# Conditional Second-order Derivative and Degree of Homogeneity

• Nov 29th 2012, 06:11 AM
sev
Conditional Second-order Derivative and Degree of Homogeneity
I have a function $\displaystyle f(a,b)$ with first-order derivatives $\displaystyle f_a>0, f_b>0$ and second-order derivatives $\displaystyle f_{aa}<0, f_{bb}<0, f_{ab}>0$.

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

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

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

Why is this so?

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

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

Now, I would have to show that $\displaystyle f_{aa} < f_{ab} \frac{\partial b}{\partial a}$ but I don't know how to do that using $\displaystyle f_b=C$ and the information about the degree of homogeneity.
• Dec 3rd 2012, 06:12 AM
sev
Re: Conditional Second-order Derivative and Degree of Homogeneity
Just a small correction. It should read:

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