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Beginning with the fact that there seems to be no price for carts/wagons/covered wagons listed within the book, continuing to the part where I haven't been able to find any rules regarding drag values relative to bulk values, and ending with the part where I understand this is reasonably arbitrary.
Honestly it's mostly about the pricing of a cart not being there that surprised/irked me. In my own campaign I'm pretty sure I made it the proportional pricing of renting for a day vs buying a pack animal applied to 25 miles of carriage rental (around a day of 30ft movement travel, which is an encumbered horse's speed) and then divided by four because carts are very much not carriages. ((10*(2*5))/4)
The price winds up being 25 silver by this logic/math.
The other question I can't see having been answered is how much my players can shove into said cart but since it seems finding out how much a horse or pack animal can carry is an adventure unto itself I'll be sparing myself from that activity.
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Well, ya put stuff in em, see, and then ya hitch em to a horse and say "giddyup" and then they just go...
Really though, I wouldn't worry too much about this unless maybe it's a push cart and the operator is a gnome, in which case, assume the cart is similar in bulk to a piano (16 bulk) and calculate the gnome's drag capacity from there.
Assuming a horse-drawn cart, it's what horses excel at. They can just about pull any load your cart's axle can reasonably bear.
So size of the cart compared to size of what's in it is the more practical consideration. And our Bulk system doesn't really explain item sizes too much. That means it's ultimately up to the GM. How many longswords can you throw into a cart? I dunno, let's say 200 and call that good enough.
As for pricing them, your logic seems reasonable but I think you got the ratio wrong on the price of pack animals - they cost 100x more to purchase than to rent. Also, I like to compare to caravan travel rather than carriage travel, both because its more similar to being a wagon (since that's what caravans usually are) and because carriages are pretty much strictly an in-town thing.
So:
Buying a horse, 80sp. Renting a horse per day, 1sp. That's an 80:1 ratio. So buying a cart should cost about 80x what renting one costs. Or, using warhorse it's 30x, or using a pack animal, it's 100x. Averaging those, it seems that 70x is the mean.
Renting a cart might be compared to traveling in a caravan. So 3cp per 5 miles or 15cp per day. x70 for buying that vehicle outright = 1050 CP or 105 SP or 10.5 gp to purchase a cart (or wagon, probably, so maybe half that for a cart).
But that gets crazy when you realize you can buy a piano for 16sp or full plate for 300sp. Seems like those things should cost more than a cart.
Back to "Oh, whatever, seems like 200sp should cover a cart or 300sp for a wagon. And, oh, no, you won't be buying a piano for only 16sp. Ever."
But that's just me.
I used the wagon prices and entries from Ultimate Equipment, since the rental prices were fairly similar to the entries in the playtest rulebook. Doing so made them comparatively much more expensive. However I felt this was an appropriate adjustment considering the party benefits.