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)