This question has been doing the rounds in MBB interviews a lot lately.

One approach to this would involve the following steps:

• Estimate typical revenue for a cab driver in NYC by taking assumptions on # of rides, ride length (avg. fare), working hours etc. (Keep in mind that taxi drivers spend a fair amount of idle time at the taxi ranks)
• Estimate typical costs for a cab driver including fuel, repairs, food, personal living expenses, car mortgage etc.
• The difference in above two would be the net cash flow
• The theoretical price for the taxi medallion should be the PV of all future cash flows
• You can assume a relevant discount rate for US/NYC and also assume some relevant inflation

Give it a go - take some assumptions - feel free to DM and discuss your full solution.

I have known people who have arrived at relatively close estimates to the actual known auction prices.