I'm trying to understand the following paper: Hsieh & Ossa: A global view of productivity growth in China (2016). The pdf can be found here: https://faculty.chicagobooth.edu/chang-tai.hsieh/research/hsieh_ossa_jie.pdf
I struggle to understand the utility maximization problem right at the beginning of the paper. Can someone show me how to maximize the following utility with respect to $x_{ijs}$:
$U_j=\prod_s^S\left(\sum_i^N \int_0^{M_is^e}x_{ijs}(\nu_{is})^{\frac{\sigma_s-1}{\sigma_s}}d\nu_{is} \right)^{\frac{\sigma_s}{\sigma_s-1}\mu_{js}}$
The end result is:
$x_{ijs}=\frac{p_{ijs}^{-\sigma_s}}{P_{js}^{1-\sigma_s}}\mu_{js}E_j$
where $P_{js}=\left(\sum_i^N M_{is}^ep_{ijs}^{1-\sigma_s}\right)^\frac{1}{1-\sigma_s}$
$N$: number of countries
$S$: number of industries
$M_{is}^e$: number of entrants in industry s of country i
$x_{ijs}$: quantity of an industry s variety from country i consumed in country j
$\mu_{js}$: fraction of country j income spent on industry s varieties
$\sigma_s>1$: elasticity of substitution between industry s varieties
$p_{ijs}$: price of an industry s variety from country i in country j
$P_{js}$: ideal price index in industry s of country j
$E_j$: total expenditure in country j
