Specifically, I am looking at this paragraph
The fractions of the aggregate endowment assigned to each individual are independent of the realization of s^t . Thus, there is extensive cross history and cross-time consumption sharing. The constant-fractions-of consumption char- acterization comes from two aspects of the theory: (1) complete markets and (2) a homothetic one-period utility function.
The paragraph is taken from paragraph from section 8.6 in Sargent and Ljungqvist.
I understand that the fractions are independent of $s^t$, the state/complete history of realizations. However, I don't see how this leads to the second sentence -- there is extensive cross-history and cross-time consumption sharing.
So my question is: why does the fractions of aggregate endowment assigned to each individual being independent of $s^t$ lead to cross-time and cross-history consumption sharing?