[RPG] How much gold would the construction of a forge cost

The rules are ambiguous, but I think they lean to "yes."

As you know, it's not explicitly spelled out in the DMG. But I think they lean your way, and here's why:

1. On p.128 "Crafting a Magic Item" lays out the prerequisites for a character to craft a magic item: (a) a formula for the construction; (b) a spellcaster with spell slots--Magic Initiates need not apply--and knowledge of any required spell; (c) character level minimum.
2. Later, near the end of that section, we find a few paragraphs dealing with ancillary matters: lifestyle while crafting, custom items, and assistance. The paragraph on assistance calls out the character-level prerequisite for any assistants, but doesn't mention the other two. The designers could easily have said "if each of them meets the requirements to create the item," but instead mentions only requirement (c) from above. Admittedly, an argument from omission is pretty thin.

Unfortunately, there's a decent counter-argument of nearly the same strength: "anyone who assists is a co-creater, and thereby must meet all the creator's requirements. The level requirement is simply called out to make it clear that another spellcaster with the right spell but not of the right level could not assist." Of course, this line of reasoning would moot "if each of them meets the level prerequisite," which I find intolerable.

But it doesn't matter, because it's explicitly your call. (Of course, all things are ultimately your call, but this one's called out as such in the DMG.)

Back it up to the very beginning of "Crafting a Magic Item":

As an option, you can allow player characters to craft magic items. (loc. cit., emphasis mine)

Obviously we all understand the "golden rule" of GM-ery: Make decisions and adjudications that enhance the fun of the adventure when possible. (Repeated at the start of all of Wizards' published adventures, and a good paraphrasing of "The Dungeon Master" from the introduction to the DMG.)

But I'd suggest a broader understanding of published materials: what we purchase in the core set is not (just) a game. It's a system, with a game as a worked example included. The rules presented provide you with a default game playable out of the box, but also everything you need to construct a game after that fashion. Every rule, in my opinion, is preceded by a silent "and here's one way to do ___:"

The guidelines (they're not hard rules by any stretch of the imagination) for creating new magic items appear on page 284 of the Dungeon Master's Guide.

The table "Magic Item Power by Rarity" advises that items creating a 3rd-level spell effect once per day are at least Uncommon. However if the effect is continuous, the advice is that the item rarity should be much greater -- no less than Rare. Assuming the 3rd-level effect to be a least a step above this baseline, it would be at least a Very Rare item.

The suggested price range for Very Rare items (DMG page 135) is 5,001-50,000 gp. The cost for crafting them (DMG page 129) is 50,000 gp and 2,000 days (5½ years) of work. It's suggested that characters should be roughly level 11 before the GM introduces items of this potency.