sasogeek
  • sasogeek
what is a homogeneous equation and how do you know if it displays constant returns to scale?
Mathematics
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chestercat
  • chestercat
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UnkleRhaukus
  • UnkleRhaukus
?
UnkleRhaukus
  • UnkleRhaukus
homogenous means a few things
sasogeek
  • sasogeek
ok i'll ask the real question so that you can see where i'm coming from, i'd like to figure out somethings about it on my own though..... one sec.

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sasogeek
  • sasogeek
Show that the production equation \(\huge Q=A[bK^a+(1-b)L^a]^\frac{1}{a} \) is homogeneous and displays constant returns to scale
UnkleRhaukus
  • UnkleRhaukus
Q(?,?)
amistre64
  • amistre64
spose you scale the variables by some constant amount (t is the usual generic that ive seen); if you can factor out the scalar completely, then the equation is homogenous
amistre64
  • amistre64
if f(tx,ty) = t*f(x,y) its homogenous
amistre64
  • amistre64
hmm, that kind of reminds me of the definition of an odd function .... i wonder if they are related
UnkleRhaukus
  • UnkleRhaukus
Q(A,K,L) ?
sasogeek
  • sasogeek
i do not know, i wasn't at the college when this assignment was given but it was given to me today and it's due on monday.
UnkleRhaukus
  • UnkleRhaukus
looks like expansion of length due to heat
sasogeek
  • sasogeek
it'd be nice to know which letters are variables and which ones are constants in this given function :/
amistre64
  • amistre64
I think convention has it that capitals are constants \[\large Q=A[tbK^{ta}+(1-tb)L^{ta}]^\frac{1}{ta}\] \[\large Q=A[tbK^{ta}+L^{ta}-L^{ta}tb]^\frac{1}{ta}\] \[\large Q=A[tbK^{ta}+L^{ta}-L^{ta}tb]^\frac{1}{ta}\]
amistre64
  • amistre64
and im just going on an idea here, not really sure if itll pan out
amistre64
  • amistre64
\[\large Q=[tb(AK)^{ta}+(AL)^{ta}-tb(AL)^{ta}]^\frac{1}{ta}\] \[\large Q^{ta}-(AL)^{ta}=tb(AK)^{ta}-tb(AL)^{ta}\]
UnkleRhaukus
  • UnkleRhaukus
oou,
amistre64
  • amistre64
if A not=0 i wonder if another route would have been easier ...
amistre64
  • amistre64
any idea if im even on teh right track with this idea?
sasogeek
  • sasogeek
how come \(\large Q^{ta}\) ?
amistre64
  • amistre64
say ta=3 Q = N^(1/3) [Q = N^(1/3)]^3 Q^3 = N^(3/3) Q^3 = N
sasogeek
  • sasogeek
oh, i see :)
amistre64
  • amistre64
but i wonder if it would be prudent to separate t and a in that .... hard to tell
UnkleRhaukus
  • UnkleRhaukus
aahh,
sasogeek
  • sasogeek
but from what you have there, what are you to factor out to confirm if it's homogeneous or not? :/
amistre64
  • amistre64
some exponential factor of t; if I can get rid of any semblense of the "t" such that it becomes a scalar instead ... then the equation would be definined as homogenous. Assuming i have the right definition of homogeneity to begin with
sasogeek
  • sasogeek
what if from the beginning, t wasn't even supposed to be mentioned and maybe your t, is same as the b, or a ? :/ i'm not sure cos i have no idea about homogeneity and i was just presented with this exercise lol, i've got quite some reading to do :/
UnkleRhaukus
  • UnkleRhaukus
where are the numbers ; (
amistre64
  • amistre64
\[\large Q=A[tbK^{ta}+L^{ta}-L^{ta}tb]^\frac{1}{ta}\] \[\large \frac QA=[tbK^{ta}+L^{ta}-L^{ta}tb]^\frac{1}{ta}\] \[\large \left(\frac QA\right)^{ta}=tbK^{ta}+L^{ta}-L^{ta}tb\] \[\large \left(\frac QA\right)^{ta}=tb(K^{ta}-L^{ta})+L^{ta}\] t is just a generic setup, it doesnt matter what it equals to. If we make it more specific, than all we do is prove that it works or does not work for a specific case.
amistre64
  • amistre64
im trying to recall ways that logs might be useful to us .... since ive got t stuck in an exponent
sasogeek
  • sasogeek
why did t go into the exponent in the first place?
amistre64
  • amistre64
because im assume that a and b are variables in this setup; so we have to attach a generic scalar to the variables and see if we can pull it out
sasogeek
  • sasogeek
interesting :)
amistre64
  • amistre64
but then again, Q would be variable as well since it is defined by the inputs ....
amistre64
  • amistre64
maybe Q, A, K, and L are the variables?
