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

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UnkleRhaukus
 3 years ago
Best ResponseYou've already chosen the best response.0homogenous means a few things

sasogeek
 3 years ago
Best ResponseYou've already chosen the best response.0ok 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.

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

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

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

sasogeek
 3 years ago
Best ResponseYou've already chosen the best response.0i 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
 3 years ago
Best ResponseYou've already chosen the best response.0looks like expansion of length due to heat

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

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

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

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

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

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

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

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

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

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

amistre64
 3 years ago
Best ResponseYou've already chosen the best response.2some 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
 3 years ago
Best ResponseYou've already chosen the best response.0what 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
 3 years ago
Best ResponseYou've already chosen the best response.0where are the numbers ; (

amistre64
 3 years ago
Best 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.

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

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

amistre64
 3 years ago
Best ResponseYou've already chosen the best response.2because 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

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

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

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

UnkleRhaukus
 3 years ago
Best ResponseYou've already chosen the best response.0lower case are scalars

sasogeek
 3 years ago
Best ResponseYou've already chosen the best response.0well 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
 3 years ago
Best ResponseYou've already chosen the best response.2what class is this for?

sasogeek
 3 years ago
Best ResponseYou've already chosen the best response.0computational mathematics

amistre64
 3 years ago
Best 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?

sasogeek
 3 years ago
Best ResponseYou've already chosen the best response.0I 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
 3 years ago
Best ResponseYou've already chosen the best response.2i 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
 3 years ago
Best ResponseYou've already chosen the best response.0thanks :) 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?

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

amistre64
 3 years ago
Best ResponseYou've already chosen the best response.2if 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
 3 years ago
Best ResponseYou've already chosen the best response.0how about if you end up with t^a?

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

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

amistre64
 3 years ago
Best ResponseYou've already chosen the best response.2for 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
 3 years ago
Best ResponseYou've already chosen the best response.0ohhhhh :) nice! i think i'm getting the hang of this, so all that matters is if you know what the variables are.... :)

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

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

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

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

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

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

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

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

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

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

amistre64
 3 years ago
Best 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

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

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

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

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

sasogeek
 3 years ago
Best ResponseYou've already chosen the best response.0with 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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