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Math Help - Please check the working for this differentiation problem.

  1. #1
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    Please check the working for this differentiation problem.


    Hi- 90% sure this is correct . . .

    Firstly, consider the following equation;

    U=I^(1-ε)/(1-ε)L^(1-l)/(1-l)

    L and I are functions of one another, regulated by this condition;

    L+I/w = H. H and w are constants.

    We need to solve for ε; expressed as a function of h, when U is maximised.

    I think I have it solved.

    Firstly express the function above as

    f(I,L,ε,l)=f(I,ε)f(L,l)

    differentiate to

    (du/u)/dh f(I,L,ε,l)=(du/u)/dh f(I,ε)+(du/u)/dh f(L,l)


    (du/u)/dh =0 due to maxisation constraint, hence

    (du/u)/dh f(I,ε)=-(du/u)/dh f(L,l)


    we then decompose the differtial terms;


    (du/u)/dh = (dI/I)/dh * (dU/U)/(DI/DI)


    (dL/L)/dh = (dL/L)/dh * (dU/U)/(dL/L)

    substituting and solving gives;

    (1/h 1/(ε+1))=(1/(H-h)1/(l+1))


    ε=((H-h).(l+1)/h))-1


    pretty sure that is right, but not sure about the clumsy proof.

    Thanks in advance, sorry if it is not clear.


    Last edited by frustrated; August 10th 2011 at 04:35 AM. Reason: formatting
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