The Vikings at War book by Kim Hjardar and Vegard Vike provides one general frame of reference for cost of arms. The book has several references to a 7th-century Frankish legal text, Lex Ribuaria.
Wikipedia says this document is from the Franks, located in northern Europe, more specifically it was from around what is modern Cologne, Germany. It was written about 630 A.D. It would thus provide a reference point from within Europe about 100 years before the start point of the Viking Age.
My guess is relative pricing of weapons in relation to each other would be sort of somewhat comparable to a few centuries later in the middle of the Viking Age, however, the prices in relation to animals is probably lower here than in Scandinavia because of the cost of transport.
My wild guess is the cost in relation to a dairy cow would be higher in Scandinavia.
In location 3935 of Vikings at War, the text states that a helmet cost as much as a shield, a spear and a sword together. Only a coat of mail cost more, at around double the price of a helmet. In algebraic form, this would be:
- Helmet = shield + spear + sword
- Mail coat = 2 helmets
- Mail coat = helmet + shield + spear + sword
In location 4050, Vikings at War give some specific prices from Lex Ribuaria spelling out the amount to pay when restitution was called for.
The law book gave these payout amounts:
- Chainmail coat – 12 solidi
- Helmet – 6 solidi
- Sword – 3 solidi
- Sword and scabbard – 7 solidi (the premium of 4 solidi for merely a simple scabbard doesn’t make sense relative to the extensive, specialized work needed to create a sword)
- Shield and spear – 2 solidi
For an indication of comparative value, the ancient law-book provides the following amounts for compensation of animals:
- Cow – 1 solidus
- Ox – 2 solidi
- Horse – 7 solidi
For a reference point that moves across time and locations, the unit of measure of a solidi is a coin with value equal to one cow.
Let’s substitute some prices into those relative pricing comments just mentioned, using “s” as abbreviation for the solidi count:
- Helmet (6s or 6 cows) = shield+spear (2s) + sword (3s)
- Which is approximately equal since that would produce point estimates of 6s ~= 5s
- Mail coat (12s or 12 cows) = 2 helmets (2x6s)
- Which works since 12s = 12s
- Mail coat (12s or 12 cows) = helmet (6s) + shield+spear (2s) + sword (3)
- Which approximately works since 12s ~= 11s
Here is an approximation of the investment needed for different levels of armament for one warrior:
- Minimum armament of shield and spear = 2 solidi = 2 cows
- Better armed warrior with sword, shield, and spear = 5 solidi = 5 cows — this would be a wealthy warrior indeed. Who could afford to spare wealth equal to an extra 3 cows, which would otherwise be incredibly valuable to make sure your family survives the brutal winter?
- Petty king, super rich yarl, or larger landholder with chainmail (12s), helmet (6s), sword & scabbard (7s) , shield & spear (2s) = 27 solidi = 27 cows. Only the super rich could ever afford this much armament.