# An Extension to CES Demand

I am reading a paper with an extended CES final good setting:

• The representative household consists of a continuum of members, indexed by $$k$$
• $$C_{t}=\int C_{k, t} d k$$
• $$C_{k, t}=\left(\int_{j \in \Omega_{t}} \theta_{k, j}^{\frac{1}{\eta}} c_{k, j, t}^{\frac{\eta-1}{\eta}} d j\right)^{\frac{\eta}{\eta-1}}$$, where $$\theta_{k, j}$$ is a utility weight summarized by a CDF $$F_{j}\left(\theta_{k, j}\right)$$
• Define utility weight for good $$j$$ at the household level $$\kappa_{j}\left(s_{j, t}\right)=\int \theta_{k, j} d F_{j}\left(\theta_{k, j}\right)$$
• Then $$C_{t}=\left(\int_{j \in \Omega_{t}}\left[\kappa_{j}\left(s_{j, t}\right)\right]^{\frac{1}{n}} c_{j, t}^{\frac{\eta-1}{\eta}} d j\right)^{\frac{\eta}{\eta-1}}$$

I don't understand how to derive the last equation? Why can I simply change the order of integration when the inner integral is to a power?

Thanks.