Might it ever make sense to require someone to pay in order to take a job? Explain, again using an example or mathematical model as you prefer

Answer: Yes

The value of taking a job is the benefit minus the cost. The cost is usually the lost time from having to work instead of doing something else. The benefit is both your salary and the increased skills you get from the experience of working which increases your future wages.

A job can thus require them to pay you if the benefit of the experience increases you future salary enough. For example, assume an airline had a job who's experience will allow you to get another job later at a high enough salary to cover all costs in the short-run now. The airline can charge you to work there and you will still take the job. Another example is college. People are essentially paying to work although the work adds no value to the economy and is just homework but still you pay for the increased skills through work.