I wanted to redo the example I did earlier because, as was pointed out, I had a couple numbers wrong and I'd like to better manage the no-whip case to target a similar hammer output to whipping (to see what it truly sacrifices).
First the whipping case:
Code:
Turn Pop Basket Surplus Comm. Hammers
1 4 34/28 5 12 96
2 5 25/30 5 16 0
3 6 15/32 5 20 0
4 6 20/32 5 20 0
5 6 25/32 5 20 0
6 6 30/32 5 20 0
7 7 19/34 5 24 0
8 7 24/34 5 24 0
9 7 29/34 5 24 0
10 7 34/34 5 24 0
I've taken 10 turns to return to my previous state. At this point I'd whip again.
In the no-whip city, I'm going to allow them to make use of 2 grassland hills tiles and 2 plains hills tiles. The goal is to convert the excess food from the food resource into hammers as efficiently as possible. To do this, I'm going to work all four mines for all but two turns. During those last two turns, I'm going to, first, work nothing but the food resource and some hamlets in order to grow back most of what was spent working the mines and then I'm going to stop working the plains hills to leave the net food change at 0. The simulation looks like this:
Code:
Turn Pop Basket Surplus Comm. Hammers
1 8 17/36 -1 12 14
2 8 16/36 -1 12 14
3 8 15/36 -1 12 14
4 8 14/36 -1 12 14
5 8 13/36 -1 12 14
6 8 12/36 -1 12 14
7 8 11/36 -1 12 14
8 8 10/36 -1 12 14
9 8 9/36 5 28 0
10 8 14/36 3 20 6
So after summing the output columns of both simulations, it looks like this:
Whip: 204C, 96H
No-Whip: 144C, 118H
You'll note that the mining city converted food to hammers more efficiently (1:2.33 vs. 1:1.92). However in order to achieve those results, it had to spend population turns performing the conversion. Those population turns were turns not spent generating commerce. Total population turns available in both cases are 61 for the whip and 80 for the no-whip. However, the no-whip had to use 34 of those pop-turns working hills, converting food into hammers. The whip city, essentially, only sacrificed 19 pop-turns performing the conversion. Notice that the difference in non-sacrificed pop turns (15) multiplied by the commerce output of those pop-turns (4) is exactly the difference in commerce output between the two trials.
If you had substituted grassland hills for the plains hills in the no-whip scenario, you would have gotten a higher food-to-hammer conversion efficiency, but you would have sacrificed even more pop-turns, leading to an even larger drop in commerce. The advantage of slavery is that it allows you to convert your excess food in to hammers at a competitive rate while not sacrificing nearly as many pop-turns and therefore commerce.
Edit: Also of note... Pop-turns spent converting food to hammers by working mines cost more (in civic and city maintenance) than pop-turns lost to slavery.
Edit 2: Changed everything to use 7/8 pop. The food totals in the two trials are more in-line in this scenario.