Why is there consumption sharing in Arrow-debreu time-zero complete markets trading model?

Question

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?

Thanks

Answer 1

I guess it just follows by definition.

Assume that in the high state (for household $i$, aggregate endowment is 10, and household $i$ consumes $1/5 = 2$ of this, while having an endowment of $5$. Now consider what happens if we were in the low state (for household $i$), where household $i$ only gets $1$ of whatever endowment. However, if this is a high state for some other household(s), such that the aggregate endowment is still $10$, since the fraction is constant household $i$ still consumes $2$, even though they have a smaller endowment.

Similarly for across time consumption sharing (although a bit weird since the model has no storage...). I guess it works that, if aggregate endowments are the same across two time periods, a household will consume the same amount regardless of whether their endowment was large or small (basically, they contribute to others when they have a lot (they lend), and they take contributions when they have little (they borrow). Thus, consumption smoothing.