amistre64
  • amistre64
which is what unkle alluded to at the start :)
UnkleRhaukus
  • UnkleRhaukus
lower case are scalars
sasogeek
  • sasogeek
well we never know until we try it out to find out how things work out :/ i'm new to this anyway so anything to simplify the situation :)
amistre64
  • amistre64
what class is this for?
sasogeek
  • sasogeek
computational mathematics
amistre64
  • amistre64
... never heard of it :/ what have you been learning in prior chapters and do they relate to this?
sasogeek
  • sasogeek
I haven't had that class at all, I spent the whole week with the admissions and faculty office. I just received this exercise though so I'm yet to read about homogeneous functions but thought i'd ask here to start with :/
amistre64
  • amistre64
i hope my framework is at least on the right track :) Itd prolly take me about a week trying to read thru the material for the class to be sure tho. good luck with it
sasogeek
  • sasogeek
thanks :) i'll try to do what you say and see what comes off it. attach t to the variables and try to factor it out. if it works, it's homogeneous, if not, it's not :) right?
amistre64
  • amistre64
correct
sasogeek
  • sasogeek
this is some sort of calculus, right?
amistre64
  • amistre64
if you can get rid of all the ts you put in; spose you end up with t^2 after factoring it all, that is acceptable as well. Not to sure how much of this has to do with calculus.
sasogeek
  • sasogeek
how about if you end up with t^a?
UnkleRhaukus
  • UnkleRhaukus
t ?
amistre64
  • amistre64
if "a" was one of the variables to begin with ... im not sure.
amistre64
  • amistre64
http://www.sosmath.com/diffeq/first/homogeneous/homogeneous.html
amistre64
  • amistre64
for example: f(x,y) = x + y^2 f(tx,ty) = tx + (ty)^2 = tx + t^2y^2 = t(x + ty^2) since we cant get rid of all the ts in the original setup, this equation would not be considered homogenous
sasogeek
  • sasogeek
ohhhhh :) nice! i think i'm getting the hang of this, so all that matters is if you know what the variables are.... :)
amistre64
  • amistre64
that does help, yes
sasogeek
  • sasogeek
ok so usually, the function would have 2 variables right?
UnkleRhaukus
  • UnkleRhaukus
at least
sasogeek
  • sasogeek
ahhh, i was going to go ahead and say that since _b and _a are the only small letters, they're possibly the variables cos there's only 2 small letters :/ if we should consider AKL, that's 3 and rather odd, i think :/
amistre64
  • amistre64
this looks like its on the same line as yours http://books.google.com/books?id=H92Z6yfhxk8C&pg=PA287&lpg=PA287&dq=is+homogeneous+and+displays+constant+returns+to+scale&source=bl&ots=U0qs5wcM5l&sig=MLZeYGYPupb1Xj3AeQF87vmSduY&hl=en#v=onepage&q=is%20homogeneous%20and%20displays%20constant%20returns%20to%20scale&f=false
sasogeek
  • sasogeek
is this correct? \(\large logAB^c=clogAB \) ?
amistre64
  • amistre64
\[\large Q(A,K,L)=tA[tbK^{a}+(1-b)tL^{a}]^\frac{1}{a}\] \[\large Q(A,K,L)=tA[t(bK^{a}+(1-b)L^{a})]^\frac{1}{a}\] \[\large Q(A,K,L)=tt^aA[bK^{a}+(1-b)L^{a}]^\frac{1}{a}\] \[\large Q(A,K,L)=t^{(a+1)}~[A[bK^{a}+(1-b)L^{a}]^\frac{1}{a}]\]
amistre64
  • amistre64
if A is a base, then yes
sasogeek
  • sasogeek
no, A is not a base :/ and oh, looks like you solved it and it appears homogeneous :D
amistre64
  • amistre64
that google book helped me to get the variables right :)
amistre64
  • amistre64
but i might have pulled out the wrong t exponent
sasogeek
  • sasogeek
I'd have to purchase it though :(
amistre64
  • amistre64
t^(1/a) pulls out, not t^a t*t^(1/a) = t^(1+1/a) =t^((a+1)/a) typoes it :)
sasogeek
  • sasogeek
yh i just noticed :) thanks for pointing it out xD seems like an interesting topic though
amistre64
  • amistre64
"displays constant returns to scale" the google book seems to be saying that: when the exponent value of t is less than 1, it displays a decreasing scale when the exponent value of t is equal 1, it displays a constant scale when the exponent value of t is greater than 1, it displays an increasing scale
sasogeek
  • sasogeek
so it doesn't display a constant scale.. :/
amistre64
  • amistre64
recheck my math to make sure theres not a mistake :)
sasogeek
  • sasogeek
yes i'm trying to solve it myself on paper right now :) thanks again though xD
amistre64
  • amistre64
good luck, thats about all i can do for it ;)
sasogeek
  • sasogeek
with the rest of the work I believe their questions i can solve on my own, basic algebra and statistics. thanks again though, can't thank you enough :)))) <3

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