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sasogeek Group Title

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

  • 2 years ago
  • 2 years ago

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  1. UnkleRhaukus Group Title
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    ?

    • 2 years ago
  2. UnkleRhaukus Group Title
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    homogenous means a few things

    • 2 years ago
  3. sasogeek Group Title
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    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.

    • 2 years ago
  4. sasogeek Group Title
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    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

    • 2 years ago
  5. UnkleRhaukus Group Title
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    Q(?,?)

    • 2 years ago
  6. amistre64 Group Title
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    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

    • 2 years ago
  7. amistre64 Group Title
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    if f(tx,ty) = t*f(x,y) its homogenous

    • 2 years ago
  8. amistre64 Group Title
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    hmm, that kind of reminds me of the definition of an odd function .... i wonder if they are related

    • 2 years ago
  9. UnkleRhaukus Group Title
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    Q(A,K,L) ?

    • 2 years ago
  10. sasogeek Group Title
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    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.

    • 2 years ago
  11. UnkleRhaukus Group Title
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    looks like expansion of length due to heat

    • 2 years ago
  12. sasogeek Group Title
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    it'd be nice to know which letters are variables and which ones are constants in this given function :/

    • 2 years ago
  13. amistre64 Group Title
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    I think convention has it that capitals are constants \[\large Q=A[tbK^{ta}+(1-tb)L^{ta}]^\frac{1}{ta}\] \[\large Q=A[tbK^{ta}+L^{ta}-L^{ta}tb]^\frac{1}{ta}\] \[\large Q=A[tbK^{ta}+L^{ta}-L^{ta}tb]^\frac{1}{ta}\]

    • 2 years ago
  14. amistre64 Group Title
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    and im just going on an idea here, not really sure if itll pan out

    • 2 years ago
  15. amistre64 Group Title
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    \[\large Q=[tb(AK)^{ta}+(AL)^{ta}-tb(AL)^{ta}]^\frac{1}{ta}\] \[\large Q^{ta}-(AL)^{ta}=tb(AK)^{ta}-tb(AL)^{ta}\]

    • 2 years ago
  16. UnkleRhaukus Group Title
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    oou,

    • 2 years ago
  17. amistre64 Group Title
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    if A not=0 i wonder if another route would have been easier ...

    • 2 years ago
  18. amistre64 Group Title
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    any idea if im even on teh right track with this idea?

    • 2 years ago
  19. sasogeek Group Title
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    how come \(\large Q^{ta}\) ?

    • 2 years ago
  20. amistre64 Group Title
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    say ta=3 Q = N^(1/3) [Q = N^(1/3)]^3 Q^3 = N^(3/3) Q^3 = N

    • 2 years ago
  21. sasogeek Group Title
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    oh, i see :)

    • 2 years ago
  22. amistre64 Group Title
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    but i wonder if it would be prudent to separate t and a in that .... hard to tell

    • 2 years ago
  23. UnkleRhaukus Group Title
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    aahh,

    • 2 years ago
  24. sasogeek Group Title
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    but from what you have there, what are you to factor out to confirm if it's homogeneous or not? :/

    • 2 years ago
  25. amistre64 Group Title
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    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

    • 2 years ago
  26. sasogeek Group Title
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    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 :/

    • 2 years ago
  27. UnkleRhaukus Group Title
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    where are the numbers ; (

    • 2 years ago
  28. amistre64 Group Title
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    \[\large Q=A[tbK^{ta}+L^{ta}-L^{ta}tb]^\frac{1}{ta}\] \[\large \frac QA=[tbK^{ta}+L^{ta}-L^{ta}tb]^\frac{1}{ta}\] \[\large \left(\frac QA\right)^{ta}=tbK^{ta}+L^{ta}-L^{ta}tb\] \[\large \left(\frac QA\right)^{ta}=tb(K^{ta}-L^{ta})+L^{ta}\] 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.

    • 2 years ago
  29. amistre64 Group Title
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    im trying to recall ways that logs might be useful to us .... since ive got t stuck in an exponent

    • 2 years ago
  30. sasogeek Group Title
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    why did t go into the exponent in the first place?

    • 2 years ago
  31. amistre64 Group Title
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    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

    • 2 years ago
  32. sasogeek Group Title
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    interesting :)

    • 2 years ago
  33. amistre64 Group Title
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    but then again, Q would be variable as well since it is defined by the inputs ....

