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Math Help - Profit Maximization Cobb Douglas Production function

  1. #1
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    Profit Maximization Cobb Douglas Production function

    I am having trouble with the algebra how do get from

    y=(pay/w)^a * (pby/v)^b .... first I factor out y

    y=(pa/w)^a * (pb/v)^b * y^(a+b) but how do I get to this result....

    y=(pa/w)^a/(1-a-b) * (pb/v)^b/(1-a-b)

    I know I could just assume the short cut...eliminate y and subtract each of its exponent variables from the denominator of the other variable exponents but I would really like to understand the long hand algebra... Thanks
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  2. #2
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    Hello, reneebu!

    What an ugly problem!


    y \:=\:\left(\dfrac{pay}{w}\right)^a\cdot\left(\dfra  c{pby}{v}\right)^b

    \text{First I factor out }y\!:\;\;y\:=\:\left(\dfrac{pa}{w}\right)^a\cdot\l  eft(\dfrac{pb}{v}\right)^b\cdot y^{a+b} [1]

    \text{but how do I get to this result? }\;y\:=\:\left(\dfrac{pa}{w}\right)^{\frac{a}{1-a-b}} \cdot \left(\dfrac{pb}{v}\right)^{\frac{b}{1-a-b}} [2]

    In [1] we have: . y\:=\:\left(\dfrac{pa}{w}\right)^a\cdot\left(\dfra  c{pb}{v}\right)^b\cdot y^{a+b}


    Divide by y^{a+b}\!:

    . . \displaystyle \frac{y}{y^{a+b}} \;=\;\left(\frac{pa}{w}\right)^a\cdot\left(\frac{p  b}{v}\right)^b \quad\Rightarrow\quad y^{1-a-b} \;=\;\left(\frac{pa}{w}\right)^a\cdot\left(\frac{p  b}{v}\right)^b


    Raise both sides to the power \frac{1}{1-a-b}\!:

    . . \displaystyle \left(y^{1-a-b}\right)^{\frac{1}{1-a-b}} \;=\;\bigg[\left(\frac{pa}{w}\right)^a\cdot\left(\frac{pb}{v}  \right)^b\bigg]^{\frac{1}{1-a-b}}

    \displaystyle\text{Simplify: }\qquad y \;=\;\left(\frac{pa}{w}\right)^{\frac{a}{1-a-b}}\cdot\left(\frac{pb}{v}\right)^{\frac{1}{1-a-b}}

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