1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Derivatives Question

  1. May 31, 2009 #1
    Hey everyone! This is an Economics question that's stumping me; but it requires a lot of calculus. Any help would be appreciated!

    1. The problem statement, all variables and given/known data
    Y = A[K^α][L^(1-α)/2][H^(1-α)/2] where 0 < α < 1.

    A) At what level of L is APL (average productivity of labour) minimized?
    B) Show K and L are complementary inputs in that more capital increases MPL and APL.

    2. Relevant equations

    MPL = marginal productivity of labor = ∂Y/∂L
    APL = average productivity of labor = Y/L

    3. The attempt at a solution

    A) APL = Y/L = A[K^α][L^(α/2)][H^(1-α)/2]
    minimize: therefore, take derivative and set to 0
    ∂APL/∂L = α/2[A][K^α][L^(α-1)/2][H^(1-α)/2] = 0
    (I have no idea how to set solve this; would it also be possible to use the quotient rule to solve ∂APL/∂L from APL=A[K^α][L^(1-α)/2][H^(1-α)/2]/L ?)

    B) prove that an increase in K causes and increase in MPL and APL
    Y = A[K^α][L^(1-α)/2][H^(1-α)/2]
    condition 1: assume A = 1, α = 0.6, K = 2, L = 3, H = 2
    MPL = ∂Y/∂L = (1-α)/2 [A][K^α][L^(α/2)][H^(1-α)/2]
    MPL = (0.25)(1)(1.41)(1.32)(1.19)
    MPL = 0.55
    APL = Y/L = A[K^α][L^(α/2)][H^(1-α)/2]
    APL = (1)(1.41)(1.32)(1.19)
    APL = 2.21

    condition 2: assume A = 1, α = 0.6, K = 4, L = 3, H = 2
    MPL = ∂Y/∂L = (1-α)/2 [A][K^α][L^(α/2)][H^(1-α)/2]
    MPL = (0.25)(1)(2)(1.32)(1.19)
    MPL = 0.79
    APL = Y/L = A[K^α][L^(α/2)][H^(1-α)/2]
    APL = (1)(2)(1.32)(1.19)
    APL = 3.14

    therefore, K and L are complements because APL and MPL increase as K increases

    Any help would be appreciated! Thanks guys!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jun 1, 2009 #2

    CompuChip

    User Avatar
    Science Advisor
    Homework Helper

    For the first one, I think you have made a small mistake. If you take L^(1-α)/2 and divide it by L, you get L^(-α)/2 instead of L^(α)/2. Then taking the derivative you will get something like
    1/L^(-α-1)
    which can only become zero if α < -1.

    For the second one, you could take the derivative w.r.t K and show that it is positive everywhere (I don't know if you are allowed to plug in numbers, in mathematics that usually means checking for a specific case instead of proving it generally).
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Derivatives Question
  1. Derivative Question (Replies: 5)

  2. Derivative Question (Replies: 8)

  3. Derivative Question (Replies: 5)

  4. Derivative question (Replies: 1)

  5. Derivate question (Replies: 6)

Loading...