1. Consider a consumer who consumes goods $i = 1, \dots, n$ and has preferences that are represented by the utility function
$u(x) = \left(\sum_{i=1}^n (a_i^\rho (x_i)^\rho) \right)^{\frac{1}{\rho}}$
where $\rho \neq 1$, $\rho > 0$, $a_i > 0$, and $\sum_{i=1}^n a_i = 1$.
(a) Show that the utility function is strictly quasi-concave assuming $\rho > 1$.
In the solution, note carefully where you use $\rho > 1$, and also where you use that $a_i > 0$!
The other case, $\rho < 1$ is very similar and you can see how to do it once you have understood the case $\rho > 1$.
Hints: (1) try to simplify by using a monotonic transformation to obtain $v(x) = \phi(u(x))$, where $v(\cdot)$ has a simpler form. Explain why $v(\cdot)$ is strictly quasi-concave if and only if $u(\cdot)$ is strictly quasi-concave. (2) quasi-concavity is implied by concavity which is often easier to show.
(b) Derive Marshallian demand for all goods. (You may assume that $u(x)$ is strictly concave for this derivation.)
