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

1. UnkleRhaukus

?

2. UnkleRhaukus

homogenous means a few things

3. 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.

4. 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

5. UnkleRhaukus

Q(?,?)

6. 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

7. amistre64

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

8. amistre64

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

9. UnkleRhaukus

Q(A,K,L) ?

10. 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.

11. UnkleRhaukus

looks like expansion of length due to heat

12. sasogeek

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

13. amistre64

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

14. amistre64

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

15. amistre64

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

16. UnkleRhaukus

oou,

17. amistre64

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

18. amistre64

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

19. sasogeek

how come $$\large Q^{ta}$$ ?

20. amistre64

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

21. sasogeek

oh, i see :)

22. amistre64

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

23. UnkleRhaukus

aahh,

24. sasogeek

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

25. 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

26. 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 :/

27. UnkleRhaukus

where are the numbers ; (

28. 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.

29. amistre64

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

30. sasogeek

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

31. 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

32. sasogeek

interesting :)

33. amistre64

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

34. amistre64

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

35. amistre64

which is what unkle alluded to at the start :)

36. UnkleRhaukus

lower case are scalars

37. 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 :)

38. amistre64

what class is this for?

39. sasogeek

computational mathematics

40. amistre64

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

41. sasogeek

42. 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

43. 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?

44. amistre64

correct

45. sasogeek

this is some sort of calculus, right?

46. 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.

47. sasogeek

how about if you end up with t^a?

48. UnkleRhaukus

t ?

49. amistre64

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

50. amistre64
51. 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

52. sasogeek

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

53. amistre64

that does help, yes

54. sasogeek

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

55. UnkleRhaukus

at least

56. 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 :/

57. amistre64
58. sasogeek

is this correct? $$\large logAB^c=clogAB$$ ?

59. amistre64

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

60. amistre64

if A is a base, then yes

61. sasogeek

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

62. amistre64

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

63. amistre64

but i might have pulled out the wrong t exponent

64. sasogeek

I'd have to purchase it though :(

65. amistre64

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

66. sasogeek

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

67. 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

68. sasogeek

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

69. amistre64

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

70. sasogeek

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

71. amistre64

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

72. 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