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Couple of tricky questions Im having trouble with...

  • 29-07-2008 3:01pm
    #1
    Peyton Manning
    Closed Accounts Posts: 16,658 ✭✭✭✭


    1. Solve the following equation without using logarithms:
    oneqa8.png

    2. Determine if the following production function is homogeneous and if it is homogeneous, state the degree of homogeneity and relate the degree of homogeneity to the returns to scale:
    twodb0.png


    Thanks guys :)


Comments

  • #2
    Delphi91
    Registered Users, Registered Users 2 Posts: 1,501 ✭✭✭Delphi91


    Not sure about the second one, but if you write 256, 4 and 32 in terms of powers of 2, you should be able to get an answer:

    (2^8)^(13x) = 2^2 . (1/[2^5])^x

    which gives

    2^(104x) = 2^2 . 2^(-5x)

    2^(104x) = 2^(2-5x)

    Therefore
    104x = 2-5x
    109x = 2
    x = 2/109

    Think this is correct.


  • #3
    Peyton Manning
    Closed Accounts Posts: 16,658 ✭✭✭✭Peyton Manning


    Thanks man, i think thats the method i was to use!

    Anyone else can help with the 2nd question, id love it :D


  • #4
    LeixlipRed
    Closed Accounts Posts: 6,081 ✭✭✭LeixlipRed


    Hmmm, without using logarithms for the first one. Isn't that how you're equating the powers? Old algebra rusty so may be wrong. And second one, haven't a clue :D


  • #5
    Delphi91
    Registered Users, Registered Users 2 Posts: 1,501 ✭✭✭Delphi91


    LeixlipRed wrote: »
    Hmmm, without using logarithms for the first one. Isn't that how you're equating the powers? Old algebra rusty so may be wrong. And second one, haven't a clue :D

    I would have thought I was using something like:

    If a^e = a^f, then e = f.


  • #6
    Michael Collins
    Moderators, Science, Health & Environment Moderators Posts: 1,852 Mod ✭✭✭✭Michael Collins


    LeixlipRed wrote: »
    Hmmm, without using logarithms for the first one. Isn't that how you're equating the powers? Old algebra rusty so may be wrong. And second one, haven't a clue :D

    I don't think so really. Delphi's just using a few rules of indices and saying that if the two sides are equal, and the number you are exponentiating (the base) is the same on both sides, then the exponents must be equal.


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  • #7
    LeixlipRed
    Closed Accounts Posts: 6,081 ✭✭✭LeixlipRed


    Yes of course. Perfectly ok


  • #8
    genesis.exodus
    Closed Accounts Posts: 1 ✭genesis.exodus


    Homogeneous function satisfies below condition
    f(t*x,t*y)=(t^n)*f(x,y) and n is the degree ..
    f(L,K) = Q = 15*L^(4/7)*K^(7/15) - 35*L^(13/35)*K^(2/3) ..
    f(t*L,t*K) = 15*(t*L)^(4/7)*(t*K)^(7/15) - 35*(t*L)^(13/35)*(t*K)^(2/3)
    = t^(4/7+7/15)*15*L^(4/7)*K^(7/15) - t^(13/35+2/3)*35*L^(13/35)*K^(2/3)
    = t^(109/105)*15*L^(4/7)*K^(7/15) - t^(109/105)*35*L^(13/35)*K^(2/3)
    = t^(109/105)* ( 15*L^(4/7)*K^(7/15) - 35*L^(13/35)*K^(2/3) )
    = t^(109/105)* Q
    Then, the function is homogeneous of degree 109/105.
    also 109/105>1 so that means increasing returns to scale..


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