Doing a bunch of research, I've started to hammer out a system of gem values. Any feedback (or resources closer to 1300) would be welcome.
The first really useful resource I found was the Rapaport Diamond Report. This gives modern values for gems, but provides a lot of insight into how gems are valued. Simplifying the categories a bit (especially through removing distinctions beyond Medieval technology), we can see some basic patterns. From this, I created some approximate gem quality levels:
1. Flawless: x3 value (This has no inclusions (flaws) visible with a jeweler's loupe)
2. Superior: x2 value (This has only small, difficult to find inclusions using a jeweler's loupe)
3. Average: x1 value (Inclusions are easily spotted with a loupe, but not the naked eye)
4. Flawed: x1/2 value (Inclusions are visible to the naked eye)
Added to this, from the same source, we have a color value. The Rapaport Diamond Report is obviously meant for diamonds, but given other gems have ideal colors, I generalized for the sake of simplicity:
1. Colorless (Diamond) / Rich Color (Other): x1.5 value
2. Faint Color (Diamond) / Average Color (Other): x1 value
3. Yellow (Diamond) / Faint Color (Other): x1/2 value
I found some very useful sources on understanding diamond pricing for color and clarity, which tied into the values for Rapaport.
For size, my first instinct from the above source was to use a quadratic equation. A helpful source confirmed that, c. 441
1592, this was an estimate being used in the trade. Although the source notes that it is no longer completely true for diamonds, it's apparently still used for other stones. So effectively:
the size factor is the square of the carats of the gemstone.
The main thing left, then, is to figure out the base values of the gems. Although the above source provides some numbers from a 16th century Italian goldsmith, it calls these numbers into question, with the ruby provided being either exceptional, or possibly simply exaggerated. Further, with other sources indicating that sapphires were valued above diamonds, the rock-bottom sapphire prices seem unlikely (or possibly a temporary market crash).
Outside that, however, there was little in the way of sources to figure out a base price. I opted then to take a page out of the Medieval Price Guide, where a gold ruby ring is priced at 26 shillings. Taking out a chunk for workmanship and profit, I used 20 shillings as an estimate for a base price for a 1 carat ruby. Given the mystique surrounding rubies, and their place at the top of the pyramid, I doubt it should be valued any less than this (though if it were a small or flawed stone, I wouldn't be surprised if an average stone would be higher).
Although it's suggested that a diamond would be worth 1/8 of a ruby, this seemed too low for my uses, so I increased it to 1/4, perhaps indicating its increasing price in the 14th century. This would give a base diamond value of about 5 shillings for an average 1 carat stone.
Amethyst was a very valuable gem back then, said to be valued equally with diamonds. This source is somewhat suspect, however, due to placing amethyst above ruby in value, so let's go slightly lower, with a value of 4 shillings for a 1 carat stone.
Sapphire is widely disagreed on, everywhere from our Italian goldsmith rating it as 1/10 the value of diamond, to others placing it neck-and-neck with rubies. Here I went with the value from <PERSON> of "twice the price of amethyst" ("amethyst was half the price of sapphire" in the original). That would place sapphire at a price of 8 shillings for an average 1 carat stone.
The only thing left is emerald, which is generally rated close to the top, so let's estimate 75% the price of a ruby (a value I got for a modern ring). That would be about 15 shillings.
For completeness, let's estimate any semi-precious stone (garnet, topaz, etc.) as about half that of the lowest stone listed before: Amethyst. That would give us 2 shillings for 1 carat.
The only thing remaining is the difference between an uncut gem and a cut one. Based on the amount of volume you're likely to use from gemcutters, let's estimate that the value of an uncut stone is 20% of a cut stone. 130