> They often lobby for minimum occupancy levels in law (effectively requiring by law a certain percentage of citizens to be marked as criminals)
I'm not sure that any actually do that. The ones I've seen turned out to be misunderstandings of the contract. The contracts were of the form that the private prison would be paid a base amount of $B to house up to C prisoners, plus an additional amount of $V per prisoner for each prisoner over C. C is typically something like 90 or 95% of the maximum capacity of the prison.
The contract actually makes more profit for the private prison if it has fewer prisoners, with the total profit going down for each additional prisoner up to C. Whether profit goes up or down for additional prisoners beyond that depends on whether $V is less than or greater than $B/C. If it is greater, than profit will go up as the last 5 or 10% of the capacity is used, but it will still be lower than the profit when the prison was at around 80 to 85%.
From the state's point of view the total cost is $B to house anywhere from 0 to C prisoners, and $B + (n-C) $V if the number of prisoners, n, is > C. From the state's point of view, $B is essentially a sunk cost.
This does provide some incentive to politicians to keep the private prisons occupied up to at least C prisoners, because if they have n < C prisoners, the per prisoner cost to the state is $B/n. If that is too high, the opponents are going to say the politician is spending too much on prisons. Increasing n lowers $B/n, so the politician might be tempted (and it also makes them look tough on crime). That's probably easier for the politician that trying to explain to voters that the total is $B regardless of many or few prisoners are there, and so per prisoner cost is not really what you should be looking at.
Also, note that this kind of incentive goes the same way with public prisons. A public prison also has a bunch of fixes costs, and so if you look at cost per prisoner you are going to see a lower cost per prisoner with higher occupancy because of those fixed costs.