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 Post subject: Subadditivity
PostPosted: Tue Oct 18, 2016 9:40 pm 

Joined: Tue Oct 18, 2016 9:36 pm
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Suppose the joint production of goods $X$ and $Y$ is described by the following cost function:

$C(qx,qy) = 0$ if $qx,qy = 0$

and
$100 + qx0.5 + qy0.5 + (qx + qy)$ if $qx,qy > 0$

Over what values of $qx$ and $qy$, if any, will $C(qx,qy)$ be subadditive?

Can somebody help me how I can solve this question?


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 Post subject: Re: Subadditivity
PostPosted: Tue Oct 18, 2016 11:00 pm 
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To be the function \[C(x,y)=\begin{cases}
100+1.5\,qx+1.5\,qy\,, & qx>0,qy>0\\
0\,, & qx=qy=0
\end{cases}\] subadditive it should be: \begin{align*}
C\big((qx_1,qy_1)+(qx_2,qy_2)\big)\leqslant C(qx_1,qy_1)+C(qx_2,qy_2)
\end{align*} for all $(qx_1,qy_1),(qx_2,qy_2)\in X\times Y$. But to proceed, we should know first if
\[C\big((qx_1,qy_1)+(qx_2,qy_2)\big)=C(qx_1+qx_2,\,qy_1+qy_2)\] holds. Does it?

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 Post subject: Re: Subadditivity
PostPosted: Wed Oct 19, 2016 3:49 am 

Joined: Tue Oct 18, 2016 9:36 pm
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Grigorios Kostakos wrote:
To be the function \[C(x,y)=\begin{cases}
100+1.5\,qx+1.5\,qy\,, & qx>0,qy>0\\
0\,, & qx=qy=0
\end{cases}\] subadditive it should be: \begin{align*}
C\big((qx_1,qy_1)+(qx_2,qy_2)\big)\leqslant C(qx_1,qy_1)+C(qx_2,qy_2)
\end{align*} for all $(qx_1,qy_1),(qx_2,qy_2)\in X\times Y$. But to proceed, we should know first if
\[C\big((qx_1,qy_1)+(qx_2,qy_2)\big)=C(qx_1+qx_2,\,qy_1+qy_2)\] holds. Does it?



Thanks for the help!

I am not quite sure how to calculate the function of only producing one good for the equation $$C((qx_1,qy_1)+(qx_2,qy_2)\big)=C(qx_1+qx_2,\,qy_1+qy_2)$$

As the first function is about producing both goods jointly, I am not sure hoe to figure out the costs for producing the goods separately.

Is it simply $100 + qx^{0.5} + qx$ and $100 + qy^{0.5} + qy$ ?


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 Post subject: Re: Subadditivity
PostPosted: Wed Oct 19, 2016 5:31 am 
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masaky wrote:
...Is it simply $100 + qx^{0.5} + qx$ and $100 + qy^{0.5} + qy$ ?

In your fist post you wrote \[C(x,y)=\begin{cases}
100+qx\,0.5+qy\,0.5+qx+qy\,, & qx>0,qy>0\\
0\,, & qx=qy=0
\end{cases}\]
which is different from \[C(x,y)=\begin{cases}
100+qx^{0.5}+qy^{0.5}+qx+qy\,, & qx>0,qy>0\\
0\,, & qx=qy=0
\end{cases}\]
Which one is the right function?

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 Post subject: Re: Subadditivity
PostPosted: Wed Oct 19, 2016 4:04 pm 

Joined: Tue Oct 18, 2016 9:36 pm
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Grigorios Kostakos wrote:
\[C(x,y)=\begin{cases}
100+qx^{0.5}+qy^{0.5}+qx+qy\,, & qx>0,qy>0\\
0\,, & qx=qy=0
\end{cases}\]
Which one is the right function?


This one is the right one. With ^0.5. I had first problems with visualizing the formulae.

So it should be first derived, right?

0.5 qx^0.5 + 0.5 qy^0.5 = ? But then, the 100 will disappear.
I am confused.


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 Post subject: Re: Subadditivity
PostPosted: Thu Oct 20, 2016 12:14 am 
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If you write the subadditive property for this function, it would help.

Is this \[C\big((qx_1,qy_1)+(qx_2,qy_2)\big)\leqslant C(qx_1,qy_1)+C(qx_2,qy_2)\] the right one?

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 Post subject: Re: Subadditivity
PostPosted: Thu Oct 20, 2016 4:13 am 

Joined: Tue Oct 18, 2016 9:36 pm
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Yes I understand what do to. But I have serious problems with seperating the cost function.
But there is no necessity of deriving those?


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