OP - It is the responsibility of the last delivering air carrier, in other words, the carrier that flew you into Paris to deliver your bags to you and to compensate you for what you need in the meantime.

What carrier is that and what is that carrier telling you?

On a route such as ORD-CDG it is most likely that your bags will catch up with you in a day or perhaps even less. Hopefully you have several days' clothing in your carry-ons anyway, but if you don't, check with the carrier. Some will give you a preloaded prepaid card which you can use to purchase necessities and others will give you a set limit. Others will tell you to keep it reasonable.

If the latter, keep it reasonable. Keep receipts and don't spend anything you wouldn't buy if you don't get reimbursed.

On that note, what about lost baggage insurance? Do you carry it? Does your credit card? Anything else?

In the remote chance that your bags are truly lost, after 21 days, you are each entitled to fair market value up to a maximum of approx. $1,600. But, that really is a remote possibility.

So, back to basics, what carrier and what is the carrier telling you?