Question

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

Answer 1

?

Answer 2

homogenous means a few things

Answer 3

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.

Answer 4

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

Answer 5

Q(?,?)

Answer 6

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

Answer 7

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

Answer 8

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

Answer 9

Q(A,K,L) ?

Answer 10

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.

Answer 11

looks like expansion of length due to heat

Answer 12

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

Answer 13

I think convention has it that capitals are constants
\[\large Q=A[tbK^{ta}+(1-tb)L^{ta}]^\frac{1}{ta}\]
\[\large Q=A[tbK^{ta}+L^{ta}-L^{ta}tb]^\frac{1}{ta}\]
\[\large Q=A[tbK^{ta}+L^{ta}-L^{ta}tb]^\frac{1}{ta}\]

Answer 14

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

Answer 15

\[\large Q=[tb(AK)^{ta}+(AL)^{ta}-tb(AL)^{ta}]^\frac{1}{ta}\]
\[\large Q^{ta}-(AL)^{ta}=tb(AK)^{ta}-tb(AL)^{ta}\]

Answer 16

oou,

Answer 17

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

Answer 18

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

Answer 19

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

Answer 20

say ta=3

Q = N^(1/3)

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

Q^3 = N^(3/3)

Q^3 = N

Answer 21

oh, i see :)

Answer 22

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

Answer 23

aahh,

Answer 24

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

Answer 25

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

Answer 26

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 :/

Answer 27

where are the numbers ; (

Answer 28

\[\large Q=A[tbK^{ta}+L^{ta}-L^{ta}tb]^\frac{1}{ta}\]
\[\large \frac QA=[tbK^{ta}+L^{ta}-L^{ta}tb]^\frac{1}{ta}\]
\[\large \left(\frac QA\right)^{ta}=tbK^{ta}+L^{ta}-L^{ta}tb\]
\[\large \left(\frac QA\right)^{ta}=tb(K^{ta}-L^{ta})+L^{ta}\]
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.

Answer 29

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

Answer 30

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

Answer 31

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

Answer 32

interesting :)

Answer 33

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

Answer 34

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

Answer 35

which is what unkle alluded to at the start :)

Answer 36

lower case are scalars

Answer 37

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 :)

Answer 38

what class is this for?

Answer 39

computational mathematics

Answer 40

... never heard of it :/

what have you been learning in prior chapters and do they relate to this?

Answer 41

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 :/

Answer 42

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

Answer 43

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?

Answer 44

correct

Answer 45

this is some sort of calculus, right?

Answer 46

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.

Answer 47

how about if you end up with t^a?

Answer 48

t ?

Answer 49

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

Answer 50

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

Answer 51

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

Answer 52

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

Answer 53

that does help, yes

Answer 54

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

Answer 55

at least

Answer 56

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 :/

Answer 57

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

Answer 58

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

Answer 59

\[\large Q(A,K,L)=tA[tbK^{a}+(1-b)tL^{a}]^\frac{1}{a}\]
\[\large Q(A,K,L)=tA[t(bK^{a}+(1-b)L^{a})]^\frac{1}{a}\]
\[\large Q(A,K,L)=tt^aA[bK^{a}+(1-b)L^{a}]^\frac{1}{a}\]
\[\large Q(A,K,L)=t^{(a+1)}~[A[bK^{a}+(1-b)L^{a}]^\frac{1}{a}]\]

Answer 60

if A is a base, then yes

Answer 61

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

Answer 62

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

Answer 63

but i might have pulled out the wrong t exponent

Answer 64

I'd have to purchase it though :(

Answer 65

t^(1/a) pulls out, not t^a

t*t^(1/a) = t^(1+1/a)
=t^((a+1)/a)

typoes it :)

Answer 66

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

Answer 67

"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

Answer 68

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

Answer 69

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

Answer 70

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

Answer 71

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

Answer 72

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