Note that a private blockchain doesn't have to be expensive. The "cost" comes from the proof-of-work problem (the Concensus Algorithm) being made more difficult

Note that a private blockchain doesn't have to be expensive. The "cost" comes from the proof-of-work problem (the Concensus Algorithm) being made more difficult to solve over time, which occurs because there is an arms race between the miners to solve the block first. If you have a private blockchain with just a handful of miners on it who are not racing to win, they can use much simpler computing power and keep the difficulty of the proof of work problem low. In fact, in a private blockchain you don't even need to use proof of work. You can come up with some other concensus algorithm which is even cheaper when it comes to CPU (proof of stake is one popular algorithm, one can think up of many more) The benefits of a private blockchain is that it's a way for multiple companies/organization to share specific data with each other without them having to rely on a traditional database which can only be maintained by one of them. Why would they not want to rely on a single database? It's high risk: One one company maintains it, so the others are relying on that company to keep it up. This isn't easy. If that database goes down for some reason the other companies can only wait while the database is fixed. This waiting can cost millions of dollars a minute. With a blockchain based database all the companies have their own copy of the data so one company having an outage will not affect the others