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

- sasogeek

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

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- UnkleRhaukus

?

- UnkleRhaukus

homogenous means a few things

- 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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## More answers

- 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

Q(?,?)

- 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

if f(tx,ty) = t*f(x,y) its homogenous

- amistre64

hmm, that kind of reminds me of the definition of an odd function .... i wonder if they are related

- UnkleRhaukus

Q(A,K,L) ?

- 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

looks like expansion of length due to heat

- sasogeek

it'd be nice to know which letters are variables and which ones are constants in this given function :/

- 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

and im just going on an idea here, not really sure if itll pan out

- 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

oou,

- amistre64

if A not=0 i wonder if another route would have been easier ...

- amistre64

any idea if im even on teh right track with this idea?

- sasogeek

how come \(\large Q^{ta}\) ?

- amistre64

say ta=3
Q = N^(1/3)
[Q = N^(1/3)]^3
Q^3 = N^(3/3)
Q^3 = N

- sasogeek

oh, i see :)

- amistre64

but i wonder if it would be prudent to separate t and a in that .... hard to tell

- UnkleRhaukus

aahh,

- sasogeek

but from what you have there, what are you to factor out to confirm if it's homogeneous or not? :/

- 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

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

where are the numbers ; (

- 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

im trying to recall ways that logs might be useful to us .... since ive got t stuck in an exponent

- sasogeek

why did t go into the exponent in the first place?

- 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

interesting :)

- amistre64

but then again, Q would be variable as well since it is defined by the inputs ....

- amistre64

maybe Q, A, K, and L are the variables?

- amistre64

which is what unkle alluded to at the start :)

- UnkleRhaukus

lower case are scalars

- 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

what class is this for?

- sasogeek

computational mathematics

- amistre64

... never heard of it :/
what have you been learning in prior chapters and do they relate to this?

- 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

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

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

correct

- sasogeek

this is some sort of calculus, right?

- 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

how about if you end up with t^a?

- UnkleRhaukus

t ?

- amistre64

if "a" was one of the variables to begin with ... im not sure.

- amistre64

http://www.sosmath.com/diffeq/first/homogeneous/homogeneous.html

- 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

ohhhhh :) nice! i think i'm getting the hang of this, so all that matters is if you know what the variables are.... :)

- amistre64

that does help, yes

- sasogeek

ok so usually, the function would have 2 variables right?

- UnkleRhaukus

at least

- 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

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

is this correct?
\(\large logAB^c=clogAB \) ?

- 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

if A is a base, then yes

- sasogeek

no, A is not a base :/ and oh, looks like you solved it and it appears homogeneous :D

- amistre64

that google book helped me to get the variables right :)

- amistre64

but i might have pulled out the wrong t exponent

- sasogeek

I'd have to purchase it though :(

- amistre64

t^(1/a) pulls out, not t^a
t*t^(1/a) = t^(1+1/a)
=t^((a+1)/a)
typoes it :)

- sasogeek

yh i just noticed :) thanks for pointing it out xD seems like an interesting topic though

- 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

so it doesn't display a constant scale.. :/

- amistre64

recheck my math to make sure theres not a mistake :)

- sasogeek

yes i'm trying to solve it myself on paper right now :) thanks again though xD

- amistre64

good luck, thats about all i can do for it ;)

- 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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