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sasogeek

  • 2 years ago

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

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  1. UnkleRhaukus
    • 2 years ago
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    ?

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

  3. sasogeek
    • 2 years ago
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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.

  4. sasogeek
    • 2 years ago
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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

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

  6. amistre64
    • 2 years ago
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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

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

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

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

  10. sasogeek
    • 2 years ago
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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.

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

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

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

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

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

  16. UnkleRhaukus
    • 2 years ago
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    oou,

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

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

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

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

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

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

  23. UnkleRhaukus
    • 2 years ago
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    aahh,

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

  25. amistre64
    • 2 years ago
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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

  26. sasogeek
    • 2 years ago
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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 :/

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

  28. amistre64
    • 2 years ago
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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.

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

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

  31. amistre64
    • 2 years ago
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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

  32. sasogeek
    • 2 years ago
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    interesting :)

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

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

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

  36. UnkleRhaukus
    • 2 years ago
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    lower case are scalars

  37. sasogeek
    • 2 years ago
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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 :)

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

  39. sasogeek
    • 2 years ago
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    computational mathematics

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

  41. sasogeek
    • 2 years ago
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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 :/

  42. amistre64
    • 2 years ago
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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

  43. sasogeek
    • 2 years ago
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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?

  44. amistre64
    • 2 years ago
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    correct

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

  46. amistre64
    • 2 years ago
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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.

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

  48. UnkleRhaukus
    • 2 years ago
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    t ?

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

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

  51. amistre64
    • 2 years ago
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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

  52. sasogeek
    • 2 years ago
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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.... :)

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

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

  55. UnkleRhaukus
    • 2 years ago
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    at least

  56. sasogeek
    • 2 years ago
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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 :/

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

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

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

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

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

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

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

  64. amistre64
    • 2 years ago
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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 :)

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

  66. amistre64
    • 2 years ago
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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

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

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

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

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

  71. sasogeek
    • 2 years ago
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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

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