CES Production Function with $\rho>1$ In using CES production functions of the form $f(x_1,x_2)=(x_1^\rho+x_2^\rho)^{1/\rho} $, we always assume that $\rho\leq1$. Why do we make that assumption? I understand that if $\rho&gt;1$, the production function won't be concave anymore (and hence production set will not be convex), but what does that imply about profit and cost functions?
The problem with $\rho&gt;1$ is that it means the marginal product of factors is not decreasing ($\rho&lt;1$) or constant ($\rho=1$) but increasing, which is an odd assumption. Such functions yield isoquants that are concave, and might lead to only one factor being used (as BKay said).

As in any generic CES, the marginal product of factor $x_i$ is

$$ MP_i = \left(\frac{y}{x_i}\right)^{1-\rho} $$

The derivative of this MP with respect to $x_i$ is, after some rearranging,

$$ (\rho-1) \left(\frac{y}{x_i}\right)^{1-\rho}\left(\frac{x_{-i}}{x_iy^{\rho}}\right) $$

For $\rho&gt;1$, this expression is positive, which means that the productivity of a factor increases as more of that factor is used. 

Regarding isoquants, you can find these by rewriting the production function as $x_2=g(y,x_1)$. In the generic CES, this is

$$ x_2 = \left(y^{\rho} - x_1^{\rho}\right)^\frac{1}{\rho} $$

These are linear in the case of $\rho=1$, convex in the case of Cobb-Douglas (where the function above is $x_2=\frac{y}{x_1}$, a hyperbole), and concave in the case of $\rho&gt;1$. For example, select $\rho=2$ and you have:

$$ x_2^2 = y^2 - x_1^2 $$

which is the formula of a circle centered at $(0,0)$, with radius $y$. Normally, for production theory only $x_i \geq 0$ is interesting, which gives you the concave isoquants for different levels of $y$. The figure below shows an example, were for a given factor prices ratio, there is a corner solution (point A):

$\hskip3cm$ 

(Code for reproducing figure here)