<p>Output <span class="math-container">$z$</span> is given as <span class="math-container">$z = x + y$</span> where <span class="math-container">$x=min(a,2b)$</span> and <span class="math-container">$y = max(3c,4d)$</span>.</p>
<p>So assume that you want <span class="math-container">$x=12$</span> then <span class="math-container">$a=12$</span> AND <span class="math-container">$b=6$</span>. Since this part of the production delivers only the minimum of serveral production processes <span class="math-container">$a$</span> and <span class="math-container">$2b$</span> the producer must get insure that ALL subproductionprocesses <span class="math-container">$a$</span> and <span class="math-container">$2b$</span> deliver the minimum.</p>
<p>The producer therefore has to buy both <span class="math-container">$a$</span> and <span class="math-container">$b$</span> in the required amounts and therefore the price for <span class="math-container">$x$</span> is composite including both the price of <span class="math-container">$a$</span> and <span class="math-container">$b$</span>.</p>
<p>However, when you want <span class="math-container">$y=12$</span> you must have <span class="math-container">$12=3c$</span> OR <span class="math-container">$12=4d$</span> implying that <span class="math-container">$c=4$</span> OR <span class="math-container">$d=3$</span>. Since the productionprocess <span class="math-container">$y$</span> delivers the max of the subproduction processes the producer only has to make sure that AT LEAST ONE of the subproduction processes delivers the wanted amount <span class="math-container">$y$</span>.</p>
<p>The producer will therefore always produce <span class="math-container">$y$</span> using the &quot;cheapest&quot; input and as dependig on input prices. The input price will therefore only depend on the factor <span class="math-container">$c$</span> or <span class="math-container">$d$</span> chosen to achieve the max.</p>

CC BY-SA 4.0
2020-12-11T17:47:36.697Z
2020-12-11T17:47:36.697Z
Jesper Hybel
<p>Find the conditional input demand function and cost function for the given production function <span class="math-container">$$f(a,b,c,d)=\min\{ a,2b\} + \max\{3c,4d\} $$</span></p>
<p>In The solution, The production function is defined as <span class="math-container">$f(x,y)=x+y$</span> which is perfect substitutes.</p>
<p>Therefore, when <span class="math-container">$P_x &gt;P_y$</span>, <span class="math-container">$x=q$</span> and <span class="math-container">$y=0$</span>. And when <span class="math-container">$P_x &lt; P_y$</span>, <span class="math-container">$y=q$</span> and <span class="math-container">$y=q$</span>.</p>
<p>So far, everything is okay. The point that I don’t understand is how to define <span class="math-container">$P_x$</span> and <span class="math-container">$P_y$</span>. In the answer, these are defined as follows</p>
<p><span class="math-container">$P_x= P_a+(P_b/2)$</span> and <span class="math-container">$P_y= P_c/3 $</span> if <span class="math-container">$P_d/P_c &gt;4/3$</span> and  <span class="math-container">$P_y= P_d/4 $</span> if <span class="math-container">$P_d/P_c &lt;4/3$</span></p>
<p>Why does he sum of the prices of goods a and b to find the <span class="math-container">$P_x$</span>? But while finding <span class="math-container">$P_y$</span>, he separates it under a condition? What is the difference ? In short, how to define Px and Py? My question is this.</p>

<microeconomics><production-function><self-study>
Finding the conditional input demand function
studentp
# Finding the conditional input demand function

(Asked by: username_0 on 2020-12-11)

Find the conditional input demand function and cost function for the given production function $$f(a,b,c,d)=\min\{ a,2b\} + \max\{3c,4d\} $$

In The solution, The production function is defined as $f(x,y)=x+y$ which is perfect substitutes.

Therefore, when $P\_x >P\_y$, $x=q$ and $y=0$. And when $P\_x < P\_y$, $y=q$ and $y=q$.

So far, everything is okay. The point that I don’t understand is how to define $P\_x$ and $P\_y$. In the answer, these are defined as follows

$P\_x= P\_a+(P\_b/2)$ and $P\_y= P\_c/3 $ if $P\_d/P\_c >4/3$ and $P\_y= P\_d/4 $ if $P\_d/P\_c <4/3$

Why does he sum of the prices of goods a and b to find the $P\_x$? But while finding $P\_y$, he separates it under a condition? What is the difference ? In short, how to define Px and Py? My question is this.

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

---

## 1 answers

### Answer by username_1 on 2020-12-11. Score: 5 | Accepted answer

Output $z$ is given as $z = x + y$ where $x=min(a,2b)$ and $y = max(3c,4d)$.

So assume that you want $x=12$ then $a=12$ AND $b=6$. Since this part of the production delivers only the minimum of serveral production processes $a$ and $2b$ the producer must get insure that ALL subproductionprocesses $a$ and $2b$ deliver the minimum.

The producer therefore has to buy both $a$ and $b$ in the required amounts and therefore the price for $x$ is composite including both the price of $a$ and $b$.

However, when you want $y=12$ you must have $12=3c$ OR $12=4d$ implying that $c=4$ OR $d=3$. Since the productionprocess $y$ delivers the max of the subproduction processes the producer only has to make sure that AT LEAST ONE of the subproduction processes delivers the wanted amount $y$.

The producer will therefore always produce $y$ using the "cheapest" input and as dependig on input prices. The input price will therefore only depend on the factor $c$ or $d$ chosen to achieve the max.


---

