Consider a $n$-good industry with a representative consumer with utility function for the differentiated goods given by,
$ U=\left(\sum^n_{i=1}q_i^\beta\right)^\theta $
Suppose that the representative consumer is endowed with income $I$.
a. Derive the inverse and direct demands;
My solution: I am writing the lagrange as:
$ L=\left(\sum^n_{i=1}q_i^\beta\right)^\theta+\lambda(I-\sum^n_{i=1}p_iq_i) $
FOC:
$ \theta \left(\sum^n_{i=1}q_i^\beta\right)^{\theta-1}\beta q_i^{\beta-1}-\lambda p_i=0 $
So the inverse demand function is:
$ p_i=\frac{\theta \left(\sum^n_{i=1}q_i^\beta\right)^{\theta-1}\beta q_i^{\beta-1}}{\lambda} $
Is this right? Because the $\lambda$ seems a little bit strange. If yes, how do I isolate $q_i$, when there is this sum symbol to derive the demand function?