For what it's worth: Baseball HQ (which, IMO, is the creme of the crop) conducted a study a few years back concerning performance in the first year of a "large" FA signing. Below is text from their 2020 Baseball Forecaster. (For those unfamiliar with Baseball HQ, "BPV" is an acronym for 'base performance value', which is a formula that combines a player's raw skills for contact/K rate, power, speed, and 'batting eye'.)
Contract year performance (Tom Mullooly)
There is a contention that players step up their game when they
are playing for a contract. Research looked at contract year players
and their performance during that year as compared to career
levels. Of the batters and pitchers studied, 53% of the batters
performed as if they were on a salary drive, while only 15% of the
pitchers exhibited some level of contract year behavior.
How do players fare after signing a large contract (minimum
$4M per year)? Research from 2005-2008 revealed that only
30% of pitchers and 22% of hitters exhibited an increase of more
than 15% in BPV after signing a large deal either with their new
team, or re-signing with the previous team. But nearly half of the
pitchers (49%) and nearly half of the hitters (47%) saw a drop in
BPV of more than 15% in the year after signing.
Although the text states that the research occured between 2005-2008, Im fairly certain that their studies such as this one, are validated every few years. Not certain about that, however. Regardless, at least during this 05-08 period, there appeared to be a measurable difference in performance for players signing large FA deals.