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Dividing with exponential functions

  1. Jul 5, 2012 #1
    1. The problem statement, all variables and given/known data

    When attempting to divide the following equality (a "tangency condition" in microeconomic consumer theory), I'm puzzled by the solution derived here, please explain the procedure for arriving at the solution. Many thanks!

    2. Relevant equations

    I differentiated the Lagrangian wrt x and y:

    L = P1x1 + P2x2 + λ[U - xαy1-α]

    First order conditions

    L'(x) = P1 - λαxα-1y1-α = 0

    L'(y) = P2 - λ(1-a)xαy = 0

    The "tangency condition" requires that we divide L'(x) by L'(y): Doing so, and carrying the second half of each function to the other side, we get:

    P1/P2 = λαxα-1y1-α / λ(1-a)xαy

    3. The attempt at a solution

    Here is where I get confused. How did the we arrive at the next step:

    P1/P2 = α/(1-α) . x/y

    Please explain you get x/y from the tangency condition.
  2. jcsd
  3. Jul 5, 2012 #2
    Looks like they just simplified, but as written the factor should be y/x.
  4. Jul 5, 2012 #3


    Staff: Mentor

    I get P1/P2 = (a/1-a) * y/x

    x^(a-1) / x^a --> x^(a-1-a) --> x^(-1)--> 1/x


    y^(1-a) / y^(-a) --> y^(1-a+a) --> y^(1) --> y
  5. Jul 5, 2012 #4
    You're right, it says y/x not x/y. My mistake. Does it all look okay to you then?
  6. Jul 5, 2012 #5

    Then, can we say 1/x times y is y/x?
  7. Jul 5, 2012 #6


    Staff: Mentor

    thats a yes
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