Composite consumption index and retail firms

General DSGE Modeling
1

If C_t is a composite index in the household sector of your model, like C_t = CES(C^h_t, \;C^f_t), then you don’t need a retail firm on the production side of an NK SOE model, right?

And if there is a retail firm on the production side of your model, then you don’t need C_t = CES(C^h_t, \;C^f_t) at the household sector part of the model, right?. Regardless, they should give the same results?

2

Yes, the aggregation can either be inside of the preferences or a production function of an aggregator/bundler.