what is a homogeneous equation and how do you know if it displays constant returns to scale?

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what is a homogeneous equation and how do you know if it displays constant returns to scale?

Mathematics
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homogenous means a few things
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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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
Q(?,?)
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
if f(tx,ty) = t*f(x,y) its homogenous
hmm, that kind of reminds me of the definition of an odd function .... i wonder if they are related
Q(A,K,L) ?
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.
looks like expansion of length due to heat
it'd be nice to know which letters are variables and which ones are constants in this given function :/
I think convention has it that capitals are constants \[\large Q=A[tbK^{ta}+(1-tb)L^{ta}]^\frac{1}{ta}\] \[\large Q=A[tbK^{ta}+L^{ta}-L^{ta}tb]^\frac{1}{ta}\] \[\large Q=A[tbK^{ta}+L^{ta}-L^{ta}tb]^\frac{1}{ta}\]
and im just going on an idea here, not really sure if itll pan out
\[\large Q=[tb(AK)^{ta}+(AL)^{ta}-tb(AL)^{ta}]^\frac{1}{ta}\] \[\large Q^{ta}-(AL)^{ta}=tb(AK)^{ta}-tb(AL)^{ta}\]
oou,
if A not=0 i wonder if another route would have been easier ...
any idea if im even on teh right track with this idea?
how come \(\large Q^{ta}\) ?
say ta=3 Q = N^(1/3) [Q = N^(1/3)]^3 Q^3 = N^(3/3) Q^3 = N
oh, i see :)
but i wonder if it would be prudent to separate t and a in that .... hard to tell
aahh,
but from what you have there, what are you to factor out to confirm if it's homogeneous or not? :/
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
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 :/
where are the numbers ; (
\[\large Q=A[tbK^{ta}+L^{ta}-L^{ta}tb]^\frac{1}{ta}\] \[\large \frac QA=[tbK^{ta}+L^{ta}-L^{ta}tb]^\frac{1}{ta}\] \[\large \left(\frac QA\right)^{ta}=tbK^{ta}+L^{ta}-L^{ta}tb\] \[\large \left(\frac QA\right)^{ta}=tb(K^{ta}-L^{ta})+L^{ta}\] 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.
im trying to recall ways that logs might be useful to us .... since ive got t stuck in an exponent
why did t go into the exponent in the first place?
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
interesting :)
but then again, Q would be variable as well since it is defined by the inputs ....
maybe Q, A, K, and L are the variables?
which is what unkle alluded to at the start :)
lower case are scalars
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 :)
what class is this for?
computational mathematics
... never heard of it :/ what have you been learning in prior chapters and do they relate to this?
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 :/
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
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?
correct
this is some sort of calculus, right?
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.
how about if you end up with t^a?
t ?
if "a" was one of the variables to begin with ... im not sure.
http://www.sosmath.com/diffeq/first/homogeneous/homogeneous.html
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
ohhhhh :) nice! i think i'm getting the hang of this, so all that matters is if you know what the variables are.... :)
that does help, yes
ok so usually, the function would have 2 variables right?
at least
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 :/
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
is this correct? \(\large logAB^c=clogAB \) ?
\[\large Q(A,K,L)=tA[tbK^{a}+(1-b)tL^{a}]^\frac{1}{a}\] \[\large Q(A,K,L)=tA[t(bK^{a}+(1-b)L^{a})]^\frac{1}{a}\] \[\large Q(A,K,L)=tt^aA[bK^{a}+(1-b)L^{a}]^\frac{1}{a}\] \[\large Q(A,K,L)=t^{(a+1)}~[A[bK^{a}+(1-b)L^{a}]^\frac{1}{a}]\]
if A is a base, then yes
no, A is not a base :/ and oh, looks like you solved it and it appears homogeneous :D
that google book helped me to get the variables right :)
but i might have pulled out the wrong t exponent
I'd have to purchase it though :(
t^(1/a) pulls out, not t^a t*t^(1/a) = t^(1+1/a) =t^((a+1)/a) typoes it :)
yh i just noticed :) thanks for pointing it out xD seems like an interesting topic though
"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
so it doesn't display a constant scale.. :/
recheck my math to make sure theres not a mistake :)
yes i'm trying to solve it myself on paper right now :) thanks again though xD
good luck, thats about all i can do for it ;)
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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