I have a question that involves finding the optimal demand of $n$ goods for a consumer. However, I haven't anything like this before and I'm not sure how to proceed.
The consumer has a utility function $U(x)=\sum_{i=1}^{L}(x_i-a_i)^{b_i}$ with $a_i \ge0$ and $b_i>0$ for all $i$ and $\sum_{i=1}^{L} b_i = 1$. I am assuming that $m>\sum_{i=1}^{L} p_i a_i$ holds where $m$ is the consumer's income.
[EDIT: Apparently there's a typo. Its not $\sum_{i=1}^{L}(x_i-a_i)^{b_i}$ but $\prod_{i=1}^{L}(x_i-a_i)^{b_i}$ which in my opinion makes solving this much easier. I'll work it out and see if I come up with a solution.
So I set up the Lagrangian as follows.
$L= \sum_{i=1}^{L}(x_i-a_i)^{b_i} + \lambda(m-\sum_{i=1}^{L}x_ip_i) $
I then find the first order conditions as:-
$\frac{dL}{dx_i}= b_i(x_i-a_i)^{b_i-1} =\lambda p_i $ for a good $i$
$\frac{dL}{dx_j}= b_j(x_j-a_j)^{b_j-1} =\lambda p_j $ for a good $j$
$\frac{dL}{d\lambda}= m-\sum_{i=1}^{L}x_ip_i =0 $
I know that $\frac{b_i(x_i-a_i)^{b_i-1}}{p_i} = \frac{b_j(x_j-a_j)^{b_j-1}}{p_j}$
This is where I get stuck. I'm thinking trying to solve for $x_j$ in terms of $x_i$ and substitute that into $m-\sum_{j=1}^{L-1}x_jp_j - x_ip_i =0$ but it doesn't seem to make sense to me. I was wondering if there's a more straightforward which I am missing?