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sasogeek
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what is a homogeneous equation and how do you know if it displays constant returns to scale?
 2 years ago
 2 years ago
sasogeek Group Title
what is a homogeneous equation and how do you know if it displays constant returns to scale?
 2 years ago
 2 years ago

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UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
homogenous means a few things
 2 years ago

sasogeek Group TitleBest ResponseYou've already chosen the best response.0
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.
 2 years ago

sasogeek Group TitleBest ResponseYou've already chosen the best response.0
Show that the production equation \(\huge Q=A[bK^a+(1b)L^a]^\frac{1}{a} \) is homogeneous and displays constant returns to scale
 2 years ago

UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
Q(?,?)
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
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
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
if f(tx,ty) = t*f(x,y) its homogenous
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
hmm, that kind of reminds me of the definition of an odd function .... i wonder if they are related
 2 years ago

UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
Q(A,K,L) ?
 2 years ago

sasogeek Group TitleBest ResponseYou've already chosen the best response.0
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.
 2 years ago

UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
looks like expansion of length due to heat
 2 years ago

sasogeek Group TitleBest ResponseYou've already chosen the best response.0
it'd be nice to know which letters are variables and which ones are constants in this given function :/
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
I think convention has it that capitals are constants \[\large Q=A[tbK^{ta}+(1tb)L^{ta}]^\frac{1}{ta}\] \[\large Q=A[tbK^{ta}+L^{ta}L^{ta}tb]^\frac{1}{ta}\] \[\large Q=A[tbK^{ta}+L^{ta}L^{ta}tb]^\frac{1}{ta}\]
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
and im just going on an idea here, not really sure if itll pan out
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
\[\large Q=[tb(AK)^{ta}+(AL)^{ta}tb(AL)^{ta}]^\frac{1}{ta}\] \[\large Q^{ta}(AL)^{ta}=tb(AK)^{ta}tb(AL)^{ta}\]
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
if A not=0 i wonder if another route would have been easier ...
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
any idea if im even on teh right track with this idea?
 2 years ago

sasogeek Group TitleBest ResponseYou've already chosen the best response.0
how come \(\large Q^{ta}\) ?
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
say ta=3 Q = N^(1/3) [Q = N^(1/3)]^3 Q^3 = N^(3/3) Q^3 = N
 2 years ago

sasogeek Group TitleBest ResponseYou've already chosen the best response.0
oh, i see :)
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
but i wonder if it would be prudent to separate t and a in that .... hard to tell
 2 years ago

sasogeek Group TitleBest ResponseYou've already chosen the best response.0
but from what you have there, what are you to factor out to confirm if it's homogeneous or not? :/
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
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
 2 years ago

sasogeek Group TitleBest ResponseYou've already chosen the best response.0
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 :/
 2 years ago

UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
where are the numbers ; (
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
\[\large Q=A[tbK^{ta}+L^{ta}L^{ta}tb]^\frac{1}{ta}\] \[\large \frac QA=[tbK^{ta}+L^{ta}L^{ta}tb]^\frac{1}{ta}\] \[\large \left(\frac QA\right)^{ta}=tbK^{ta}+L^{ta}L^{ta}tb\] \[\large \left(\frac QA\right)^{ta}=tb(K^{ta}L^{ta})+L^{ta}\] 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.
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
im trying to recall ways that logs might be useful to us .... since ive got t stuck in an exponent
 2 years ago

sasogeek Group TitleBest ResponseYou've already chosen the best response.0
why did t go into the exponent in the first place?
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
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
 2 years ago

sasogeek Group TitleBest ResponseYou've already chosen the best response.0
interesting :)
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
but then again, Q would be variable as well since it is defined by the inputs ....
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
maybe Q, A, K, and L are the variables?
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
which is what unkle alluded to at the start :)
 2 years ago

UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
lower case are scalars
 2 years ago

sasogeek Group TitleBest ResponseYou've already chosen the best response.0
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 :)
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
what class is this for?
 2 years ago

sasogeek Group TitleBest ResponseYou've already chosen the best response.0
computational mathematics
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
... never heard of it :/ what have you been learning in prior chapters and do they relate to this?
 2 years ago

sasogeek Group TitleBest ResponseYou've already chosen the best response.0
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 :/
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
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
 2 years ago

sasogeek Group TitleBest ResponseYou've already chosen the best response.0
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?
 2 years ago

sasogeek Group TitleBest ResponseYou've already chosen the best response.0
this is some sort of calculus, right?
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
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.
 2 years ago

sasogeek Group TitleBest ResponseYou've already chosen the best response.0
how about if you end up with t^a?
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
if "a" was one of the variables to begin with ... im not sure.
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
http://www.sosmath.com/diffeq/first/homogeneous/homogeneous.html
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
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
 2 years ago

sasogeek Group TitleBest ResponseYou've already chosen the best response.0
ohhhhh :) nice! i think i'm getting the hang of this, so all that matters is if you know what the variables are.... :)
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
that does help, yes
 2 years ago

sasogeek Group TitleBest ResponseYou've already chosen the best response.0
ok so usually, the function would have 2 variables right?
 2 years ago

UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
at least
 2 years ago

sasogeek Group TitleBest ResponseYou've already chosen the best response.0
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 :/
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
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
 2 years ago

sasogeek Group TitleBest ResponseYou've already chosen the best response.0
is this correct? \(\large logAB^c=clogAB \) ?
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
\[\large Q(A,K,L)=tA[tbK^{a}+(1b)tL^{a}]^\frac{1}{a}\] \[\large Q(A,K,L)=tA[t(bK^{a}+(1b)L^{a})]^\frac{1}{a}\] \[\large Q(A,K,L)=tt^aA[bK^{a}+(1b)L^{a}]^\frac{1}{a}\] \[\large Q(A,K,L)=t^{(a+1)}~[A[bK^{a}+(1b)L^{a}]^\frac{1}{a}]\]
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
if A is a base, then yes
 2 years ago

sasogeek Group TitleBest ResponseYou've already chosen the best response.0
no, A is not a base :/ and oh, looks like you solved it and it appears homogeneous :D
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
that google book helped me to get the variables right :)
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
but i might have pulled out the wrong t exponent
 2 years ago

sasogeek Group TitleBest ResponseYou've already chosen the best response.0
I'd have to purchase it though :(
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
t^(1/a) pulls out, not t^a t*t^(1/a) = t^(1+1/a) =t^((a+1)/a) typoes it :)
 2 years ago

sasogeek Group TitleBest ResponseYou've already chosen the best response.0
yh i just noticed :) thanks for pointing it out xD seems like an interesting topic though
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
"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
 2 years ago

sasogeek Group TitleBest ResponseYou've already chosen the best response.0
so it doesn't display a constant scale.. :/
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
recheck my math to make sure theres not a mistake :)
 2 years ago

sasogeek Group TitleBest ResponseYou've already chosen the best response.0
yes i'm trying to solve it myself on paper right now :) thanks again though xD
 2 years ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.2
good luck, thats about all i can do for it ;)
 2 years ago

sasogeek Group TitleBest ResponseYou've already chosen the best response.0
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
 2 years ago
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