It is currently Thu Jul 18, 2019 1:54 am

 All times are UTC [ DST ]

 Print view Previous topic | Next topic
Author Message
 Post subject: Subadditivity Posted: Tue Oct 18, 2016 9:40 pm

Joined: Tue Oct 18, 2016 9:36 pm
Posts: 4
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?

Top   Post subject: Re: Subadditivity Posted: Tue Oct 18, 2016 11:00 pm
 Team Member Joined: Mon Nov 09, 2015 1:36 am
Posts: 460
Location: Ioannina, Greece
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?

_________________
Grigorios Kostakos

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

Joined: Tue Oct 18, 2016 9:36 pm
Posts: 4
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 $100 + qy^{0.5} + qy$ ?

Top   Post subject: Re: Subadditivity Posted: Wed Oct 19, 2016 5:31 am
 Team Member Joined: Mon Nov 09, 2015 1:36 am
Posts: 460
Location: Ioannina, Greece
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?

_________________
Grigorios Kostakos

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

Joined: Tue Oct 18, 2016 9:36 pm
Posts: 4
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.

Top   Post subject: Re: Subadditivity Posted: Thu Oct 20, 2016 12:14 am
 Team Member Joined: Mon Nov 09, 2015 1:36 am
Posts: 460
Location: Ioannina, Greece
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?

_________________
Grigorios Kostakos

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

Joined: Tue Oct 18, 2016 9:36 pm
Posts: 4
Yes I understand what do to. But I have serious problems with seperating the cost function.
But there is no necessity of deriving those?

Top   Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending

 All times are UTC [ DST ]

#### Mathimatikoi Online

Users browsing this forum: No registered users and 1 guest

 You cannot post new topics in this forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forumYou cannot post attachments in this forum

Search for:
 Jump to:  Select a forum ------------------ Algebra    Linear Algebra    Algebraic Structures    Homological Algebra Analysis    Real Analysis    Complex Analysis    Calculus    Multivariate Calculus    Functional Analysis    Measure and Integration Theory Geometry    Euclidean Geometry    Analytic Geometry    Projective Geometry, Solid Geometry    Differential Geometry Topology    General Topology    Algebraic Topology Category theory Algebraic Geometry Number theory Differential Equations    ODE    PDE Probability & Statistics Combinatorics General Mathematics Foundation Competitions Archives LaTeX    LaTeX & Mathjax    LaTeX code testings Meta 