    • 2 years ago
  34. amistre64 Group Title
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    maybe Q, A, K, and L are the variables?

    • 2 years ago
  35. amistre64 Group Title
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    which is what unkle alluded to at the start :)

    • 2 years ago
  36. UnkleRhaukus Group Title
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    lower case are scalars

    • 2 years ago
  37. sasogeek Group Title
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    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 :)

    • 2 years ago
  38. amistre64 Group Title
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    what class is this for?

    • 2 years ago
  39. sasogeek Group Title
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    computational mathematics

    • 2 years ago
  40. amistre64 Group Title
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    ... never heard of it :/ what have you been learning in prior chapters and do they relate to this?

    • 2 years ago
  41. sasogeek Group Title
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    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 :/

    • 2 years ago
  42. amistre64 Group Title
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    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

    • 2 years ago
  43. sasogeek Group Title
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    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?

    • 2 years ago
  44. amistre64 Group Title
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    correct

    • 2 years ago
  45. sasogeek Group Title
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    this is some sort of calculus, right?

    • 2 years ago
  46. amistre64 Group Title
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    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.

    • 2 years ago
  47. sasogeek Group Title
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    how about if you end up with t^a?

    • 2 years ago
  48. UnkleRhaukus Group Title
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    t ?

    • 2 years ago
  49. amistre64 Group Title
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    if "a" was one of the variables to begin with ... im not sure.

    • 2 years ago
  50. amistre64 Group Title
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    http://www.sosmath.com/diffeq/first/homogeneous/homogeneous.html

    • 2 years ago
  51. amistre64 Group Title
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    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

    • 2 years ago
  52. sasogeek Group Title
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    ohhhhh :) nice! i think i'm getting the hang of this, so all that matters is if you know what the variables are.... :)

    • 2 years ago
  53. amistre64 Group Title
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    that does help, yes

    • 2 years ago
  54. sasogeek Group Title
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    ok so usually, the function would have 2 variables right?

    • 2 years ago
  55. UnkleRhaukus Group Title
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    at least

    • 2 years ago
  56. sasogeek Group Title
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    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 :/

    • 2 years ago
  57. sasogeek Group Title
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    is this correct? \(\large logAB^c=clogAB \) ?

    • 2 years ago
  58. amistre64 Group Title
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    \[\large Q(A,K,L)=tA[tbK^{a}+(1-b)tL^{a}]^\frac{1}{a}\] \[\large Q(A,K,L)=tA[t(bK^{a}+(1-b)L^{a})]^\frac{1}{a}\] \[\large Q(A,K,L)=tt^aA[bK^{a}+(1-b)L^{a}]^\frac{1}{a}\] \[\large Q(A,K,L)=t^{(a+1)}~[A[bK^{a}+(1-b)L^{a}]^\frac{1}{a}]\]

    • 2 years ago
  59. amistre64 Group Title
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    if A is a base, then yes

    • 2 years ago
  60. sasogeek Group Title
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    no, A is not a base :/ and oh, looks like you solved it and it appears homogeneous :D

    • 2 years ago
  61. amistre64 Group Title
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    that google book helped me to get the variables right :)

    • 2 years ago
  62. amistre64 Group Title
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    but i might have pulled out the wrong t exponent

    • 2 years ago
  63. sasogeek Group Title
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    I'd have to purchase it though :(

    • 2 years ago
  64. amistre64 Group Title
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    t^(1/a) pulls out, not t^a t*t^(1/a) = t^(1+1/a) =t^((a+1)/a) typoes it :)

    • 2 years ago
  65. sasogeek Group Title
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    yh i just noticed :) thanks for pointing it out xD seems like an interesting topic though

    • 2 years ago
  66. amistre64 Group Title
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    "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

    • 2 years ago
  67. sasogeek Group Title
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    so it doesn't display a constant scale.. :/

    • 2 years ago
  68. amistre64 Group Title
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    recheck my math to make sure theres not a mistake :)

    • 2 years ago
  69. sasogeek Group Title
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    yes i'm trying to solve it myself on paper right now :) thanks again though xD

    • 2 years ago
  70. amistre64 Group Title
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    good luck, thats about all i can do for it ;)

    • 2 years ago
  71. sasogeek Group Title
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    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

    • 2 years ago
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