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?



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