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Math Help - Conditional Second-order Derivative and Degree of Homogeneity

  1. #1
    sev
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    Conditional Second-order Derivative and Degree of Homogeneity

    I have a function f(a,b) with first-order derivatives f_a>0, f_b>0 and second-order derivatives f_{aa}<0, f_{bb}<0, f_{ab}>0.


    Additionally I know that the degree of homogeneity of the function is larger than 0 but smaller than 1.


    In an economics paper I found the statement that given these assumptions the derivative of f_a with regard to a is negative if the first-order derivative with regard to b is held constant:


    \frac{\partial f_a}{\partial a}\mid_{f_b=C} <0


    Why is this so?


    I understand that I can decompose \frac{f_S(a,b)}{\partial a} into


    f_{aa} + f_{ab} \frac{\partial b}{\partial a}


    Now, I would have to show that f_{aa} < f_{ab} \frac{\partial b}{\partial a} but I don't know how to do that using f_b=C and the information about the degree of homogeneity.
    Last edited by sev; November 29th 2012 at 08:25 AM.
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  2. #2
    sev
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    Re: Conditional Second-order Derivative and Degree of Homogeneity

    Just a small correction. It should read:

    I know that I can decompose \frac{\partial f_a(a,b)}{\partial a} into ...
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