Suppose you have a neoclassical production function with N-inputs
$F(x_t^1,...,x_t^N)$
All input factors grow in continuous time with constant, but not identical growth rates $g^j$. Assume $g^1 \leq g^2 \leq ... \leq g^N$. The growth rate of $F$ is then
$\hat{F}=\sum_{j=1}^N \varepsilon_{F,x^j} g^j$
with $\varepsilon_{F,x^j}$ being the elasticity of $F$ with respect to $x^j$. Since $F$ is linear homogenous I know that $\sum_{j=1}^N \varepsilon_{F,x^j}=1$ holds. $\frac{\partial F}{\partial x^j}>0$ and $\frac{\partial^2 F}{\partial {x^j}^2}<0$ imply $\varepsilon_{F,x^j}>0$. Hence
$g^1 \leq \hat{F} \leq g^N$ $\forall t$
My question is: Is $\hat{F}$ going to converge as $t \to \infty$? I find it hard to imagine that the elasticities fluctuate (periodically) around some value as all input factor ratios go to zero or infinity (or stay constant the whole time in case of equal growth rates).
I tried to show that the elasticities all converge. I suspect that the result might hinge on the propety that $\frac{\partial^2 F}{\partial x_j \partial x_k}>0$ for $k \neq j$ which always holds in the two goods case, but I'm not sure about that.
Thank you all in advance for your help! If my English seems a little awkward, it's because I'm German. But I hope you understand the problem anyway. :)