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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.

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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.

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