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

1. UnkleRhaukus Group Title

?

2. UnkleRhaukus Group Title

homogenous means a few things

3. sasogeek Group Title

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 Group Title

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 Group Title

Q(?,?)

6. amistre64 Group Title

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 Group Title

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

8. amistre64 Group Title

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

9. UnkleRhaukus Group Title

Q(A,K,L) ?

10. sasogeek Group Title

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 Group Title

looks like expansion of length due to heat

12. sasogeek Group Title

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

13. amistre64 Group Title

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 Group Title

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

15. amistre64 Group Title

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

16. UnkleRhaukus Group Title

oou,

17. amistre64 Group Title

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

18. amistre64 Group Title

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

19. sasogeek Group Title

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

20. amistre64 Group Title

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

21. sasogeek Group Title

oh, i see :)

22. amistre64 Group Title

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

23. UnkleRhaukus Group Title

aahh,

24. sasogeek Group Title

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

25. amistre64 Group Title

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 Group Title

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 Group Title

where are the numbers ; (

28. amistre64 Group Title

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

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

30. sasogeek Group Title

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

31. amistre64 Group Title

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 Group Title

interesting :)

33. amistre64 Group Title

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

34. amistre64 Group Title

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

35. amistre64 Group Title

which is what unkle alluded to at the start :)

36. UnkleRhaukus Group Title

lower case are scalars

37. sasogeek Group Title

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 Group Title

what class is this for?

39. sasogeek Group Title

computational mathematics

40. amistre64 Group Title

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

41. sasogeek Group Title

42. amistre64 Group Title

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 Group Title

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 Group Title

correct

45. sasogeek Group Title

this is some sort of calculus, right?

46. amistre64 Group Title

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 Group Title

how about if you end up with t^a?

48. UnkleRhaukus Group Title

t ?

49. amistre64 Group Title

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

50. amistre64 Group Title
51. amistre64 Group Title

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 Group Title

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 Group Title

that does help, yes

54. sasogeek Group Title

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

55. UnkleRhaukus Group Title

at least

56. sasogeek Group Title

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 Group Title
58. sasogeek Group Title

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

59. amistre64 Group Title

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

60. amistre64 Group Title

if A is a base, then yes

61. sasogeek Group Title

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

62. amistre64 Group Title

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

63. amistre64 Group Title

but i might have pulled out the wrong t exponent

64. sasogeek Group Title

I'd have to purchase it though :(

65. amistre64 Group Title

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

66. sasogeek Group Title

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

67. amistre64 Group Title

"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 Group Title

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

69. amistre64 Group Title

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

70. sasogeek Group Title

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

71. amistre64 Group Title

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

72. sasogeek Group Title

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