In case of your itinerary Eurail pass hardly can be
the most economical, so let's just take a look at each leg and see what your options are:
Amsterdam-Paris: if you are ok with booking some time in advance, then it won't be a problem to catch a discounted ticket for as low as €35 per person. Though such discounted tickets will be not flexible, thus for a specific train and date + no rebooking. Such promo offer tickets can be bough via
Dutch Railways website In case you buy the same ticket before departure expect to pay >100 euro per person
Paris-Luzern: again same idea here, the earlier you buy the ticket - the more chance to get the best price. Though in case of Paris-Luzern route you have two options: a) buy one through ticket Paris-Luzern (for December it's not a problem to find tickets around 60 euro per person); b) buy first Paris-Basel ticket for the direct TGV trains (at the moment these start from 39 euro), while then just buy additionl Basel-Luzern Swiss domestic ticket (for instance SuperSaver for as low as 12 Swiss francs). Since train change in Basel is basically necessary, then there is no obstacles to choose option 'b', since it will be cheaper in total. Ticket (both to Basel or Luzern) can be bough via tgv-europe.com
Luzern-Milan: same story here, in advance purchase can get you a ticket to Milan for only 23 CHF (which is great deal). Once again in advance purchase via
SBB website is a must
Milan-Rome: and as you can guess, here discounted promo tickets are once again the best choice for the travelers who don't want to pay much.
TrenItalia offers such from 19 euro.
Resume: as you can see, at best you can manage the whole itinerary Amsterdam-Paris-Luzern-Milan-Rome for as low as 120 euro per person. Doubt that any pass can give the value like that, although, as we already said, discounted tickets are great, but have a number of limitations (no rebooking, no refund, can't change train or departure time), so you need really be sure about your itinerary before booking
So we advise you just to check out all those tickets we mentioned, and then, if you still have questions, we will be glad to help
p.s. if you trip takes place in a later weeks of December, then you might not yet be able to see or book ticket, since in Europe there is a train schedules change at this time, so in some cases you might need to wait a bit until all the railways companies agree on all the changes and open sale