Hi! The most important message first:
>>> There is no conceptual difference between profit and non-profit cases! <<<
This perception just comes from the tradition of "case frameworks" that have been spread across the globe in the last 15 years. In my experience, using these frameworks to approach cases is extremely damaging to your ability to think through issues, hence making it far more difficult to be successful in MBB interviews.
If you really understand how to tackle strategic issues, it is completely irrelevant whether your focus metric is profit or something else! It can be quality of an education system, vaccination share in a population, youth unemployment rates, or hundreds of other things. It just makes ZERO difference to the approach if you really think through it in a rigorous way.
It also has absolutely nothing to do with "knowledge" about public sector (this is NOT what MBB are testing! Kowledge is irrelevant for the assessment of generalist candidates if they show the problem solving abilities needed!).
In essence, what you always have to do in order to set up a robust approach is to, firstly, define the objective (if it is not profit, it usually needs to be clarified into a clear KPI (like, e.g., share of children going to school)).
The rest then depends on the form of the question.
- If the question is something like "How can we achieve XYZ?", then you disaggregate this metric into its conceptual drivers, usually one or two layers deep. An THEN you develop qualitative ideas how to infliuence these conceptual/numerical drivers.
- If the question is "Should the client do XYZ in order to achieve ABC", then you need to first define the criterion when ABC is attained. And then you need to assess each element of the criterion to check whether and how you can realistically meet it. This sounds a bit abstract, but it is extremely straight forward and easy to conceptualize once it is properly shown and explained.