one posted here: https://qr.ae/pvKJuI
Amit
CC BY-SA 4.0
2022-09-09T08:20:18.600Z
Using this method will give the following cost function for the above production function:

$C(w_1, w_2, w_3, w_4, q)=q\min\left\{(w_1+w_2),w_3, \dfrac{w_4}{2}\right\}$

where $w_i$ denotes the price (per unit) of input $x_i$, and $q$ is the quantity of output produced.
Amit
CC BY-SA 4.0
2022-09-09T15:12:59.903Z
I am glad that the link I provided is helpful in understanding the approach to solve this question.
Amit
CC BY-SA 4.0
2022-09-09T16:05:08.200Z
Yes for sure @Amit that question with solution is very clear and explanatory :)
studentp
CC BY-SA 4.0
2022-09-09T17:13:36.260Z
Thanks. Actually moderator of the econ-stackexchange deleted the answer giving the reason that the "link only" answers are against the rules of the econ-stackexchange. So I was just confirming.
Amit
<microeconomics><production-function><optimization><cost><cost-functions>
Find cost function for given production function
studentp
# Find cost function for given production function

(Asked by: username_0 on 2022-09-08)

I have the following production function

$$f(x\_1,x\_2,x\_3,x\_4)=max\{\min\{x\_1, x\_2), x\_3+2x\_4\}\}\ge q$$

And I want to find the cost function.

What I think

(1) $P\_1+P\_2 <P\_3$ and $P\_3/P\_4<1/2$

then $(x\_1^\*, x\_2^\*, x\_3^\*, x\_4^\*) =(q,q,0,0) $

Cost function is $c(q,w)=q[w\_1+w\_2]$

(2) $P\_1+P\_2 <P\_4$ and $P\_3/P\_4>1/2$

then $(x\_1^\*, x\_2^\*, x\_3^\*, x\_4^\*) =(q,q,0,0) $

Cost function is $c(q,w)=q[w\_1+w\_2]$

(3) $P\_1+P\_2 >P\_3$ and $P\_3/P\_4<1/2$

then $(x\_1^\*, x\_2^\*, x\_3^\*, x\_4^\*) =(0,0,q,0) $

Cost function is $c(q,w)=qw\_3$

(4) $P\_1+P\_2 >P\_4/2$ and $P\_3/P\_4>1/2$

then $(x\_1^\*, x\_2^\*, x\_3^\*, x\_4^\*) =(0,0,q/2,0) $

Cost function is $c(q,w)=qw\_4/2$

I have no idea how to solve such a production function. But I posted what I tried. Please show me the correct solution and how to solve such a function. Thank you.

**Tags:** microeconomics, production-function, optimization, cost, cost-functions

> **Comments:**
> **username_1**: Someone has deleted the answer. Let me share it here: One of the ways to solve this problem is almost the same as the one posted here: https://qr.ae/pvKJuI  
> **username_1**: Using this method will give the following cost function for the above production function:  $C(w_1, w_2, w_3, w_4, q)=q\min\left\{(w_1+w_2),w_3, \dfrac{w_4}{2}\right\}$  where $w_i$ denotes the price (per unit) of input $x_i$, and $q$ is the quantity of output produced.  
> **username_1**: I am glad that the link I provided is helpful in understanding the approach to solve this question.  
> **username_0**: Yes for sure @username_1 that question with solution is very clear and explanatory :)  
> **username_1**: Thanks. Actually moderator of the econ-stackexchange deleted the answer giving the reason that the "link only" answers are against the rules of the econ-stackexchange. So I was just confirming.  

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## 0 answers

