Thank you for your analysis. agree with your range for an unskilled slave. With a working life of 10-years taking risks and costs into account, 2-years paid up front felt about right taking into cost of living and a desired return on investment. To respect accuracy over precision I roughed the number to 150gp. That falls within your estimates for 20-30% discount rates so it's probably fine to use. Taking @CapnZapp's example of hard-bargaining, the 50% discount level could represent quite well what you can get a seller down to in the right circumstances.An unskilled laborer earns 2 SP a day. PHB pg 159.
However, you also need to subtract the cost of living. While the conditions a slave lives in certainly qualifies as "inhumane," a term used to describe the wretched standard of living (PHB pg 158), the truth is that the remainder of the description fails to line up with what we'd expect for housing a slave. The squalid living condition (also PHB pg 158) is far more accurate, and that costs 1 SP a day.
So, we have a net income of 1 SP per day. And, that's ignoring any costs for hiring out the slave, or paying people to make sure the slave doesn't run away or otherwise try to fight for its freedom (perhaps by killing its owner).
Let's use your 10 year assumption for how long they can work. So, we have 1 SP per day, 365 days per year, for 10 years: that's 3,650 SP, or 365 GP.
But, that's future income. Now you have to discount that income stream. Since there really is no good way to determine the prevailing interest rate, our best bet is to use the desired rate of return as the discount rate so we can see what a willing buyer would pay in an arms'-length fair-market transaction.
Using the formula for the present value of an annuity, where the annual net income of 365 SP is the annual rents, 10 years is the number of periods, and 10% is the desired rate of return, we get 2,242.77 SP. That's a price of 224 GP, 2 SP, and 8 CP (actually 7.7 CP, but I rounded up). Which is actually equivalent to the PHB price for 4.5 draft horses (at 50 GP each). PHB pg 157.
To give that further perspective, examine that cost if paid in trade goods.
22,428 CP = 22,428 pounds of wheat. That's more than eleven tons of wheat!
22,428 CP = 4,485.6 pounds of salt. That's more than two tons of salt!
22,428 CP = 11,214 pounds of flour, or 11,214 chickens.
22,428 CP = 224 goats, or 112 sheep, or 112 pounds of cinnamon or pepper.
22,428 CP = 75 pigs.
22,428 CP = 15 oxen.
22,428 CP = an elephant and 1 mastiff (approximately).
22,428 CP = 448.4 pounds of copper, or 448.4 square yards of cotton cloth.
Edit: of course, all that changes if you tinker with the rate of return. And, naturally, rate of return is pegged to perceived risk of investment loss (escape, death, etc). So, it really wouldn't be outrageous for a buyer to expect a rate of return that would seem obscenely high to us in modern times. I could easily see a desire for a 25% to 50% rate of return being reasonable.
And, for the record, at increased levels of risk we're talking about the following prices:
20% = 15,302 CP
25% = 13,032 CP
30% = 11,284 CP
40% = 8,809 CP
50% = 7,173 CP
A skilled slave's price is based on the 2gp/day from the book i.e. 10x that amount. Where I've struggled most is the tier-1 and tier-2 equivalent slaves. Here I believe my daily income assumption is incorrect. I initially assumed 5x the skilled income, based in part of the tier-1 spell casting services costs. I now believe that is incorrect. If you flip to PHB 157 to look at the costs of living, the next steps are doublings not quintuplings. I believe we exclude the aristocratic level because anyone at that level is not earning their income from their own labour, but rather from rents. (Hence the logic of ransoming them back to their faction or family rather than enslaving them.) Also taking other posters thoughts into account I would revise to
Slavery is common in the Underdark, being most practiced by drow, duergar and fomorians. About 1/5 such creatures own on average 5 slaves, while 1/50 own on average 50. About 1/10 slaves are skilled, of whom 1/10 have tier 1 character-class equivalence. Old slaves are rare: most are young or adult. Adult slaves in good health are priced on the basis of 2 years earnings from their labour, and are expected to last 10 years in service. A seller could be forced down to half the starting price through circumstance or hard bargaining. Prices are sometimes paid in trade goods or promises of goods such as shares in future harvests.
Menzoberranzan (pop 20,000 free)
Unskilled......40,000.........150gp based on 2sp earnings/day
Skilled...........4,000.........1500gp based on 2gp earnings/day
Tier 1...............400.........3000gp based on 4gp earnings/day
Tier 2.................40.........6000gp based on 8gp earnings/day