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Math Help - Production Function (Partial Derivatives)

  1. #1
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    Production Function (Partial Derivatives)

    1. Consider the production function f(x,y) = 60x^3/4 y^1/4 ,which gives the number of units of goods produced when utilizing x units of labor and y units of capital.

    (a) Compute ∂f/∂x and ∂f/∂y . These quantities are referred to as the ∂x/∂y
    marginal productivities of labor and of capital respectively.

    (b) If the amount of capital is held fixed at y = 16 and the amount of labor increases by 1 unit, estimate the increase in the quantity of goods produced.


    I did part a. And the partial derivatives i got were

    ∂f/∂x = 45x ^ -1/4 y ^ 1/4

    ∂f/∂y = 15 x^ 3/4 y ^ -3/4


    I got confused with part B. I know that the capital stays at y = 16 but labor increases by 1 which means x= x+1
    And I use the partial derivative rule = delta P = P capital (delta capital) + P labor (delta labor)

    The capital stays constant so delta capital should be 0...??

    Some help would be great!
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  2. #2
    Grand Panjandrum
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    Re: Production Function (Partial Derivatives)

    Quote Originally Posted by afung22 View Post
    1. Consider the production function f(x,y) = 60x^3/4 y^1/4 ,which gives the number of units of goods produced when utilizing x units of labor and y units of capital.

    (a) Compute ∂f/∂x and ∂f/∂y . These quantities are referred to as the ∂x/∂y
    marginal productivities of labor and of capital respectively.

    (b) If the amount of capital is held fixed at y = 16 and the amount of labor increases by 1 unit, estimate the increase in the quantity of goods produced.


    I did part a. And the partial derivatives i got were

    ∂f/∂x = 45x ^ -1/4 y ^ 1/4

    ∂f/∂y = 15 x^ 3/4 y ^ -3/4


    I got confused with part B. I know that the capital stays at y = 16 but labor increases by 1 which means x= x+1
    And I use the partial derivative rule = delta P = P capital (delta capital) + P labor (delta labor)

    The capital stays constant so delta capital should be 0...??

    Some help would be great!
    You want:

    \Delta P\approx \left. \frac{\partial f}{\partial x} \right|_{y=16} \Delta x

    so as \Delta x=1

    \Delta P \approx \left. \frac{\partial f}{\partial x} \right|_{y=16}=45 \frac{1}{2x^{1/4}}

    CB
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