The answer is: We don't know for sure when. FPL have their own hidden formula, most likely based (among other things) on the actual demand for a player. This may then be further calculated via net transfers with set quotas to be met from FPL gamers' transfers to ultimately determine if a player's price will rise or fall. For instance, if a Premier League footballer has had a really good gameweek his price is likely to rise due to subsequent demand via purchases. Where as a popularly selected Premier League player suffering from bad form (or a long-term injury) will eventually be transferred out, have decreased demand and almost certainly see his price fall in the immediate short-term. According to some, price increases/decreases are capped at 0.3 per gameweek. Courtesy of my friends at FPL Dugout, I was referred to the following website (FISO Crack the Code) which can provide estimated projections* of player price increases and decreases.
*Disclaimer: They are not 100% correct and their projections should only be used as a guideline.
The benefit of knowing of price changes is crucial as it will ultimately affect your team value and your spending power. Thus you will benefit if a player from your FPL team increases in value. But you may also be short changed if a player from your FPL team decreases in value. This isn't to say that a player with a declining price will not bounce back or a player rising in price will not go through a bad patch in the future. As with all FPL decisions it's all about calculated probability, some luck, a bit of skill on your part and some hunches, etc. Do bare the above in mind when making FPL transfers in the future.