How do the economics behind Rent the Runway and Black Tux work? How would you calculate breakeven turnover and inventory requirements?

Ok, so I'm not an expert in fashion but I know finance.

Here is my take:
These two would be considered "fast fashion" retailers or better yet, e-tailers. Fast fashion simply means that unlike Coach or American Eagle or Levis, these FF retailers don't have to try to predict fashion 6 months in advance risking a big flop and having to heavily discount items that don't sell. FF retailers simply 'scout' runway shows and buy wholesale from these designer labels. A lot of designers would like this because they are essentially getting a guaranteed sale plus added exposure. Another thing is that these FF retailers don't keep their inventory for months at a time, they do so in cycles of weeks. If a designer sells out, chances are they retailer will continue to come back for more designs from them.
They are purchased wholesale, on cash basis account, payable on credit of 30 days or 90 days.
The economics as you state it are a bit more complex that what I care to explain here, but essentially if you were to 'replicate & improve' what I would do is scout and offer purchase orders to designers, just like they do. First order completed as 50/50 paid in full/credit term payable 60 days or so (assuming you already have a store ready to move inventory and not waste those 60 days setting up). Aim to sell all inventory before 60 days and pay balance with revenues. Extend credit term to 90 days at increased inventory, aim to sell by 60 days and keep that cycle going. What this will allow you to do is to always have inventory being paid for by customers before they are due for you essentially having the clients pay for your expansion in inventory. The break even is simple, don't sell for less than what your wholesale amount is.
Typical increase from wholesale commodity goods is 30%, try that margin. If you have to discount "heavily" at 15% or 25% you still get at least 5% safe margin

I recently read a great HBS case study on Rent the Runway, and we've recently completed a tuxedo rental e-commerce site for another competitor of the Black Tux.

The economics of it might look something like this:

Middle-Tier Tux:
Retail price of tuxedo $500
Wholesale cost of tuxedo $215
Cost of tuxedo to online retailer $150 (negotiated)
Revenue per rental $60 (just the tuxedo, not accessories)
Variable cost per rental $20 (shipping, stains, cleaning, etc.)

So if you're paying $150 for the tuxedo, and you rent it out just 4 times, you'll be above break even ($60/rental - $20/rental/fixes = $40 profit / rental, x 4 = $160).

This is a real world example FYI. In terms of inventory requirements, that's going to depend upon your business. Rent the Runway has TONS of dresses. But they started out with a limited supply and grew from there. The Black Tux keeps their selection limited and high quality, which I'm sure helps them manage their inventory easily, run numbers constantly and make sure they are running everything appropriately from an inventory standpoint. I wouldn't be surprised if they had product ready to be added to their distribution at the drop of a hat, especially for busier seasons such as prom or summer (weddings).

I hope that helps. Happy to answer more questions for you.

Black Tux and Rent the Runway operate on a similar business model, offering rental services for high-end clothing and accessories. Let's check the main aspect of their economics:
1)Revenue Model: Both of them generate revenue through rental fees which is a fraction of the item's retail price.
2)Inventory Acquisition: They acquire their inventory through various means.
3)Effective inventory management
To calculate breakeven turnover and inventory requirements, the following points are to be considered:
1)Overhead Costs
2)Rental Fee and Rental Duration
3)Breakeven Turnover Calculation=Total Costs / (Average Rental Fee - Variable Costs per Rental)
4)Cost of Inventory