Before reading my thoughts please watch this game theory video.
Now before I get started let me throw a disclaimer out there: I appreciate Matt's work and research into his videos but as with any human he too is sometimes wrong.
Ok so Game Theory's prime point of research for this video was he looked at the dates for the various songs heard on the ingame radio. He then looked at the latest song's date and used that as a point of diversion from the real world timeline. The reason why this is wrong is that the Fallout timeline actually diverted at a much earlier time between 1945 and 1961 to be specific. (We know this because just about every event of the Space Race played out differently in the fallout universe) The first major event that contradicts the real world was when (As described in the Fallout 4 intro) a fictional soldier left the military in the hopes of creating more peaceful uses of nuclear power. This soldier's achievements launched the world on a completely different technological path. Instead of miniaturization of computers we had miniaturization of nuclear power. this lead our world into super technologies only dreamed of years earlier.
Now we can assume GT had not played FO4 by the time this video was made so I will use a different piece of evidence. First off the way he calculated bottle cap price was based on the ingame radio dates this is Cultural not Economic. At the time of 1970s in the fallout universe the resource wars were beginning the cause for these wars was as one might guess resources. Prime resources such as Gasoline and Metal were in short supply (another major reason for the nuclear technology being developed). As we know bottle caps are made from very cheap metal. Given the metal shortages of the world the prices would also skyrocket.
So long story short we don't have an exact price for fallout bottlecaps but whatever they are worth it is more than real world ones.