# utility maximization with nested Cobb–Douglas–CES preferences

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

• I dont understand why is there an integral in the utility function. I see that the amount of output purchased by j's consumers from the s-th sector in i ($x_{ijs}(ν_{is})$) depends on the number of entrants in s in i. But I dont understand why do they integrate all the values if$x_{ijs}$ from $x_{ijs}(0)$ to $x_{ijs}(M^e_{is})$, since they only consume $x_{ijs}(M^e_{is})$. Could you explain? – Grada Gukovic Aug 22 '19 at 0:05