An Adult Gold Dragon has Change Shape as one of its actions (emphasis mine):
Change Shape. The dragon magically polymorphs into a humanoid or beast that has a challenge rating no higher than its own […]
In a new form, the dragon retains its […] proficiencies, […] and Intelligence, Wisdom, and Charisma scores, as well as this action. Its statistics and capabilities are otherwise replaced by those of the new form, except any class features or legendary actions of that form.
With this action, the dragon can magically polymorph itself into an Archmage. While the dragon is an Archmage, what are its saving throw and skill bonuses? (To be clear, I'm not talking about its spell save DC here.1) Does the fact that its proficiency bonus is different in Archmage form have an effect on these bonuses? Similarly, do changes in the dragon's Strength, Dexterity, and Constitution scores caused by being an Archmage matter?
1 It probably wouldn't have a spell save DC, because non-innate Spellcasting is a class feature.
Best Answer
What are proficiencies?
The dragon retains its proficiencies, but what counts as a proficiency? In the Character Creation chapter, a character's proficiencies are defined as follows:
The term is bolded in the original text, as this is the place where it is first defined. A couple sections later, the concept of a proficiency bonus is introduced (emphasis mine):
Proficiency bonus is not mentioned in the list of proficiencies above, and it is noted separately from the proficiencies themselves. Therefore, a character's proficiencies indicate instances where the proficiency bonus is added, but the bonus itself does not count as one of the character's proficiencies. When a character reaches level 5, their proficiencies (usually) do not change; instead, their proficiency bonus increases to +3.
Does this apply to monsters?
While monsters have many abilities unavailable to PCs, they are typically built using the same set of rules. The introduction to the monster rules refers back to the player rules in the sections for Size, Alignment, Hit Points, Speed, Ability Scores, etc. What about saving throws and skills? Are monsters proficient in them?
The Skills section comes right out and says it (emphasis mine):
The Saving Throws section almost says the same thing:
To me, the intent is clear: monster saving throw bonuses also count as proficiencies, much like a PC's class saving throw bonuses do.
Comparison
When using Change Shape, an Adult Gold Dragon retains its proficiencies. I will compare its saving throw and skill bonuses in dragon form to those in Archmage form.
Adult Gold Dragon
The dragon's proficiency bonus is +6, because it is a CR 17 creature. Creatures of CR 1-20 have the same proficiency bonus as PCs of the same level.
Of particular note is the dragon's Perception bonus. Much like a rogue with Expertise, the dragon adds double its proficiency bonus to Perception checks. The Skills section tells us about this possibility:
Adult Gold Dragon → Archmage
The Archmage's proficiency bonus is +4, because it is a CR 12 creature. When an Adult Gold Dragon uses Change Shape to become an Archmage, its saving throw and skill bonuses are as follows:
In general, during Change Shape a dragon's saving throws and skills are weakened; however, there are exceptions. A Githzerai Anarch (MTF, p. 207) is a CR 16 creature (+5 proficiency bonus) with 21 Dexterity (+5), raising the dragon's Dexterity save bonus to +10.
Rules-as-written, the decision of whether a proficiency bonus should be omitted, included, or doubled is maintained, while the proficiency bonus itself changes to match the creature changed into.